An Introduction to Dynamical Systems - 2nd Edition

6. 3. Self-Excited Oscillator 229

b. Show that the xed point (a:",y") with :z:“,y‘ > 0 is repelling and the
other (two) xed points are saddles.

c. Assume that every orbit in this rst quadrant is bounded for t 2 0, prove
that there must be a limit cycle. Note: This model gives a more stable
periodic limit cycle for a predator-prey system, than the periodic orbits
for earlier models considered.

6.3. Self-Excited Oscillator

In this section, we present a class of differential equations, called Lienard equa-
tions, which have increasing “amplitude” for small oscillations and decreasing “am-
plitude” for large oscillations. The result is a unique periodic orbit or limit cycle.
Since the amplitudes increase for small oscillations without external forcing, they
are called self-excited oscillators. These equations were originally introduced to
model a vacuum tube that settled down to periodic output. Later, a similar type of
phenomenon was found to occur in many other situations, and the equations have
become a nontrivial model of a situation with an attracting limit cycle.

The simplest example of this type of system is given by

(6.1) at = -12 - (12 - 1):;-,

which is called the Van der Pol equation. When written as a system of differential
equations, the equation becomes

(6.2) :i: ='v,

1) = —:|:— ($2 — 1)v,

which is called the Van der Pol system. This system can be generalized to the
Lienard system,

(6.3) 0': = v,

if = _g($) _ f(x)v)

where assumptions on g(:|:) and f (:z:) make them similar to the corresponding factors
in the Van der Pol system. In particular, g(:|:) > 0 for a: > 0, so we have a restoring
force. The function f(a:) is negative for small and then becomes positive for
large |:c|. We give the exact requirements in what follows.

Before we change coordinates, note that if we use

2 where I

L(:c,v) = G(:c) + 1%, G(a:) =/L g(s) ds,
o

then L = v2 (1 — 2:2) is both positive and negative. It is greater or equal to zero on
a small level curve L“1(C') contained strictly in the strip

{(2,0) : -1 5a:5 1}.

However, there is no large level curve on which L is less than or equal to zero. Each
such level curve passes through the strip where L is positive. Therefore, we cannot

use just the type of argument we gave in the preceding section in the applications
of the Poincaré—Bendixson theorem. Instead, the proof discusses the total amount

230 6. Periodic Orbits

that a function like L changes when a trajectory makes a whole passage around the
origin.

Before we proceed further with the discussion, we rewrite the Liena.rd equations
in a form that is easier to analyze. Let y = v + F(:2), where

F(a:) =/ f(s) ds.
0

Then,

1' = y _' F(x):

1? = 13 + F'(¢)-'i= = —9($) — f($)v + f($)v = -9(¢)-

Thus, we get the equivalent set of equations I

(6.4) :i: = y — F(a:)',

2) = —y(¢)-

For the Van der Pol system, g(:z:) = :2 and F(:c) = 2:3 /3 _— :|:. The phase portrait of
Van der Pol system in the form of the equations (6.4) is given in Figure 7.

The assumptions we make on the equations are the following.

(1) The functions F(a:) and g(a;) are continuously differentiable.
(2) The function g is an odd function, g(—:r:) = -g(:c), with g(:|:) > 0 for a: > 0.

Thus, g is a. restoring force.
(3) The function F is an odd function, F(—z) = -F(a:). There is a single positive

value a for which F(a) = 0, F(a:) < 0 for 0 < as < a, F(:c) > 0 for :1: > a, and
F(:12) is nondecreasing for :1: 2 a,.

In Section 6.8.2, we show how the second form of the Lienard equations arises in a.
nonlinear electric RLC circuit.

3 Z!

IL‘
-3 3

-3

Figure 7. Phase portrait for Van der Pol system in the form of the equations (6.4)

6. 3. Self-Excited Oscillator 231

Theorem 6.5. With the preceding assumptions (1)—(3) on g(:n) and F(:r), the
Lienard system of equations (6.4) has a unique attracting periodic orbit.

The proof of this theorem is given in Section 6.9. For the rest of this section,
we sketch the ideas of the proof.

1/ F(=v)
Yl

YII

I

l
l

Y]

Figure 6. Trajectory path for Lienard equations

A trajectory starting at Y = (0,y), with y > 0, has :i: > 0 and Q < 0, so it goes
to the right and hits the c1u've y = F(aw). Then, it enters the region where :i: < 0
and Q < 0. It is possible to show that the trajectory must next hit the negative
y-axis at a point Y’. See Figure 8. Subsequently, the trajectory enters the region
with as < 0 and y < F'(:c), where :i: < O and y > O, so it intersects the part of the
curve y = F(:r) with both :0 and y negative. Next the trajectory returns to the
positive y axis at a point Y". This map P from the positive y-axis to itself, which
takes the point Y to the point Y", where the trajectory intersects the positive
y-axis the next time, is called the rst return map or Poincare’ map. The half-line
{(3/,0) : y > 0} is the transversal for the Poincaré map. . The orbit is periodic
if and only if Y" = Y. But, (a:(t),y(t)) is a. solution if and only if (—a:(t), —y(t))
is a. solution. Therefore, if Y’ = -Y then Y" ='—Y’ = Y and the trajectory is
periodic. On the other hand, we want to show that it is not periodic if Y’ 95 —Y.
First, assume that ||Y’ [I < ||Y||. The trajectory starting at -Y goes to the point
—Y' on the positive aads. The trajectory starting at Y’ is inside that trajectory
so ||Y"|| < ||Y’|| < ||Y[| and it is not periodic. Similarly, if ||Y'|| > ||Y||, then
Y” > ||Y’ || > Y and it is not periodic. Therefore, the trajectory is periodic if and
only if Y’ = —Y.

The idea of the proof is toluse a real-valued function to determine whether the
orbit is moving closer or farther from the origin. We use the potential function

G'(a:) = /019(3) ds.

232 6. Periodic Orbits

Since g is an odd function, with g(a:) > 0 for :1: > 0, G(:v) > 0 for all a: 96 O. Let

Lea = Ge) +

which is a measurement of the distance from the origin. The orbit is periodic if
and only if L(Y) = L(Y’).

A direct calculation shows that L = —g(a:)F(z), which is positive for -a < :1: <
a and negative for |a:| > a. The quantity L increases for a: between zta. Therefore,
the origin is repelling and there can be no periodic orbits contained completely in
the region —a 5 1: 5 a.

As the size of the orbit gets bigger, a comparison of integrals shows that the
change in L, L(Y') —- L(Y) decreases as Y moves farther up the y-axis. A further
argument shows that L(Y') — L(Y) goes to minus in nity as the point goes to
in nity along the axis. Therefore, there is one and only one positive y for which
Y = (0,y) is periodic. See Section 6.9 for details.

Exercises 6.3

1. Prove that the system

i = it
9 = —w - =3 — u(=v‘ — 1):/,

has a unique periodic orbit when u > 0.
2. The Raleigh differential equation is given by

z.. +e (512.3 —. z)+z_-0.

a. Show that the substitution :0 = 2 transforms the Raleigh differential equa-
tion into the Van der Pol equation (6.1). Hint: What is rii?

b. Also, show that the system of differential equations for the variables a: = 2
and y = —z is of the form given in equation (6.4).

c. Conclude that the Raleigh differential equation has a imique limit cycle.

6.4. Andronov—Hopf Bifurcation

In this section, we discuss the spawning of a periodic orbit, as a pa.ra.meter varies
and a xed point with complex eigenvalue makes a transition from a stable focus to
an unstable focus. Such a change in the phase portrait is called a bifurcation, and
the value of the parameter at which the change takes place is called the bifurcation
value. The word bifurcation itself is a Eench word that__is used to describe a fork
in a road or other such splitting apart.

We start with a model problem and then state a general theorem.

Example 6.11 (Model Problem). Consider the differential equations
:i:=ua:-wy+K:c(:r2+y2),

z2=W=+m/+Ky(¢2+3/2).

6.4. Andronov—Hopf Bifurcation 233

with w > 0. We consider /,4 as the parameter that is varying. The eigenvalues of
the fixed point at the origin are /\,, = u i iw. For [L = 0, the linear equations have
a center, and as u varies from negative to positive, the xed point changes from a
stable focus to an unstable focus. In polar coordinates, the equations become

1‘ = /ir + K r3,

9 = w.

The 6 equation implies that 0 is increasing and is periodic when t increases by
2"/W units of time.

We rst consider the case when K < 0. For /,4 < 0, all solutions with r(0) > 0
tend to 0 at an exponential rate. For u = 0, 1" = Kr3 < 0 for 1' > 0, so still
all solutions with 1'(0) > 0 tend to 0, but now at a slower rate. Thus, for u = 0,
the xed point is weakly attracting. _For /,4 > 0, 0 is a repelling xed point in the
1'-space, and there is a xed point at r = ,/|/‘/K|, that is attracting. See Figure 9.

r 1- r
0-4-‘ii 0——<—i 0-D—9—Gi
0 0 0 7'14

(P-) (bl (<1)

3/ U 3/

{B 3 3

(<1) _<e> <0

Figure 9. K < 0, Supercritical bifurcation. The dynamics in the r-space for

(a) p < 0, (b) u = 0, and (c) p > 0. The phase plane in (:c,y)-variables for

(d) #<0. (6) u=0.end(f)#>0-

In the Cartesian (:r,y)-space, the origin changes from a stable focus for u < 0,
to a weakly attracting focus for ,u = 0, to an unstable focus surrounded by an
attracting limit cycle for /1 > 0. Notice that, for this system, when the xed point
is weakly attracting at the bifurcation value it = O, the limit cycle appears when
the xed point is unstable, Re(/\,,) = /.L > 0, and the limit cycle is attracting. This
is called a supercritical bifurcation, because the limit cycle appears after the real
part of the eigenvalue has become positive.

Next, we want to consider the case when K > 0. The change in the eigenvalues
is the same as before. Now the xed point is weakly repelling for the nonlinear
equations at the bifurcation value u = 0. For u > 0, there is no positive xed
point in the 1' equation, and in the (a:,y)-plane, the origin is repelling. There are
no periodic orbits. Now, the second xed point in the r equation appears for /i < 0

at 1‘ = \/W, and it is unstable. Thus, the limit cycle appears before the
bifurcation, when the real part of the eigenvalue at the xed point is negative. See
Figure 10. This case is called the subcritical bifurcation.

Notice that, in both cases the limit cycle has radius \/ |u/K | for the values of
the parameter having a limit cycle.

6. Periodic Orbits

o<—o—>——'l 0->——-—T o—»—i—T

0 Tu O 0

(*1) (bl (C)

3/ 3/ U

{Z J3 I1!

(<1) (6) (F)

Figure 10. K > 0 Subcritical bifurcation. The r variables for (a) /1 < 0,
(b) /1 = 0, and (c),u > 0. The phase plane in'_(:c,y)-variables for (d) p. < 0,
(e) p=0, and (f) y>0.

The general result concerns the change of a xed point from a stable focus to an
unstable focus and the resulting periodic orbit for parameter values either before of
after the bifurcation parameter value. The key assumptions of, the theorem, which
indicate when it can be applied, are as follows:

(1) There is a system of differential equations

*=Fua

in the plane depending on a parameter [.L. For a range of the parameters, there
is a xed point x,, -.that varies smoothly. (In Example 6.11, x,, = 0 for all 1,4.)
The xed point has complex eigenvalues, /\,, = 01,, 1!; M3,, The bifurcation
value no is determined so that the eigenvalues are purely imaginary (i.e., the
real parts of the eigenvalues am, = 0, and the imaginary parts i m 76 O).

The real part of each eigenvalue is changing from negative to positive or pos-
itive to negative as the parameter /.4 varies. For simplicity of statement of the
theorem, we assumed that 04,, is increasing with ,u,, '

d
@C!#l"o > 0,

but the results could be rewritten if this derivative were negative. Thus, the
xed point is a stable focus for [L < [19, but it near #9, and it is an unstable
focus for it > pg, but it near #0.

The system is weakly attracting or weakly repelling for p = pg. We show
three ways to check this nonlinear stability of the xed point at the bifurcation
value: (i) We check most of our examples by using a test function and apply
Theorem 5.2. (ii) We give an expression for a constant Kno that can be
calculated by converting the system to polar coordinates and averaging some
of the coef cients. This constant plays the same role as K in the model
problem. (iii) There is a constant M that can be calculated using partial
derivatives in Cartesian coordinates, that is similar to K,4.

6.4. Andronov—Hopf Bifurcation 235

Theorem 6.6 (Andronov—Hopf bifurcation). Let 5: = F,,(x) be a system of dif-
ferential equations in the plane, depending on a parameter p. Assume that there
is a xed point x,, for the parameter value p, with eigenvalues /\,, = 01,, i i,6,,.
Assume that L3,, 94 0, am, = 0, and

i(a,,)‘ > 0.
411- l4=#o

a. (Supercritical bifurcation) If xm, is weakly attracting for p = pg, then
there is a small attracting periodic orbit for |p—pg| small and p > pg. The period
of the orbit is equal to 2"/pm, plus terms which go to zero as p goes to pg. See
Figure 9.

b. (Subcritical bifurcation) If xm, is weakly repelling for p = pg, then there
is a small repelling periodic orbit for |p — pg| small and p < pg. The period of the
orbit is equal to 2"/pm, plus terms that go to zero as p goes to pg. See Figure 10.

Remark 6.12. If the rest of the assumptions of the theorem are true, but

d
$0" (#0 < 0,

then the periodic orbit appears for the parameter values on the opposite side of the
bifurcation value. In particular, for the supercritical bifurcation case, the stable
periodic orbit appears for p < pg (when the xed point is unstable), and for the
subcritical bifurcation case, the unstable periodic orbit appears for p > pg (when
the xed point is stable).

A proof of part of this theorem is contained in Section 6.9.

The next theorem indicates two ways to check the weak stability or weak in-
stability of the xed point at the bifurcation value; the rst is given in terms of
polar coordinates and the second is given in terms of partial derivatives in Cartesian
coordinates. A third way is to use a test function as we demonstrate in some of the
examples.

Theorem 6.7 (Stability criterion for Andronov—Hopf bifurcation). Assume that

the system of differential equations in the plane satisfies Theorem 6. 6'. Then, there

is a constant Kg that determines the stability of the xed point at p = pg and

whether the bifurcation is supercritical or subcritical. (i) If Kno < 0, then the xed

point is weakly attracting at the lowest onder terms possible when p = pg, and

there is a supercritical bifurcation of an attracting periodic orbit of approximate

radius \/|a,,/K,,,,| for |p— pg| small and p > pg. (ii) If Km, > 0, then the xed

point is repelling at the lowest orderterms possible when p = pg, and there is a

subcritical bifurcation of a repelling periodic orbit of approximate radius \/ Ia“/ K,,°|

for |p — pg| small and p < pg. In fact, there is a change of coordinates to

(R, 9) such that the equations become

R = 01,, R + Kg R3 + higher order terms,

9 = u, + higher order terms.
(These are polar coordinates, but are not_ given in terms of 1:1 and 1'2.)

The following two parts give two formulas for the constant KM.

236 6. Periodic Orbits

a. Let xp = (a:,,_1,:r,,_2). Assume the system in polar coordinates $1 —— :|:,,,1 =
r cos(6) and $2 — a:,,_2 = r sin(0) is

r = 0,, r + C3(0, /J.) 1-2 + C4(9,;.l)1'a + higher order terms,
9 = [in + D3(0, p.) r + D4 (0,/.i) r2 + higher order terms,

where C’;-‘ and D5? are homogeneous of degree j in sin(6) and cos(6). Then, the
constant KM is given as follows:

KI-1-0-i2/”h0 ow4 ,/10)-l5o0 wa ll-*0)n(aa,/10)¢w-

-- b. Assume that the system at u = no is given by £1 = f(:c1,:1:2) and 5:2 =
g(a:1,:c;) (i. e., the first coordinate function at /10. is FM’; = f, and the second coor-
dinate function is F,_,°,2 = g). We denote the partial derivatives by subscripts, with
a subscript :2 representing the partial derivative with respect to the rst variable and
a subscript y representing the partial derivative with respect to the second variable:

6 83 82
b?j;'(x#o) = fa» (xw) = f===u» 5%(x!1o) = Quin

etc. Assume that

f= = 0. ft 9‘ 0.
y= 94 0. at = 0.

and fy g, < 0. This ensures that the eigenvalues are :l:i M. Then,

_ 1 iv 1/2 gr 1/2 fu 1/2 gz 1/2

Kuq - 16 ( .9: , .f:|:=a:'l' fy .f:|:yy'l‘ gr gszy 'l' fu $1;/yy

1 Q2 1/2 91 Q2:
+ 16 gr 7; (fmyfzm + fa fzyfyy + E fuyilyy)

1 ft "2 1. 1'l' i1.6_fy g_s_
(g:y§1/u 'l' _gz Qzy§:r::|: ‘l’ g.1t f:a:§:|:a;) -

We do not prove this theorem, but only discuss its meaning. Part (a) gives
a formula in terms of “averaging” terms given in polar coordinates. See [Cho77],
[Ch082], or [Rob99] for the derivation of the formula given in part (a). The average
of C4(0,p.g) in the formula. seems intuitively correct, but the weighted average of
C3(0, no) is not obvious.

Part (b) gives a formula in terms of usual partial derivatives and is easier to
calculate, but is less clear and much messier. The reader can check that, if only
one term of the type given by the partial derivatives in the rst four terms appears,
then the result is correct by using a test function. See Example 6.13. We have
written it in a form to appear symmetric in the rst and second variables. We have
assumed that f, = 0 = gy, but not that gm = —fy as our references do. If f, 9‘: 0
or gy gé 0, but the eigenvalues are still purely imaginary, then a preliminary change
of coordinates is necessary (using the eigenvectors) to make these two terms equal
zero. Note that, in the second line (of tenns ve through seven), the coe icient
outside the parenthesis has the same sign as gz, while in the third line (of terms

6.4. Andronov—Hopf Bifurcation 237

eight through ten), the coefficient outside the parenthesis has the same sign as fy-
ln [Mar76], this constant is related to a Poincaré map. The Poincaré map P is
formed by following the trajectories as the angular variable increases by 21r. At the
bifurcation value, P'(0) = 1 and P"(0) = 0. This reference shows that the third
derivative of the Poincaré map, P"’(0), equals a scalar multiple of this constant.
The derivation is similar to the proof of Part (a) given in the references cited. It is
derived in [Guc83] using complex variables: This calculation looks much different.

In some equations, it is possible to verify that the xed point is weakly attract-
ing or repelling at the bifurcation value by means of a test function, rather than
calculating the constant KM. This calculation shows whether the bifurcation is
supercritical or subcritical and whether the periodic orbit is attracting or repelling.
The following example illustrates this possibility.

Example 6.13. Consider the system of equations given by

H? = y.
y)= —a:—:|:3 +uy+ka:2y,

with parameter /.1. The only xed point is at the origin. The linearization at the
origin is
O1
_1 M '

The characteristic equation is A2 — /.4). + 1 = 0, with eigenvalues

A_-e2_:l:z-,/ 1 _£4.

Thus, the eigenvalues are complex (for. -2 < /.i < 2) and the real part of the
eigenvalues is an = “/2. The real part of the eigenvalues is zero for /.i = 0, so this
is the bifurcation value. The sign of the real part of the eigenvalue changes as u
passes through 0,

d1
Z0” = E > O,

so the stability of the xed point changes as /.1. passes through 0.
We need to check the weak stability of the xed point at the bifurcation value

1.1. = 0. (Notice that the xed point has eigenvalues ii, and is a linea.r center for
u = 0.) The constant in Theorem 6.7(b) is

k
K0 = gr

because f,, = 1, g, = -1, guy = 2k, and the other partial derivatives in the
formula are zero. Applying the theorem, we see that the origin is weakly attracting
for It < 0 and weakly repelling for k > 0. We will obtain this same result using a
test function in what follows.

For u = 0, the system of equations becomes

i = v.

1}: —a:—a:3+Ic:c2y.

238 6. Periodic Orbits

Because the term zzy is like a damping term, we use the test function
242

L(.1:,y)= %+$T+%

to measure the stability of the origin. The time derivative

L=(:r:+:c3):i:+y3}

= (:c+a:3)y+y(—a:—:c3 +ka:2y)

= lc :22 yz.

We rst consider the case in which k < 0. Then, this time derivative is less than
or. equal to zero. It equals zero only on

Z={(a:,y):.1:,=0ory=O}.

The only point whose entire orbit is totally within the set Z is the origin. Thus,
this system satis es the hypothesis of Theorem 5.2. The function L is positive on
the whole plane except at the origin, so we can use the whole plane for U, and
the basin of attraction of the origin is the whole plane. Thus, the origin is weakly
attracting for /.4 = 0. Notice that, for /1 < 0, we have

L= 1./2 (/1+ kw”).

which is still less than or equal to zero. Now, the set Z is only the set y = 0, but
the argument still implies that the origin is attracting. Thus, for any it 5 0, the
origin is asymptotically attracting, with a basin of attraction consisting of the whole
plane. The bifurcation theorem applies and there is a supercritical bifurcation to
a stable periodic orbit for p. > O. Notice that, for p. = O, the system is attracting
across some small level-‘curve L"1(C). This will still be true on this one level curve
for small it > 0. However, now the xed point is repelling, so the w-limit set of
a point different than the xed point must be a periodic orbit and not the xed
point. Theorem 6.6 says that there is a unique periodic orbit near the xed point
and that all points near the xed point have this periodic orbit as their w-limit sets.

Now consider k > 0. The time derivative of L is still L = y2(/1 + k 2:2), but this
is now greater than or equal to zero for /.4 Z 0. Thus, the origin is weakly repelling
for p. = 0 and strongly repelling for u > 0. Therefore, there are no periodic orbits
for any u Z O. However, by the theorem, there is a repelling periodic orbit for
/.i < 0. In fact, for /.1 = 0, the system comes in toward the origin across a small level
curve L"'(C) backward in time. This will still be true for small u < 0. However,
the xed point at the origin is now attracting, so a trajectory starting inside the
level curve L“(C) must have an a-limit set that is different from the xed point,
and so there must be a repelling periodic orbit.

There can be other periodic orbits for the system besides the one caused by the
Andronov—Hopf bifurcation. The following example illustrates this possibility in the
case of a subcritical Andronov—Hopf bifurcation, where there is another attracting
periodic orbit whose radius is l)Ol1I1(l6d away from zero.

Example 6.14. Consider the equations given in polar coordinates by

r = /J. r + r3 —- r5,

0=w.

6.4. Andronov—Hopf Bifurcation 239

This is a subcritical bifurcation because the coe cient of 1'3 is positive. The nonzero
xed points in the 1" equation are given by

T2: 1It\/I+4[L

There is always one positive root

1'? = g1+\/ld-all >2 1 + p..

This is not the orbit bifurcating from the xed point. The second xed point is
realwhen

T2z—_ g1—\/ri+3/'1~

is positive, or /1 < 0. Notice that 1'2 z \/Til, and r1 2: T z 1. The 'r equation
has xed points at 0 < 1'2 < 1'1 for p_< O, and at 0 < 1'1 for /J > 0. The xed point
at 1-1 is always stable. The xed point at 1'2 is unstable when it exists. The xed
point 0 is stable for /.1 < 0 and unstable for /.4 > 0.

O T T T
0+ 0i>iO'(—
0 T2 T1 U 1'1 O 1'1

(*1) (bl (C)
y y
y

(B 13 9.‘

(<1) (6) (fl

Fig-ure 11. Example 6.14. r variables for: (a) it < 0, (b) it = 0, (C) u > 0.
Phase plane in (:c,y) variables for (d) p. < O, (e) p. = 0, (f) p. > O. In each
case, the outer periodic orbit is attracting. There is an inner repelling periodic
orbit for 44 < 0.

In terms of the Cartesian coordinates, there is always an attracting periodic

orbit of radius approximately For [J < 0, the xed point at the origin is

stable, and there is another repelling periodic orbit of radius approximately

See Figure 11.

Example 6.15 (Vertical Hopf). Consider an oscillator with friction

51 = y.
£1 = —== + ml-

This has an attracting xed point for p. < 0 and a repelling xed point for _u > 0.
In any case, there are no periodic orbits for /.1 yé 0. For pt = 0, the whole plane
is lled with periodic orbits. These orbits are not isolated, so they are not called
limit cycles. They all appear for the same parameter value. See Figure 12.

240 6. Periodic Orbits
U 3/ 3/

(8) (b) (C)

Figure 12. Phase plane in (:|:,y)-variables for vertical Hopf of Example 6.15

A small change of the preceding example is given by the penduhun equation
with friction,

fir = 1/.

3] = —sin(z) + /4y.

Again, for /,1 = 0, there is a whole band of periodic orbits. For ,u. 76 0, there are
no periodic orbits. Thus, this system has neither a subcritical nor a. supercritical
bifurcation.

Exercises 6.4

1. Consider the system of differential equations

iv = 1;.
y = —:c—2a:3 —3:c5 +y(2/i +22 +3/2).

a. What are the eigenvalues at the xed point at the origin for di erent values
of p?

b. For /.4. = 0, is the origin weakly attracting or repelling?
c. Show that there is an Andronov—Hopf bifurcation for p. = 0. Is it a sub-

critical or supercritical bifurcation, and is the periodic orbit attracting or
repelling? Explain your answer.
2. Consider the system

i=1/+#~r.
1J=—==+/1u*w’y-

a. Show that the origin is weakly attracting or repelling for it = 0 by using a
Lyapunov function.

b. Show that there is an Andronov—Hopf bifurcation as /,1 varies.
c. Is the bifurcation subcritical or supercritical? Is the periodic orbit attract-

ing or repelling?

Exercises 6.4 241

3. Consider the variation of a predator—prey system

:i:=a:(1+2:c—:r:2—y),

g=l/(I_a')r

with a > 0. The a: population is the prey; by itself, its rate of growth increases
for small populations and then decreases for :0 > 1. The predator is given by
y, and it dies out when no prey is present. The parameter is given by 0.. You
are asked to show that this system has an Andronov—Hopf bifurcation.

a. Find the xed point (ma, ya) with both ma and ya positive.
b. Find the linearization and eigenvalues at the xed point (sc,-,, ya). For what

parameter value, are the eigenvalues purely imaginary? If aa is the real
part of the eigenvalues, show that

£01..) ¢ 0

for the bifurcation value.
c. Using Theorem 6.7, check whether the xed point is weakly attracting or

repelling at the bifurcation value.
d. Does the periodic orbit appear for a. larger than the bifurcation value or

less? Is the periodic orbit attracting or repelling?

4. Consider the system of Van der Pol differential equations with parameter given

by

:i:=y+/.¢:c—:r3,

j(]=—:|3!

a. Check whether the origin is weakly attracting or repelling for pt = 0, by
using a test function.

b. Show that there is an Andronov—Hopf bifurcation as p. varies.
c. Is the bifurcation subcritical or supercritical? Is the periodic orbit attract-

ing or repelling?

5. Mas-Colell gave a Walrasian price and quantity adjustment that undergoes an
Andronov—Hopf bifurcation. Let Y denote the quantity produced and p the
price. Assume that the marginal wage cost is 1/1(Y) = 1 + 0.25Y and the

demand function is D(p) = -0.025;? + O.75p2 - 6p + 48.525. The model

assum that the price and quantity adjustment are determined by the system
of differential equations

p=0.75[D(p) —Y],

Y=blp—¢(Y)1-

pa. Show that p‘ = 11 and Y‘ = 40 is a xed point.
b. Show that the eigenvalues have zero real part for b‘ = 4.275. Assuming

the xed point is weakly attracting for b‘, show that this is an attracting
limit cycle for b near b‘ and b < b‘.

242 6. Periodic Orbits

6. Consider the Holling-Tanner predator-prey system given by

“W’=/"<’(»“:§>‘§»%?>)'

a. Find the xed point (:r:",y") with 2:‘ > 0 and y‘ > 0.
b. Find the value of /1' > 0 for which the eigenvalues for the xed point

(:c‘, y‘) have zero real part.
c. Assuming the xed point is weakly attracting for ,u,', show that this is an

attracting limit cycle for it near p‘ and /4 < it‘.

6.5. Homoclinic Bifurcation .

Example 5.11 gives a system of differential equations for which the w-limit set of
certain points was equal to a homoclinic loop. In this section, we add a parameter
to this system to see how a periodic orbit can be born or die in a homoclinic loop
as a parameter varies. After considering this example, which is, not typical, but
which illustrates the relationship between periodic orbits and homoclinic loops, we
present a more general theorem. We consider this bifurcation mainly because it
occurs for the Lorenz system, which is discussed in Chapter 7.

Example 6.16. Consider the system of differential equations

5: = ya

q_= 1 - $2 + y(6a:2 - 413 - 61,12 +12t),

where /,4 is a parameter.._- If the term containing y in the 3) equation were not there,
there would be an energy function

2 a2

L = -5 3- 3-.

For the complete equations, we use L as a bounding function. See Figure 13 for a
plot of some level curves for L. Its time derivative is

Z0 whenL</1,

L=y2(6:|:2—4a:3—6y2+12p.)= —12y2(L—/J) =0 when L-——'/J.,

50 whenL>p..

First we consider the situation when the parameter p = 0. Then, there is a
homoclinic loop

PD = {(a:iy) 3 L(3ry) = or 3 2
For (:v0,y0) with L(a:0,y0) < 0 and 1:0 > 0, w(:z:0,yo) = F0, as we discussed before.
See Figure 14(b).

Next, consider the situation when the parameter "1/5 < /1 < 0. Then, the level
curve

Fl-l = {(3)1/) : L($iy) : /L» 3 >
is invariant, and is a closed curve inside the loop F0. Since it is invariant and
contains no xed points, it must be a periodic orbit. If (:c0,yo) is a point with
2:0 > 0 and ‘1/6 < L(:c0,y0) < /1, then, again, L 2 0 and w(a:o,y0) = 1",, Similarly,

6.5. Homoclinic Bifurcation 243

ll ta

:\

> ‘I

Figure 13. The level curves L“1(—-0.5), L‘1(—0.1), 171(0), and L“'(0.1)
for the test function for Example 6.16

if /,1, < L(a:0,y0) 3 0 and 2:0 > 0, then L 3 0, the orbit is trapped inside L"(0),
and so w($q, yo) = I‘,,. In particular, the branch of the unstable manifold of the
saddle point at the origin with st > 0 must spiral down onto the periodic orbit.
The periodic orbit l",, is an attracting limit cycle. See Figure l4(a). If (:1:0,y0),
with mg > 0, is on the stable manifold of the origin, the quantity L(¢(t; (:::o,yO)))
increases as t becomes more negative, ¢(t; (a:q,y0)) must go outside the unstable
manifold of the origin, and ¢(t; (:r:0,y0)) goes off to in nity. (In Figure 14(a), we
follow this trajectory until it intersects the positive y axis.) In particular, there is
no longer a homoclinic loop,

W“(0) Fl W’(0) = {O}.

and the stable and unstable manifolds intersect only at the fixed point. See Figure
14(a). As ,u goes to zero, the limit cycle l‘,, passes closer and closer to the xed
point at the origin; it takes longer and longer to go past the xed point, and so the
period goes to in nity.

Next, we consider p. > 0. The invariant curve L_1([J.) is unbounded with no
closed loops. In fact, we argue that there are no periodic orbits. On the unstable
manifold of the origin, the value of L starts out near 0 for a point near the origin,
which is less than /,1. The derivative L is positive and the value of L increases.
Therefore, the orbit carmot come back to the origin, but goes outside the stable
manifold of the origin. Thus, the homoclinic loop is destroyed. In fact, if (:|:0,y0) is
any point inside P0 but not the xed point (1, 0) (i.e., 0:0 > 0 and 0 > L(z0,y0) >
-1/6), then (:ro,yQ) is either on the stable manifold of the origin or eventually
d>(t; ($0, yq)) goes off to in nity as t increases. The time derivative L is positive on
the loop F0 = L“1(0). Therefore, no trajectory can enter the inside of this loop
as time increases, and any point starting outside the loop must remain outside the
loop. In fact, it is not hard to argue that points not on the stable manifold of the
origin must go off to in nity. Therefore, there are no periodic orbits. See Figure
14(c) for the full phase portrait of this case.

Remember that the divergence of a vector eld F(x) is given by

71 8171'

244 6. Periodic Orbits
1/
a:

(a)

y\

I

(b)
:1

J3

(C)

Figure 14. Homoclinic bifurcation for Example 6.16

For a system of differential equations x = F(x), we often call the divergence of
the vector eld F the divergence of the system of di erential equations. The main
difference between the preceding example and the generic situation is that the
divergence of the vector eld in the example is O at the xed point when it = O. The
typical manner in which such a homochnic bifurcation occurs is for the divergence
to be nonzero at the xed point, as assumed in the next theorem.

Theorem 6.8 (Homoclinic bifurcation). Assume that the system of di erential
equations x = F,,(x), with parameter ,u,, has a saddle xed point p,, and satis es
the following conditions."

6. 5. Homoclinic Bifurcation 245

(i) For u = no, one branch FL of the unstable manifold W"(p,,,,,F,,,,) is
contained in the stable manifold W‘(p,,,, , F,,,,).

(ii) For p. = no, assume that the divergence of FM at pm, is negative. Then
orbits starting at qo, inside Fjo but near FIB, have

°-’(q0»F;1ol={Puo}U Fio-

(iii) Further assume that the homoclinic loop Pjo is broken for /.i 96 1,1,0, with the
unstable manifold of the xed point p,, inside the stable manifold for it < /.40 and
outside the stable manifold for p. > uo.

Then, for it near /so and u < /1.0, there is an attracting periodic orbit I‘,, whose
period goes to in nity as /.1. goes to 0, and is such that PM converges to Pjo.

If we have the same situation except that the divergence of FM at p,,,, is positive,
then there is an unstable periodic orbit for u near no and /.t > uo.

The creation of a periodic orbit from a homoclinic loop as given in the preceding
theorem is called a homoclinic bifurcation.

We explain the reason that the homoclinic loop is attracting when the diver-
gence is negative at the xed point. By differentiable linearization at a saddle xed
point as given in Theorem 4.14 or 4.16, there is a differentiable change of variables,
so the equations are

¢=an

y = _by>

where 0. — b < 0, or a < b. (We are writing these equations as i.f the coefficients do
not depend on the pa.rameter u.) Then, a trajectory starting at the cross section
y = yq has initial conditions (a:,y0) and solution x(t) = a:e“‘ and y(t) = yo e"". If
we follow the solutions until a: = :c1, then the time t = -r satis es

:01 = :1: e°',

e" = II]/0' :21/°.

The y value for t = r is

?J(T) = yo (eqlb

= yo 13;‘,/0 :13"/°.

Thus, the Poincaré map past the xed point is

1/ = Pu,1(1l=) = 11011?/art’/at
which has an exponent on :1: that is greater than one. The derivative is P,1,,(a:) =
yo zcfb/° a:‘1*”/“, so P,’,_1(0) = 0 and |P,’,'1(a:)| << 1 for small a:. The Poincaré
map a: = P,,,2(y) from a: = 2:1 back to y = yo is a differentiable map. Since the
time to travel between these two transversals is bounded, PL’, (y) 76 0 and is nite.
The point y = 0 is on the unstable manifold, so P,,_2(0) is the intersection of the
unstable manifold with the transversal at y = yo. This point changes with u, so we
let

P,,_2(0) = 6,“

246 6. Periodic Orbits

which has the same sign as /,4. The composition of P,,_1 and P,,_g is the Poincaré
map all the way around the loop from y = yo to itself,

11(1) = P/».2(Pp.1(¢)) = P,».2(1"’/“)-

By the chain rule,

P,i(I) = 1’,’i,2(1=°’“lP,1,1 (I), and

P,1(0) = P,1,2(0)P,1_1(0) = 0,

|P,1<=»>| < 1

for all small :2. Using the mean value theorem, for p small and :0 small, we have

-- lP»(I) - Pt(0)l = |P,1(='=')l ' ll‘ - 0|-

Forp=0andsmalla:>0,- ‘ '_

0 = P0(0) < P0(:i') < 5:.

For /1 > 0 and sufficiently small,

0 < 6,, = P,,(0) < P,,(:E) < 5:,

and PP contracts the interval [0,5] into itself. Therefore, there is a point 22,, with
P,,(a:,,) = a:,, and P,1(a:,,) < 1. (See Theorem 9.3 and Lemma 10.1 for more details
on this type of argument to get a xed point.) This results in an attracting periodic
orbit for /4 > 0, as we wanted to show.

If a > b, the same argument applies, but P,j_1(0) = oo, and

P,i($) = P,',.2($b/°)P,1,1($) > 1
for small ac. Thus, for /.I7,< 0,

P,,(0) = 6,, < 0 < :5 < P,,(:E),

and P“ stretches the interval [0, 5:] across itself. Therefore, there is a point :z:,, with
P,, (:r:,,) = a:,, and P,’,(:c,,) > 1. This gives a repelling periodic orbit for /1 < 0, as we
wanted to show.

Exercises 6.5

l. Consider the system of differential equations

i = 2/.
s'/==v—2w3+u(/1+1’ -1‘ -21’),

where /.t is a parameter.
a. Using the test function
-1’ + $4 + y’

‘L(z1 = +1

show that there are (i) two homoclinic orbits for p = 0, (ii) one attracting
limit cycle for /1 > 0, and (iii) two attracting limit cycles for —% < p < 0.
b. What are the w-limit sets for various points for p < 0, p. = 0, and it > 0.
c. Draw the phase portrait for it > 0, it = 0, —% < /.t < 0, and p. < —%

6.6. Rate of Change of Volume 247

6.6. Rate of Change of Volume

In this section, we consider how the ow changes the area or volume of a region
in its domain. In two dimensions, this can be used to rule out the existence of a
periodic orbit in some part of the phase plane. This result is used in Section 7.1 to
show that any invariant set for the Lorenz equations must have zero volume. We
also use the Liouville formula from this section to calculate the derivative of the
Poincaré map in the next section.

The next theorem considers the area of a region in two dimensions or volume
in dimensions three and 1a.rger. By volume in dimensions greater than three, we
mean the integral of dV = d:c1~--da:,, over the region. Although we state the
general result, we only use the theorem in dimensions two and three, so this higher
dimensional volume need not concern the reader.

Theorem 6.9. Let 1': = F(x) be a system of di erential equations in IR", with ow
¢(t;x). Let 9 be a region in IR", with (i) nite n-volume and (ii) smooth boundary
bd Q. Let 9(t) be the region formed by owing along for time t,

90) = {¢(t;Xo) 1X0 6 9}-

a. Let V(t) be the n-volume of 9(t). Then

d
—dt Vt =[g(t) V-F (X) dV,

where V - F(x) is the divergence of F at the point x and a'V = dz; ---da:,, is the
element of n volume. If n = 2, then V(t) should be called the area and dV should
be replaced by the element of area dA = da:1da:2.

b. If the divergence of F is a constant (independent of x), then

V(t) = V(O) e‘V'F.

Thus, if the vector eld is divergence free, with V - F E 0, then the volume is
preserved; if V - F < 0, then the volume decays exponentially; if V - F > 0, then
the volume grows exponentially.

In the next chapter, we use the version of the preceding theorem with constant
divergence to discuss the Lorenz equations. This system of differential equations
in three dimensions has a constant negative divergence. It also has a positively
invariant region 9 that is taken inside itself for positive time. The volume V(t)
Of 9(t) = ¢(t; 9) has to decrease as t increases. Thus, this set converges to an
invariant attracting set that must have zero volmne. (It is less than any positive
number by the preceding theorem.)

We will also use the manner in which the determinant of the derivative of the
ow varies along a solution curve. This formula is called the Liouville formula.
Since the determinant of D,,¢(,;x°) is the volume of an infinitesimal n-cube, this
formula is related to the preceding theorem.

248 6. Periodic Orbits

Theorem 6.10 (Liouville formula). Consider a system of di erential equations
x = F(x)

in IR" with both F(x) and -g-§(x) continuous. Then, the determinant of the matria:

of partial derivatives of the flbw can be written as the exponential of the integral of
the divergence of the vector eld, V - F(x), along the trajectory,

t

det (D=<¢c.x@>) = <=><P (I0 (V - F)<¢<=»=».»>> '18) -

The equation for the derivative with respect to time of the determinant of
l)x¢(,,,°) follows by combining the rst variation formula given in Theorem 3.4
with the formula for the determinant of a fumdamental matrix solution of a time-
dependent linear differential equation given in Theorem 2.3. We give another proof
in Section 6.9 based on the change of volume given in the preceding theorem.

We can also use these theorems to show that it is impossible to have a periodic
orbit completely contained in a region of the plane.

Theorem 6.11. Suppose >1 = F(x) is a di_ 'erential equation in the plane. Let 'y
be a periodic orbit which surrounds the region Q, and the differential equation is
de ned at all points of 9.

a. (Bendixson Criterion) Then,

//9(v-F)(,,, dA=0.

In particular, if Q,is a fregion with no holes and divergence has one sign in Q, then
there can be no periodic orbit that can be completely contained in Q.

b. (Dulac Criterion) Let g(a:,y) be any real-valued function de ned and
di erentiable at least on the points of Q. Then,

//9V-(gF)(,,) dA=0.

In particular, ifQ is a region with no holes and the divergence of g(:r, y) F(:r, y) has
one sign in Q, then there can be no periodic orbit that can be completely contained
in Q.

Part (b) follows from Part (a) because x = g(a:, y) F(:v,y) has the same trajec-
tories as it = F(:r, y); they just travel at different speeds.

This theorem can be applied to some examples to show there are no periodic
orbits.

Example 6.17. Consider the Van der Pol equation given in the form

if = v.
27 = —w.+u(1 — I2):/,

This has divergence equal to pi (1 — :02). Therefore, for u 75 0, there cannot be any
periodic orbit contained completely in the strip -1 5 a: 5 1. This same fact can
be shown using the test function L(:z:, y) = ($2 + y2)/2, as we did for the Lienard
equation in Section 6.3.

Exercises 6.6

Example 6.18. Consider the equations

:i:=a:(a—by—f:1:),

i)=v(—¢+@@=—hi/).

where all the parameters a, b, c, e, f, and h are positive. Notice that this is just
the predator-prey system, with a negative impact of each population on itself. We
assume that °/f > °/e, so there is a xed point (a:',y') in the rst quadrant. The
rst quadrant is invariant, because :i: = 0 along a: = 0 and y = 0 along y = 0. The
divergence does not have one sign in the rst quadrant:

(V-F)(,,,,,) =a—by-2fa:—c+ea:—2hy.

However, if we let g(:|:,y) = 1/rry, then

h
V'(gF)(:z:,y)=_£_5v

which is strictly negative in the rst quadrant. Therefore, there cannot be any
periodic orbits inside the rst quadrant. Note that g(:r,y) = 1/wy is not de ned
when either :1: or y equals zero, but it would be well de ned inside any periodic
orbit contained entirely in the rst quadrant.

In the next section, we give another argument that this example does not have
any periodic orbits by using the Poincaré map. See Example 6.24.

Exercises 6.6

I

1. Use Theorem 6.11 to show that the following system of di erential equations
does not have a periodic orbit in the rst quadrant:

:i: = a: (a — by),

29 = v (-6 — iw)-

Both a and b are positive parameters.
2. Consider the predator—prey system of differential equations

£=$(a_I—y))
9:1/(_3a'+$)1

where a is a positive parameter.
a. Find the xed points and determine the stability type of each xed point
(saddle, attracting, repelling, etc.).
b. Use Theorem 6.11 to show that the system does not have a periodic orbit
in the rst quadrant,
c. What is the basin of attraction of any attracting xed point?
d. Sketch the phase portrait of the system using the information from parts
(a—c) and the nullclines. Explain the location of the stable and unstable
manifolds of any saddle xed points.

250 6. Periodic Orbits

3. Show that the following system of differential equations does not have a periodic
orbit:

1.: = _$ + yzn

iv = I2 — va-

4. Consider the forced damped pendulum given by

+ = 1 (mod 2 1r ),
:i: = y (mod 21r ),
1'1 = — S'H1(='=) — v + <=<>S(T)-
a. What is the divergence of the system of equations?
b. Show that the region Q = {(r,z,y) 1 lyl S 3} is a positively invariant
region. Hint: What is the sign of y when y = :l:3?
c. Let Q(t) = ¢(t;Q). What is the volume of Q(t) in terms of the volume
of Q? What happens to the volume of Q(t) as t goes to in nity‘?
5. Consider the system of differential equations given by

I = 2/.

9 = at _ 21:8 — 1/»

with ow ¢(t;x),
a. What is the divergence of the system?
b. If 9 is a region in the plane with positive area, what is the area of Q(t)
in terms of the area of Q?

6. Consider the system of differential equations

w=v.

y=—:r—y+:c2+y2.

Using the scalar ftmction g(a:, y) = e4” in the Dulac Criterion, show that the
system does not have a periodic orbit.
7. Rikitake gave a model to explain the reversal of the magnetic poles of the earth.
The system of differential equations is

:i: = —u x + y z,

v= -I11/+(Z-‘1)I,

2 = 1 — :1: y,

Wltl'l/t>08.1'l(lG>0.
a. Show that the total system decreases volume as t increases.
b. Show that there are two xed points (a:',y',z"), which satisfy y‘ = 1/1:‘.

and Z’ = #(r")2~ (1=‘)’ — (I‘)" = 0/I1»

c. Show that the characteristic equation of the linearization at the xed
points is

/\3+2;.t/\2+ Z/\+2a=0.

Show that A1 = —2p. is one eigenvalue, and the other two eigenvalues are
ii V (°/u)-

6. 7. Poincaré Map 251

6.7. Poincaré Map

The self-excited oscillator given by the Lienard system of differential equations was
shown to have a unique attracting periodic orbit by following the trajectories from
the y-axis until the rst time they return to the y-axis. This rst retmn map is an
example of a. Poincaré map.

We start this section by calculating the Poincaré map for the simple example of
the rst section of this chapter, for which the solutions can be calculated using polar
coordinates. After this example, we turn to the general de nition and examples in
which the Poincaré map has more variables. We give a condition for the periodic
orbit to be asymptotically stable in terms of the eigenvalues of the derivative of the
Poincaré map. We also relate the eigenvalues of the derivative of the Poincaré map
to the eigenvalues of the derivative of the ow. In the case of two dimensions, there
is a condition in terms of the divergence which can be used to determine whether a
periodic orbit is attracting or repelling. We apply this to the Van der Pol equation
to give another veri cation that its periodic orbit is attracting.

Example 6.19 (Return to the Poincaré map for Example 6.4). The differ-
ential equations are

:t=y+a:(1—a:2—y2),

v=—r+v(1—=2—1/2).

OT

r='r(1—r2),

9=-L

in polar coordinates. In Example 6.4, we showed that the rst return of trajectories
from the half-line {(a:, 0) : 1: > 0} to itself, the Poincaré map, is given by

P(r) = r[r2 + e'4"(1— r2)]'1/2
= r[1 — (1 — e_4")(1 - 'r2)]_1/2
= [1 + e_4"(i-'2 — 1)]'1/2.

The derivative of the rst form of the Poincaré map, is

I_ e—41r _ e—41fP(7.)3

P (T) _ [T2 + 6-41r(1 _,.2)]a/2 _ 7.3 > O"

For the rest of the discussion of this example, we use the Poincaré map to
show that the periodic orbit r = 1 is orbitally asymptotically stable, and attracts
all orbits other than the xed point at the origin. This can be seen from the
differential equations directly, but we want to derive this from the properties of the
Poincaré map in order to introduce the relationship of the Poincaré map and the
stability of the periodic orbit.

(a) The points r = 0 and r = 1 return to themselves, P(O) = 0 and P(1) = 1,
88 can be seen from the rst equation for P(r).

252 6'. Periodic Orbits

(b) Next, we consider 'r > 1. The denominator in the second representation of
P(1') has

1 - (1 - e-"')(1 - T2) > 1, so

P('r) < 1'.

Similarly, the denominator in the third representation of P(1‘) has

1 + e"“'(1-'2 — 1) < 1, so
1 < P(1').

Combining, we have 1 < P(7') < r, and the orbit returns nearer the periodic orbit
r = 1. See the graph of P(r) in Figure 15. Note that the graph of P(r) is below
the diagonal ('1', 1') for 1' > 1.

P(1')

0 e_4'K ’I‘
1

Figure 15. Poincaré map for Example 6.4

If 1'0 > I, then 1 < 1'1 = P('r0) < ro. Continuing by induction, we see that
1 < r,, = P(1',,_1) < 1',,_|, so

1<7' = P(1‘n_1)<7‘n_1 = P(Tn_2) < < T2 = P(7'1) < 7'1 =P(1'g) < T0.

This decreasing sequence of points must converge to a xed point of P, so r,,
converges to 1.

In fact, the mean value theorem can be used to show this convergence more
explicitly. For 1' > 1, PM/r < 1, so

—-4 3

P'(1') = _i"T_1;(T") < 6-4" <1.

For the preceding sequence of points 1‘,-, there are points 1‘; between rj and 1 such
that

P(1'1)— P(1) = P'(1"})("j - 1),

6. 7. Poincaré Map 253

SO
1',, — 1 = P(1‘,,_1) — P(1) = P'(1‘;,_1)(1',,_1 — 1)
< e"'“‘ (1-,,_1 — 1) = e_'"’(P(1‘,,_2) — P(1))
= e_4"P’(1{,_2)(1',,_2 — 1) < e_4"e_4" (1',,_2 — 1) = e_2'4” (1',,_2 — 1)

< C_n4 (T9 -

Since e‘4" < 1, e"“” goes to zero as n goes to in nity, and 1",, must converge to
1. In fact, this gives an explicit bound on the rate at which the sequence of return
positions r,, converges to 1.

(c) Finally, we consider 0 < 1' < 1. The quantity 0 < (1 — e“4")(1 — 1'2) < 1,
so the denominator

0 < [1 - (1 - e-'")(1 - r’)]‘/2 <1 and

0 < 1- < P(r).

Using the second equation for P(r), we have

1 < 1 + e_4"('r'2 — 1), so

P(1-) = [1 + e-"(T-2 - 1)]"/2 < 1.

(We can also see that P(1~) < 1 because two di erent trajectories cannot cross and
1' = 1 retmns to P(1) = 1.) Combining, we get

0<1'<P(1')<1,

and the orbit returns nearer the periodic orbit 1- = 1. Taking an 1'0 with 0 < To < 1,
we have

0<1'0<1'1 <---<'r,,_1<1',,<1.
This increasing sequence of points must converge to a xed point of P, so 1;,
converges to 1.

We can also apply the mean value theorem as in the earlier case. For 0 < 1' < 1,
P(r) < 1, so

P'(1') § e_4'1‘_3.

Therefore, 0 < P’ (1) < 1 for 1 > e‘4"/3. On the other ha.nd, P’ (0) = e2" > 1, and
so P’ (1') > 1 for small 1' > 0. These estimates on the derivative indicate that the
graph of the Poincaré map P('r) indeed looks like the graph in Figure 15. Once the
sequence gets in the range where P’ (1') < 1, the argument using the mean value
theorem can be used again to get an explicit bound on the rate of convergence to
1.

The derivative of the Poincaré map at the periodic orbit is P’ (T) = e"” < 1.
Because P’ (1) is a single number, it can be thought of as the eigenvalue of the 1 >< 1
matrix whose entry is P’ (1).

Before giving the general conditions, we discuss a speci c example of a non-
homogeneous linear di 'erential equation. The Poincaré map for such an equation
requires the evaluation of the fundamental matrix solution at the period of the

254 6. Periodic Orbits

forcing. For the example, the derivative of the Poincaré map can be given fairly
explicitly in terms of the mdamental matrix of the homogeneous linear equations.
This allows us to discuss the conditions on the eigenvalues of the Poincaré map.

Example 6.20. Consider the nonhomogeneous linear system

$1 : $21

5:2 = —a:1 — 1:2 + cos(w t).

The forcing term cos(w t) has period 2"/1.1 in t, cos (w(t + 2"/w)) = cos(wt + 21r) =
cos(w t). In order to change this into a system that does not depend on time
explicitly, we introduce the variable 1 whose time derivative is identically one and
is-taken as a periodic variable with period 2"/1., (the period of the forcing term):

+ = 1 (niod 21/., )1,

ii = $2,
£132 = -111 — I2 + cos(w'r).

Here we write “( mod 2"/1., )" as an abbreviation for modulo 2 7(;), which means
that we subtract the integer multiples of 2"’/,,, and leave only the remaining part,
which is a fraction of 2"/w. Since the variable 1 is periodic,_we can consider it
an “angle variable". We put the differential equation for 1 rst so the results look
more like the general case considered later in the section. Taking initial conditions
(0, :c1_o, 22,0), we know by variation of parameters that the solution is given by

v(t) t

§$llW@$+a~~Qt$e)

where

A-(A0 -11 )

is the matrix of the li.near system. The variable 1' comes back to itself at time 2"/w.

We can take the transversal to be

' >3 = {T = o }.

Writing only the x-coordinates on 2, the Poincaré map from 2 to itself is given

by

P(x0) = em"/“X0 + v,

where

V;-[git 6»-(<2»/~)-~>< 0 )d_.,

O cos(ws)

is a constant vector. The derivative of the Poincaré map is DP(,,) = eA2"/"’.

The eigenvalues of A are /\* = —% :l: Hg, and those of em"/“ are e2"’\*/“’ =
e"’/“’e*'. ‘/5”/“’, which have absolute value equal to e"'/"’. The point x‘ is xed
by the Poincaré map, provided that

em"/”x' + v = x’ or
x' = (I — elm"/“’) -1 v.

6.7. Poincaré Map 7 255

Thus, there is one xed point for P and one periodic orbit for the system of differ-
ential equations. For any other initial condition,

P(xq) — x‘ = P(x0) — P(x') = eA2"/“’x0 + v — (eA2"/“'x‘ + v)
: eA21r/u (X0 _ X.) = 6-11/ue(A+§I)21r/w (X0 _ x.)_

Continuing by induction, the nil‘ iterate x,-, = P(x,,_1) = P"(xo) satis es

P"(x0) — x" = P"(x0) — P"‘(x') = e""'/“’e(A"'§n"2"/“’ (xo — x‘).

The eigenvalues of e(A+51)"2"/‘*’ are ei ‘"‘/5"/” and the points
e(A+§1)1~.21r/..,(xo _ X.)

move on an ellipse. The term e"""/“’ goes to zero as n increases. Therefore, for
any initial condition, P"(x0) converges to the xed point x‘ of P, which is on
the periodic orbit of the ow. The fact that makes this work is that e""/"’ < 1.
But e_"/“’ equals the absolute value of each eigenvalue of the matrix em”/"’. The
matrix em"/“’, in turn, is the matrix of partial derivatives of the Poincaré map.

In terms of the whole flow in three dimensions, the time 2"/1., map of the flow is

1r T +2?"
¢ (?;;(To,X0)) = (;(x°))

Tg+%
= CA2-rr/wxo +J‘%§' eA((21r/w)-s)( 0 ) is) .

° cos(w s + 10)

The derivative with respect to the initial conditions in all the variables (10, X0) is

1U
D(-ru,xol¢(21r/wi(-m,xo)) = (* DP(xo)) »

where the entry designated by the star is some possibly nonzero term. Therefore,
the eigenvalues of the derivative with respect to initial conditions of the ow after
one period are 1 and the eigenvalues of DP(X°).

We now state a general de nition of a Poincaré map.

De nition 6.21. Consider a system of differential equations x = F(x), and a point
x‘ for which one of the coordinate functions F;,(x') 75 0. The hyperplane through
x‘ formed by setting the k"h coordinate equal to a constant,

E={x::z:k=:z:;,},

is called a tmnsversal, because trajectories are crossing it near x‘. Assume that
¢(1-"‘;x") is again in E for some 1' > 0. We also assume that there are no other
intersections of ¢(t;x") with E near x‘. For x near x‘, there is a nearby time 1'(x)
such that ¢(1(x);x) is in E. Then,

P(x) = ¢(r(X):><)

is the Poincaré map.

256 6. Periodic Orbits

We use the notation P"(x0) = P(P""1(x0)) = P o P o - - - o P(xQ) for the nu‘
iterate. Thus, P" is the composition of P with itself n times. If x,- = P(x,--1) for
j = 1,...,n, then x.,, = P"(x0).

We next state the result that compares the eigenvalues of the Poincaré map
with the eigenvalues of the period map of the flow.

Theorem 6.12. Assume that x‘ is on a periodic orbit with period T. Let E be a
hyperplane through x‘, formed by setting one of the variables equal to a constant;
that is, for some 1 3 k 3 n with F1, (x") 76 0, let

Z={x:a:;,=a:f,}.

Let P be the Poincaré map from a neighborhood of x’ in E back to E. Then, the n
eigenvalues of D,,¢(q~,,,-) consist of the "n - 1 eigenvalues of DP(,,-), together with
1, the latter resulting from the periodicity of the orbit.

Idea of the proof. We showed earlier that

D><¢('r;x-)F(X') = F(¢(T;X')) = Fix‘)-

Therefore, 1 is an eigenvalue for the eigenvector F(x').

The Poincaré map is formed by taking the time r(x) such that ¢(-r(x);x) is

back in E:

P(x) = ¢(1'(='=); X)-

Therefore, if we take a vector v lying in E, we have

DP(X')v:,= Dx¢("'(x')ix')v + 3¢ 3
(1.(x.).x.)) ,6) V

='"'_Dx¢(T;x~)V + F(X‘) (DT(x.)) V.

In the second term,

8r

(Dw->) " = 5

is a scalar multiplied by the vector eld F(x"') and represents the change in time
to return from E back to E. Thus, if we take a vector v lying in E, we get

Dx¢(T;x-)V = DP(x-)V — F(X') (DT(x-)) V.
Therefore, using a basis of F(xU ) and n — 1 vectors along E, we have

_ 1 —DT(x.)

DX¢(T;X') _' DP(x_)) 1

and the eigenvalues are related as stated in the theorem. El

De nition 6.22. At a periodic orbit, the (n — 1) eigenvalues of the Poincaré
map DP(,,.) are called the cha.racteri.9tic multipliers. The eigenvalues of D,,¢(7-,,,-)
consist of the (n— 1) characteristic multipliers together with the eigenvalue 1, which
is always an eigenvalue.

If all the characteristic multipliers have absolute value not equal to one, then
the periodic orbit is called hyperbolic.

Now, we can state the main theorems.

6. 7. Poincaré Map 257

Theorem 6.13. Assume that x‘ is on a periodic orbit with period T.
a. If all the characteristic multipliers of the periodic orbit have absolute values

less than one, then the orbit is attracting e.o,rbitally asymptotically stable).
b. If at least one of the characteristic multipliers of the periodic orbit has an

absolute value greater than one, then the periodic orbit is not orbitally Lyapunov
stable (i.e., it is an unstable periodic orbit).

It is only in very unusual cases that we can calculate the Poincaré map ex-
plicitly: nonhomogeneous linear system or simple system where the solutions for
the variables can be calculated separately (in polar coordinates). In other cases, we
need to use a more indirect means to determine something about the Poincaré map.
In the plane, the characteristic multiplier of a periodic orbit can be determined from
an integral of the divergence.

Theorem 6.14. Consider a differential equation in the plane, and assume that x‘
is on a. periodic orbit, with period T. Let (V - F)(,,) be the divergence of the vector
eld. Let y be the scalar variable along the transversal E, and y‘ its value at x‘.
Then, the derivative of the Poincaré map is given by

Fly‘) = QXP (ATW ' F)(¢(i;><-)) 11¢)-

Proof. In two dimensions, using equation (6.5), we have

d€I1(Dx¢(T;x-)) = P'(y-),

where we use prime for the derivative in one variable. By the Liouville formula,

Theorem 6.10, we have '

T
det(D,,¢(q-,,,.)) = exp (‘A (V - F)(¢(,;,,-)) dt) det(D,,¢(0,,,-))

T

= exp (fa (V " F)<¢<¢.><->1 ii) -

because D,,¢((,,,,.) = I. Combining, we have the result. El

We give a few nonlinear examples that use the preceding theorem.
Example 6.23. Consider the time-dependent differential equation

:i: = (a + b cos(t)) :c — 1:3,
where both a and b are positive. We can rewrite this as a system which does not
explicitly depend on t:

9 = 1 (mod 21r),
:i: = (a — b cos(0)) :1: — 23.
The ow starting at 0 = 0 can be given as

¢(¢; (0, 110)) = (i.¢(i; $11)).

258 6. Periodic Orbits

where 0(t) = t. We can write the Poincaré map from 0 = 0 to 0 = 21r as
¢(21r;(O,:1:0)) = (21r, P(a:0)) or
¢(27I'; Z170) = P($q).

Since st = 0 for as = 0, 1/;(t; 0) E 0, P(O) = 0, 0 is a xed point of the Poincaré
map, and (0, 0) is on a periodic orbit for the system of differential equations.

For:r>\/a+b,

:i:<(a.+b):r:—a:3=:n(a+b—a:2)<0,

so 0 < '¢(t;a:) < :1: and 0 < P(x) < :2. Similarly, for a: < —\/a + b, 0 > P(a:) > sc.

Therefore, any other xed points for P must lie between :t\/a + b.

The matrix of partial derivatives is" -

00
(—b sin(0) a: a + b cos(0) — 32:2)?

Note that the equation for 9 is a constant, so it has zero derivatives. Therefore, the
divergence is a + b cos(9) — 322.

Along ¢(t; (0, 0)) = (t,0), the divergence is a + b cos(t). Using Theorem 6.14,

we have '

P/(0) = cf”21r a+b coa(t)dt = ea21r > 1.

Therefore, 0 is a repelling xed point of P, and an unstable periodic orbit for the
ow.

Next we consider a periodic orbit with a:¢, 96 0, so P(a;o) = 2:0. Then, 1/2(t; $0) 74
0 for any t, by uniqueness of the solutions. We can write the divergence as

a + b cos(t) Q 31:2 = 3 (a + b cos(t) — :22) — 2a - 2b cos(t)

= 3-: — 2a — 2b cos(t).

I

Then,

P/($0) =.ef°"' 3&7/;|;—2a.—2b coa(t) dt = e3ln|P(:|:°)|—3ln|:|:g| 6-G41! : e—a.41r < 1

Now, P'(0) > 1, so P(:r1) > 2:1 for small positive azl. But, < :1: for

:1: > \/0. + b. Therefore, there has to be at least one $0 between 0 and \/a + b for

which P(a:O) = 1:0. At each of these points, P’ (2:0) < 1 and the graph of P is

crossing from above the graph of a: to below this graph as z increases. Therefore,

there can be only one such point, and so, only one positive xed point of P; this

x point corresponds to an attracting periodic orbit for the system of differential

equations.

For :1: < 0, a. similar argument shows that there is a unique xed point of P, so
there is a unique periodic orbit for the system of differential equations.

The periodicity of the equation means that we cannot explicitly nd the solu-
tions, but the basic behavior is similar to that of the equation with the periodic
term dropped.

We describe an application of the preceding theorem to show that a predator-
prey system, with limit to growth, has no periodic orbit.

6. 7. Poincaré Map 259

Example 6.24. Consider the system

i=Iv(¢1—by—fI),
1': = 2/(—¢+e¢— by).

where all the parameters a, b, c, e, f, and h are positive. Notice, that these are just
the predator—prey equations, with a negative impact of each population on itself.
We assume that “/1 > °/C, so there is a xed point (a:‘,y') in the rst quadrant.
For these values, the xed point satis es the equations

0=a—by‘—fa:",

0 = —c+ ea:‘ — h.y".

The matrix of partial derivatives at (:|:‘, y’) is

—fa:‘ —ba:‘
ey" —hy‘ '

This matrix has a negative trace, so the xed point is attracting.
In the preceding section, we proved that this equation does not have any pe-

riodic orbits by applying the Dulac criterion. We now show this same fact using
an argument involving the Poincaré map. To show that there is no periodic orbit,
we use Theorem 6.14 to determine the derivative of the Poincaré map at a periodic
orbit, if it exists. The divergence is given by

V-F(,_,,)=(a—by—f:z:)—fa:+(—c+e:r-hy)—h.y

—_Em+y2_ fa: _ hy.

To apply the theorem, we consider the Poincaré map P from

3={(<=.y')===>=='}

to itself. If the system has a periodic orbit and P(a:1) = 1:1, then the integral giving
the derivative of P can be calculated as

T Ti

I0 (V ' F)(¢(¢=<:'.~->>> <1‘ =[o -1 + -3/ - fx - by it

= l"($(T)) - l11(I(°)) +1r1(1/(T))—lI1(1/(0))- AT fw + by it

T
=—/ f:r:+hydt

0
<0,

since :r:(T) = :|:(0) = :01 and 3/(T) = 1/(0) = y‘ on a periodic orbit. Thus,

P'(I) = EXP T

' F)(¢(t;(:1‘v-))) (it) <

The Poincaré map has P(O) = 0 and 0 < P’ (:0) < 1, so the graph of P(a:) is
increasing, but below the diagonal. Therefore, there are no periodic orbits. If

(Ia,y0) is an initial condition not at the xed point (a:',y") and with 2:0 > 0 and
U0 > 0, then the backward orbit cannot limit on any periodic orbit or xed points
8-nd so must be unbounded.

260 6. Periodic Orbits

Hale and Kocak, [Hal91], apply Theorem 6.14 to the Van der Pol system to
give another proof that the periodic orbit is stable.

Theorem 6.15. Any periodic orbit for the Van der Pol system is stable. Therefore,
there can be only one periodic orbit.

Proof. We write the Van der Pol equations in the form

i = 1/.

Q = —:c+ (1 —:r:2)y.

The divergence of F is given as follows:
V-F=%3_(y)+€y6(—z+(1fz2)y)=1—:n2.

Consider the real-valued function L(a:, y) = %(a:2 + y?). Then,

L = y2(1 — :02).

If L has a minimum along a periodic orbit ¢(t; x‘) at a time t1, then L(¢(t1;x")) =
0,and:c=:l:1ory=0. If:::7é:t1,1—:r2 a.ndsoLdonot changesign,soit
cannot be a minimum. Therefore, the minimum along a periodic orbit occurs where
a: = :l:1. The minimum has y 96 0 because the points (i1,0) are not on a periodic
orbit. Therefore,

L(¢(w<')> 2 L<¢(¢1;><')) > L(1.0) =

and 2L — 1 > 0 alonglthe periodic orbit. We want to relate the divergence and the
time derivative of L, L = y2(1— 1:2). Using that yz = 2L — 0:2 = 2L — 1 + (1 — 2:2),

L=(1-1.-1)(21.-1+(1-<=2))
= (V - F)(2L— 1) + (1 - $2)?

Since 2L — 1 > 0, we can solve for V - F:

V ' F * KL “ "(1 —_aT:2)~2

By Theorem 6.14,

Iu T T T(1_x2)2

P(")'°""(h ‘V'F)"“"""“)=e""(h §FI"‘"/0 fid‘ -

The rst integral under the exponential is

§<11=(21><¢<T;x'>> - 1) —1n<2L<¢<o.=='>> - 1»,

which is zero, since ¢(T;x‘) = x‘ = ¢(0;x‘). Therefore,
P'(x. )-_exp(__/-LT (12—L_I12)2 dt <1.

This shows that any periodic orbit is attracting. Since two attracting periodic
orbits must be separated by a repelling one, there can be only one periodic orbit.

Exercises 6. 7 261

Notice that we were able to determine the sign of the integral of the divergence

without actually calculating it. The divergence can be positive and negative, but

we split it up into a term that is strictly negative and one that integrates to zero

around a periodic orbit. El

Exercises 6.7

1. Consider the 21r-periodically forced scalar differential equation

:i: = -2: + sin(t).

a. Find the solution of the equation by the variation of parameters formula.
(In one dimension, the variation of parameters formula is usually called
solving the nonhomogeneous linear equation by using an integration fac-
tor.)

b. Find the Poincaré map P from t = 0 to t = 21:.
c. Find all the xed points of the Poincaré map and determine their stability.

2. Consider the 21r-periodically forced scalar differential equation

:i: = (1 + cos(t)):r — 1:2.

a. Show that :0 = 0 lies on an unstable periodic orbit.

b. Show that there are no periodic orbits with :1: < 0. Hint: Show that a': < 0.

c. Let P be the Poincaré map from t = 0 to t = 21r. Show that < a; for

:1: 2 1. Show that there are no periodic orbits with :1: > 2 and that there

is at least one periodic orbit with 0 < 2:0 < 2.

d. Using Theorem 6.14, show that any periodic orbit with 0 < mo is orbitally

stable. Hint: Use a method similar to that of Example 6.23.

e. Conclude that there are two periodic orbits, one stable and one unstable.

3. Consider the forced equation

-f' = 1 (mod 21r),
:i: = :r(1— as) [1 + cos(-r)].

Notice that there are two periodic orbits: (r,a:) = (-r,0) for 0 § 'r 3 21r and
(1,2) = (T, 1) for 0 $ T 3 21r.

a. What is the divergence of the system of equations?
b. Find the derivative of the Poincaré map at the two periodic points ('r, as) =

(0, 0) and (T,:c) = (0,1). Hint: Use the formula for the derivative of the
Poincaré map

211'
P'(a:) = exp (‘A V - F¢(¢,(o_,,,)) dt) ,

where V -F is the divergence of the differential equations and the expo-
nential exp(u) = e“.

262 6. Periodic Orbits

4. Consider the system of differential equations

for /1 > 0. $=y1
a. Show that
i1=—w—2w3+y(#—y2—$’—~'v“)

7={<:;) :;.i=y2+I2+:1:4}

is a periodic orbit. Hint: Use a test function.
b. Show that the divergence of the system is —2y2 along the periodic orbit

7. Hint: Use that )1 — yz — :22 — 2:4 = 0 along 7.

c. Let E = :_:c > 0}, and P be- the Poincaré map from E to it-

self. Show that the periodic orbit 7 is orbitally asymptotically stable
(attracting) by showing that the derivative of the Poincaré map satis es

0 < P’(:1:0) < 1 where (160) E 7. Hint: Use Theorem 6.14.

5. Consider the system of differential equations

d:= —y+:v(1-a:2—y2),

z)= w+w=(1 -12-:/2).

2 = —z.

a. Show that 1:2 + yz = 1 is a periodic orbit.
b. What are the characteristic multipliers for the periodic orbit $2 + yz = 1?

6. Assume that xo is on a periodic orbit 7 for a flow and that E is a transversal
at xo, with Poincaré map P 'om an open set U in E back to E. Then, xo
is a xed point for P. This xed point is called L-stable for P provided that,
for every e > 0, there is a 5 > 0 such that, if x in U has ||x — x0|| < 6, then
||P5 (x) — P7 (x0)|| < e for all j Z 0. This xed point is called asymptotically
stable provided that it is L-stable and that there is a 60 > 0 such that, for any
X in U with ||x — Xoll < 60,

j1j1_§°||P’(X) — Pi (Xo)ll = 0-

a. Prove that, if xo is L-stable for the Poincaré map P, then the periodic
orbit 7 is orbitally L-stable for the ow. .

b. Prove that, if X0 is asymptotically stable for the Poincaré map P, then
the periodic orbit 7 is orbitally asymptotically stable for the ow.

6.8. Applications

6.8.1. Chemical Oscillation. In the early 1950s, Boris Belousov discovered a
chemical reaction that did not go to, an equilibrium, but continued to oscillate
as time continued. Later, Zhabotinsky con rmed this discovery and brought this
result to the attention of the wider scienti c community. Field and Noyes made a
kinetic model of the reactions in a. system of differential equations they called the
“Oregonator," named after the location where the research was carried out. For

6.8. Applications 263

further discussion of the chemical reactions and the modeling by the differential
equations see [Mur89], [Str94], or [Enn97]. In this subsection, we sketch the
reason why this system of differential equations with three variables has a periodic
orbit. This presentation is based on the treatment in [Smi95].

The Oregonator system of di erential equations can be written in the form

e:i:=y—1:y+a:(1—q:c),

(6-6) 11=—y—1=y+2fz.

2 = 6 (:1: — z),

where the parameters e, q, f, and 6 are positive. We restrict this discussion to the
case where 0 < q <1.

A positively invariant region is given by the cube

.‘?Z'={(a:,y,z):1§:z:§q", %i—q; gy g, 1§z§q'1

(The reader can check the signs of :i:, 1}, or 2 on each of the faces of the boundary.)

The xed points are the solutions of the three equations

_ _2f:z: _ 2f:r: 2
z-a:, y-1+1}, and 0-(1 a:)1+$+a:—qa:.

Multiplying the third equation by ‘(I + fl/4;, we obtain
0=q:r2+(2f+q—1):n—(2f+1).

This last equation has one positive solution,

=='=§(<1-q-2r>+[<1.q-2r>’+4q<2/+1>1"’)-

2U

Let z‘ = a:" and y‘ = The other xed points are (0,0,0) and one with

negative coordinates.

The matrix of partial derivatives is

e_1(1— y — 2qa:) c'1(1-— 0
—y —(1+:1:) 2f .
60 -6

A calculation shows that the determinant at (a:"‘, y"‘, 2") is

d°t(DF(=-,y-,1-)) = ";6(2yi + 17$I +¢I(1=U )2),

which is negative. Therefore, either all three eigenvalues have negative real parts,
or one has a negative real part and two have positive real parts. It is shown in
[Mur89] that both cases can occur for different parameter values.

For the case in winch two of the eigenvalues of (a:',y', z‘) have positive real
P9-rts, [Smi95] argues that there must be a nontrivial periodic orbit. First, the
System can be considered "competitive" by considering the signs of the off-diagonal
terms of the matrix of partial derivatives. The (1,2) and (2, 1) terms are negative
for the points in .92. The sign of the (2,3) and (3, 1) terms are positive, but by using
“Z, which multiplies the third row and third column by -1, these terms become

264 6. Periodic Orbits

negative. The other two terms, (1,3) and (3,2), are zero. Thus, the system for
(x, y, -2) is competitive by his de nition.

Smith in [Smi95] has a theorem that shows that a limit set of a competitive
system in R3 is topologically equivalent to a How on a compact invariant set of
a two-dimensional system. The system may not have a derivative, but only be
of Lipschitz type, but this is good enough to permit application of the Poincaré-
Bendixson theorem. See Theorem 3.4 in [Smi95].

The results of these arguments in the next theorem.

Theorem 6.16. Suppose that the parameter values for system (6.6) are chosen so
that the red point (:z:‘,y',z‘) has two eigenvalues with positive real parts, so the
‘stable manifold W”(zU ,yi ,zC ) is one-dimensional. If q is an initial condition in
the previously de ned region 3?, but not on W"(a:", y’, z‘), then w(q) is a periodic
orbit.

Remark 6.25. Smith mentions that much of the work on competitive systems
given in his book started with the work of Morris Hirsch in the 19805.

6.6.2. Nonlinear Electric Circuit. In this subsection, we show how the Van
der Pol equation can arise from an electric circuit in a single loop with one resistor
(R), one inductor (L), and one capacitor (C). Such a circuit is called a RLC circuit.
The part of the circuit containing one element is called a branch. The points where
the branches connect are called nodes. In this simplest example, there are three
branches and three node. See Figure 16. We let ‘lg, i;,, and ic be the current in the
resistor, inductor, and capacitor, respectively. Similarly let 1:3, vL, and 12¢ be the
voltage drop across the three branches of the circuit. If we think of water owing
through pipes, then the current is like the rate of ow of water, and the voltage
is like water pressure. Kirchhoff’s current law states that the total current owing
into a node must equal the current owing out of that node. In the circuit being
discussed, this means that |iR| = |iL| = |ic| with the correct choice of signs. We
orient the branches in the direction given in Fig'u.re 16, so

_ a: =11, = ‘ll, =10.

Kirchhoff’s voltage law states that the sum of the voltage drops around any loop is
zero. For the present example, this just means that

vg+v;,+'v¢-=0.

Next, we need to describe the properties of the elements and the laws that
determine how the variables change. A resistor is determined by a. relationship
between the current in and voltage 113. For the Van der Pol system, rather than a
linear resistor, we assume that the voltage in the resistor is given by

v;q=—iR+i§;=—:r:+:r3.

Thus, for small currents the voltage drop is negative (antiresistance) and for large
currents it is positive (resistance).

A capacitor is characterized by giving the time derivative of the voltage, d-35-,

in terms of the current ig,

d'lJ(_','

C —at = ‘'C ,

6. 8. Applications 265

R/‘S

Figure 16. RLC electric circuit

where the constant C > 0 is called the capacitance. Classically, a capacitor was
constructed by two parallel metal plates separated by some insulating material. To
make the equation look like the form of the Van der Pol equation, we let y = —v¢
and the equations becomes

dy_
-(73-13.

An inductor is characterized by giving the time derivative of the current, %,
in terms of the voltage 12¢: Faraday’s law says that

Lgd.=..=-1..-...=..-<-.+.==>.

where the constant L > 0 is called the. inductance. Classically, an inductor was
constructed by making a coil of wire. Then, the magnetic eld induced by the
change of current in the coil creates a voltage drop across the coil.

Summarizing, system of differential equations is

da:

LE =y—(—.'B'l'$3),

C —ddyt = —z.

This system is just the Lienard system in the form given in equation 6.4 with g(:i:) =
:0/C and F(:c) = —:z:/L + 2:3/L. These functions clearly satisfy the assumptions
needed on the functions g and F from that context.

By changing the voltage drop for the resistor to include a parameter in the
coe icient of iR,

1);; = —;iiR +i§; = —/.ia: +13,
this system has an Aiidronov-Hopf bifurcation at ii = 0, as the reader can check.

6.8.3. Predator—Prey System with an Andronov—Hopf Bifurcation.
I11 [Wa183] and [Bra01], a variation of the predatorhprey system is given that
exhibits an Andronov—Hopf bi.f1u'cation; The so called Rosenzweig-MacArthur model

266 6. Periodic Orbits

is given by

:b=a: (1—a:——Ly >,
1+2a:

__ 2:1: _
y_y 1+2:c ll"

(We have changed the labeling of the variables to make them conform to the no-
tation used in other examples. Also we have picked speci c values for some of the
parameters given in [Wal83].) The :1: population is the prey; by itself, it has a
positive growth rate for small populations and has a carrying capacity of 2: = 1,
with negative growth rates for :1: > 1. The predator is given by y, and it dies
out when no prey is present. The interaction between the two species involves the
product my, but the bene t to the predator decreases with increasing population of
the prey: The net growth rate for the predator approaches 1 — /J as :1: approaches
in nity. The parameter is given by ii, which we assume is positive, but less than
one, so the growth rate for the predator is positive for :1: su iciently large.

To nd the xed points with cc and y positive, the g = 0 equation yields

21:‘ =;i(1+2:i:‘),

(2—2/.i)a:‘=a, andso

”._ in-I-4/i> _

For future use, notice-that

-1 and
1+2:z:‘=glM'—=—1——u
1_$._2—2#—#_ 2-3;»

' 2(1—u) '2(1—#)’

The value of ;i/ at the xed point is

y, : (1 —:c‘)(1+2:|:") = (1—a:‘)

2 2(1—#)

_ 2-3/.1 1 _ 2—3i.i

" 20-») ' 2(1"/1) “4<1-/1)’

The linearization at this xed point is _1+i212_:¢‘ -

1 _ 2°'-* _ i(_1+22yi¢’ -)2 0

2y
(1 + 21:‘)?

The determinant is

A = _(1 +-2_@~)-=* > o.

6.8. Applications 267

The trace is

'r=1—2a:. —214‘

2—2 2 2-3

2<1—/ll) _2(1f/.~.)_2(1—/f)"(1_”)2

2—2;i—2;i—(2—5p.+3;i2)

2(1—#)
/1(1—3#)
2(1-/¢)'

The eigenvalues are purely imaginary, :ti \/K, when r = 0, or 1.1.0 = 1/3. For
/10 = 1/3|

::'=l, 3/‘=3, and 1+2a:‘=§.
4 16 2

The real pa.rt of the eigenvalue is 01,, = "'/2, so

a Z #(1 -3/1): 3#(1—#)+2(1—#)—2

“ 4(1—#) 4(1—#)

31 1
=—4” +—2 —2—(1i-—ii)

and

m; = _ _ __id _]

du #=1/3 2(1 — I-‘)2 i»=1/3

iP~OJi§OO @<O

3
——§<0.

Therefore, the real part of the eigenvalues is decreasing with ii.

To check the stability at ii = 1/3, we apply Theorem 6.7. In terms of the
notation of that theorem,

2y‘ 2:12‘ 1

I Z 1 —2 ~— ii Z) Z —j-i Z ——’

f x + 2$‘)2 ly,=l/3 0 fy 1+ 2:11‘ p=1/3 3

2;; 2(3/16) 1 1
9:=(i1-ai-2$-")=2 a=9/4—6, 9" =1— ” —1—+2—w-i,.=1/3 0=,

___J¢P%® __ §QQQ_;Eii=:E
f“' 2 (1+2£lZ")3l;i=l/3_ 2+ 27/9 _ 9 9'

f _ 8z‘(—3)(2)| _ _48(3/16) _ _E

m‘ (1+2¢-)4 ,.=1/3‘ (81/81) _ 9’

268 6. Periodic Orbits

f __ __ 2 ' _ _ 2 _ _§

1” (1 +22‘)? ,,=i/a _ (9/4) _ 9'

8 8 64

f"" _ (1 + 2a:')3 l,.=1/3 _ (21/s) _ ii’

fay = O frail»

,u=s2 d‘ _—2hQ2 =_8l9L1J6 :__4

(1+ 2$ ) i4=1/3 3

m n8g"—lg3g2 l =4_89Q16n=4_8 =_16
I ) p=1/3 93

gs - 2 - W2 ; r 8§'

9 =2eam' =_ 8 =_g

“" (1 + 2¢*)=* ,.=i/3 (27/s) 27'

9w = 0 = 99:/11 = 91w»

and other partial derivatives are zero. Therefore,

f i/2 9 1/2 f 1/2 - 9 i/2
16K1/3 = ( 5:‘ fax: 'l' 7:; f:eyy+ I: gz:xy+ T: gyi/y

+gis@fy mQ=1: ==/== +%fy r=v/W+! @fv /i/119w)

+ 5": gzygyy + gzrygzx + 5% f==9===) -

=\/§(%63)+0+\/§(--g) +0

+%(<~%)(~%‘>+@+@)
—%(O+no (S) (-§)~(~%> (—%>)

= g (-4s-64+ 112+ 192-ass)

144,/5

=--T7—<0.

Therefore, the xed point is weakly attracting at it = 1/3, and the bifurcation is
supercritical with a stable periodic orbit appearing for /,1. < 1/3 (since the derivative
of the real part of the eigenvalue is negative).

We now give an alternative way to check that the xed point is weakly attract-
ing when /1 = 1/3, using the Dulac criterion from Theorem 6.11. We can use this
criterion to show that there a.re no periodic orbits in the rst quadrant for /1. Z 1/3-
We multiply by the function

9(w.y) = 1+2 1/“_ ‘-

6. 8. Applications 269

1 The only justi cation for this function is that it eliminates
where oi =

the ms from some terms in the :i: equation, and the power on y makes the nal

outcome (at the end of the following calculation) negative. With this choice, we

obtain _,,..] +% [,,. (2e%r2_$>)]

v.(,F,=%

1

= -l!aT—][—1(1+2r)+2(l—e'z + %—-12¢-ii-2,_a]

l 2? " >l-_ 22 £l3(4117+l.)+21_1#:))$ 2(1#_#

-_—3——la—_2:L_ _4.,12 .+2$ _ ?1I_-L p)].

The quantity inside the brackets has a maximum at :1: = 1/4 and a maximum value

of 2 — Then, V - (gF) § 0 in the rst quadrant, provided that

-41 —<2(_1—ii_. p.), 1- “-<2“’ °r -31 <-”.

Thus for [1, Z 1/3, the integral of V - (gF) inside a closed curve in the positive
quadrant must be strictly negative. Therefore, by Theorem 6.11, there can be no
periodic orbits.

There is a saddle xed point at (1, 0) whose unstable manifold enters the rst
quadrant and then reaches the line :1: = :i:‘ before it gets to in nity. Then, 3) becomes
negative, so it is trapped. This forces all the points below this to have nonempty
w-limit sets, which must be (a:‘,y‘) sirlce it cannot be a periodic orbit. In fact,
with more argument, it is possible to show that any initial (a:o,yo) must have a
trajectory which intersects :i: = :i:‘ and must then become trapped. Therefore, for
ii Z 1/3 and any 2:0 > 0 and yo > 0, w(:rg,yq) = (:i:“,y‘), and the basin of attraction
includes the whole rst quadrant.

6.6.4. No Limit Cycles for Lotka—Volterra Systems. We saw in Section 5.1
that for certain predator-prey systems in two dimensions any initial condition in
the rst quadrant lies on a periodic orbit or xed point. In particular, these systems
do not have any limit cycles In this section, we will show that more general Lotka-
Volterra systems with two variables do not have limit cycles.

Theorem 6.17. Consider a LotIca— Volterra system in two variables,

2

ii = Ilii 11+ £0-ij I3" f01' i= 1,2,

i=1

where A = (ai,-) is invertible. Then there are no limit cycles.

Proof. Let it solve r + Air = 0. Then the equation can be rewritten as
2

(ii = I13; 2041' (Ij — £1") = F,'($1,I2).
.'i=1

270 6. Periodic Orbits

We calculate the divergence of F at ft:
6—F—,- =1».-0».-.-+23"-u(¢1" ria),
8165 J,

a = as/(F)(s<) = Zn,-.s..

Choose exponents c = (c1,c2)T for a rescaling factor that solve

ATC + ("'ii.'122)T = 0-
Set
__ F(x) = :i:'f"la:§"'l F(x), so

17“.-(X) = =v.~I'i’_lI§’_1 2%" (-11 - =51")-

.'i
Any limit cycle has to be in a single quadrant. Since the orbits of F are just
reparametrizations of orbits of F, if F has_no limit cycles then F has no limit
cycles. We use the divergence to show that F does not have any limit cycles.

a5pt = c,-xi‘ -1 :25’ -1 E a,-_.,- (sir, — :i:,-) + a,-, 1:, azf‘ -1 mg’ -1 ,-
J1 -

div(F) = a:‘,""1z§"l 22650;," (lij — 5-'3') + Z01,-£6;
ii

=-"=i‘_1¢'~'§’_1 - ZZQW1 ("=1 "5=1)+2"n' ("=1 -i1)+Z:'1n'ii
1 i 1'

-M= Ii firl do Ci - Q11") ($1 - =51) + 5
Iii“!
I

= :i:§"1:i:§" I-4 6.

In this calculation, we used the de nition of 6 and the fact that E, a;_.,- c, — a,-j = 0

by the choice of c. The div(F) has the same sign in any single quadrant. If 6 is

nonzero, there can be no periodic orbits by the Bendixson Criterion, Theorem 6.11.

If 6 = 0, then F preserves area, and it cannot have any orbits that are attracting

or repelling from even one side. Cl

Exercises 6.8 271

Exercises 6.8

1. A simpli ed model for oscillation in a chemical reaction given by Lengyel is

“. =“"-m4a:yi’

-_ _L
y_b$(1 1+:v2)’

witha>0andb>0.
a. Plot the nullclines and nd the equilibrium (:c“,y‘) with a:" > 0 and
y‘ > 0.
b. Show that the determinant A and trace 1 at the equilibrium (a:‘, y‘) are

A - MW5b:|:‘ > °*

T _ 3(:c‘)2 — 5 — bx‘
1 + (a:')2 '

Find an inequality on b which ensures that the xed point is repelling. -
c. Show that a box :21 3 1: 3 :2 and yl 3 y 3 y; is positively invariant

provided that 0 < 1:1 < $2, 0 < yl < 12 < yg, yg = 1 +:r§, 4:c;y1 =

(<1 - ¢,)(1 + mg), and y, < .
4Z1
d. Show that, if the parameters satisfy the conditions given in part (b), which
ensure that the xed point is repelling, then there is a limit cycle.

2. (Fitzhugh-Nagurno model for ne1u'ons) The variable v is related to the voltage
and represents the extent of excitation of the cell. The variable is scaled so v = 0
is the resting value, v = a is the value at which the neuron res, and v = 1 is
the value above which the ampli cation turns to damping. It is assumed that
0 < a < 1. The variable w denotes the strength of the blocking mechanism.
Finally, J is the extent of external stimulation. The Fitzhugh-Nagumo system
of differential equations are

1':=—v('u—a.)(v—1)—w+J,

’Li)=€('U—b1.U),

where 0 < a. < 1,0 < b < 1, and 0 < c << 1. For simplicity, we take e =1. See
[Bra01] for more details and references.

a. Take J = 0. Show that (0,0) is the only xed point. (Hint: 4/5 > 4 >
(1 — a)2.) Show that (0,0) is asymptotically stable.

b. Let (vJ,w_;) the the xed point for a value of J. Using the fact that J > 0
lifts the graph of w = —'u('u - a.)(v — 1) + J, explain why the value of v_;
increases as J increases.

c. Take a = 1/2 and b = 3/3. Find the value of 11' = 'u_;- for which the xed
point (v_;-,w_;-) is a linear center. Hint: Find the value of v and not of J.

d. Keep 0. = 1/2 and b = 3/3. Using Theorem 6.6, show that there is a
supercritical Andronov—Hopf bifurcation of an attracting periodic orbit.

272 6. Periodic Orbits

Remark: The interpretation is that when the stimulation exceeds this
threshold, then the neuron becomes excited.

6.9. Theory and Proofs

Poincaré—Bendixson theorem

Restatement of Theorem 6.1. Consider a system of di ferential equations 1': =
F(x) on IR2.

a. Assume that F is de ned on all of R2. Assume that a forward orbit {¢(t; q) :
t 2 0} is bounded. Then, w(q) either: (i) contains a red point or (ii) is a periodic
orbit.

b. Assume that .12?’ is a closed (includes its'boundary) and bounded subset of
R2 that is positively invariant for the differential equation. We assume that F(x)
is de ned at all points of d and has no red point in .121. Then, given any X9 in
.21, the orbit ¢(t; X9) is either:(i) periodic or (ii) tends toward a periodic orbit as t
goes to oo, and w(x0) equals this periodic orbit.

Proof. The proof is based on the one in [Har82].
As mentioned in the main section, the proof de nitely uses the continuity of

the ow with respect to initial conditions.
Part (b) follows from part (a) because the w-limit set of a point X9 in .2! must

be contained in .2! (since it is positively invariant) and cannot contain a xed point
(since there a.re none in .01), and so must be a periodic orbit.

Turning to part (a), the limit set w(x0) must be nonempty because the orbit
is bounded; let q be a point in w(x0). The limit set w(xq) is bounded because the
forward orbit of xo is bounded. We need to show that q is a periodic point, so

rst we look at where its orbit goes. The limit set w(q) is contained in w(x0) by
Theorem 4.1 and is nonempty because the forward orbit of q is contained in w(x0),
which is bounded. Let z be a point in w(q). Since w(x0) does not contain any xed
points, z is not a xed point.

We can take a line segment S through z such that the orbits through points
of S are crossing S. They all must cross in the same direction, since none of them
are tangent. The forward orbits of both X9 and q repeatedly come near z, so must
repeatedly intersect S. There is a sequence of times t,, going to in nity such that
x,., = ¢(t,,,x0) is in S. As discussed in the sketch of the proof, taking the piece of
the trajectory

{¢(tix-O) 3 tn S t S tn+1}|
together with the part S’ of S between x,, = ¢(t,,;x0) and x,,+1 = ¢(t,,+1;x0), we
get a closed curve T‘ which separates the plane into two pieces,

R2 \ P = gin U -gout.

See Figure 17. The trajectory going out from ¢(t,,+1;x0) enters either the region
5?,“ on the inside of F or the region .930“, on the outside of I‘. We assume that
it enters 3?,“ to be speci c in the discussion, but the other case is similar. All
the trajectories starting in S’ also enter the same region $5,, as that starting at
¢(t,,+1;xo). Therefore, ¢(t;x0) can never reenter .930“, for t > t,,+,. This means

6.9. Theory and Proofs 273

XS I I‘
X0 n

gout xn+1

Flgure 17. Curve I.‘ and regions 5?,“ and 5?,“

that the intersections of the trajectory with S occur in a monotone manner. They
must converge to z from one side, and the orbit of xo can accumulate only on one
point in S; therefore, w(x°) O S = {z} is one point.

By the same argument applied to q, there are increasing times s,, going to

in nity such that ¢(s,,;q) accumulates on z. Since the whole orbit of q is in
w(x0), these points ¢(s,,; q) must be in w(x0) F1 S = {z Therefore, all the points

¢(s..;q) = Z 81¢ the same. ¢(8..; q) = ¢($n+1;q). and q is periodic-

We have left to show that (u'(X[)) = §(q). Assume that Ld(X[)) \ 0(q) 94 (ll. By
Theorem 4.1, the set w(x0) is connected. Therefore, there would have to exist points

yj G w(xq) \ 0(q) that accumulate on a point y‘ in 0(q). Taking a. transversal S
through y‘, we can adjust the points yj so they lie on S. But, we showed above that
w(x0) Pl S has to be a single point. This contradiction shows that w(x0) \ 0(q) = fl

and w(x0) = 0(q), i.e., w(x°) is a single periodic orbit. El

Self—excited oscillator

Restatement of Theorem 6.5. With the assumptions (1)—(3) on g(a:) and F(a:)
given in Section 6.3, the Lienard system of equations (6.4) has a unique attracting
periodic orbit.

Proof. A trajectory starting at Y = (0,y), with y > 0, has rt > 0 and y < 0, so

it goes to the right and hits the curve y = Then, it enters the region where

ti < 0 and g < 0. It is possible to show that the trajectory must next hit the

negative y-axis at a point Y’. Subsequently, the trajectory enters the region with

11: < 0 and y < F (:11), where 13: < 0 and y > 0, so it intersects the part of the curve

1/ = F (:i:), with both :1: and y negative. Next, the trajectory returns to the positive y
axis at a point Y”. This rst return map P from the positive y-axis to itself, which

1&8-kes the point Y to the point Y", is called the rst return map or Poincaré map.

The orbit is periodic if and only if Y" = Y. But, (a:(t),y(t)) is a solution if and

274 6. Periodic Orbits

only if (—a:(t), —y(t)) is a solution. Therefore, if Y’ = —Y then Y" = -Y’ = Y
and the trajectory is periodic. On the other hand, we want to show that it is not
periodic if Y’ 94 —Y. First, assume that |[Y'|| < ||Y||. The trajectory starting at
—Y goes to the point —Y' on the positive axis. The trajectory starting at Y’ is
inside that trajectory so ||Y”l| < |]Y’|| < ||Yl| and it is not periodic. Similarly,
if ||Y’|| > ||Y||, then ||Y”|| > ||Y’|l > ||Y|| and it is not periodic. Therefore, the
trajectory is periodic if and only if Y’ = —Y.

To measure the distance from the origin, we use the potential function

I

G(:c) =L g(s) ds.

Since g is an odd function, with g(:i:) >,0 for :1: > 0, G(a:) > 0 for all cc 95 0. Let

' La. ll) = Ge) + ”;2.

~

which is a measurement of the distance form the origin. The orbit is periodic if
and only if L(Y) = L(Y’).

We let

Leo = L(Q) - L(P)

be the change of L from P to Q. Also, let

K(Y) = Lyyl = L(Y') — L(Y).

Then, the orbit is periodic if and only if K (Y) = 0.

We can measure K in terms of a.n integral along the trajectory,
" t(Y) _
K(Y) = / Ldt,
0

where t(Y) is the time to go from Y to Y’. But,

i = s(r)i + :1/iv = a(w) (y — F(I)) — 2/9(==) = —y(==) F(r'=) = F(=v)i/-

Thus,

K'(Y)= Lt(Y) F(:c)ydt= /YY’ F(a:)dy=—[Y- IYF(:1:)dy.

Now, let Y0 be the point such that the trajectory becomes tangent to :1: = a

and returns to the y-axis. Then, if 0 < Y < Y0, K(Y) = Lyyl > 0, so it cannot

be periodic. (We saw this earlier by considering the 1" equation for the Van der Pol

equation.) For Y above Y0, let B be the rst point where the trajectory hits :1: = a

and let B’ be the second point. See Figure 18. We give three lemmas that nish

the proof of the theorem. D

Lemma 6.18. For Y aboue Yo, Ly-B > 0 and Lglyl > 0, and Ln; + LB/Y1 is
monotonically decreasing as Y increases.

Proof. The quantity

Ly]; = LB —g(:r:) F(a:) dt = /QB (— F(:c)) dz > 0,

6.9. Theory and Proofs 275

Y1 E’
w ___ H1
Y
Y0

G

B’ ___ Hi

Y6

Y!

Yi -_’-Sq.--

Figure 18. Map for Lienard equation

since both terms in the integrand are positive. The value of y on the trajectory

starting at Y1 above Y is always larger on the trajectory from Y1 to B1, so

[y — F(a:)]"‘ and the integrand is sma.ller, and Ly,3, < Ly3. Similarly, L3:ly; <

Lgvyl. Adding, we get the result. El

Lemma 6.19. For Y above Y0, L331 is monotonically decreasing,

I/BB’ >, 113113;.
if Y1 is above Y, which is above Y0.

Proof.

BI

Lana; = -LB;B1 = -[BI F($)dl/-

I

Let H1 and HQ be the points on the trajectory starting at Y1 with the same y
values as B and B’ respectively. See Figure 18. Then F(a:) > 0 on this path, so if
we shorten the path to the one from H’, to H1, the quantity gets less negative, and

H1 B
L3,3/1 < —/ dy < —/l F(:r:)dy = L331,

I-lg B’

because —F(:c) is less negative on the integral from B’ to B than from H’, to
H1. D

Therefore, the graph of K (0,y) is positive for 0 < y 3 yo, and then is mono-
tonically decreasing. Thus, we have shown that K has at most one zero and there
is at most one limit cycle. The following lemma shows that K(0,y) goes to —oo
B8 y goes to oo, so there is exactly one zero and therefore, exactly one limit cycle.

Lemma 6.20. K(0,y) goes to —oo as y goes to oo.

276 6. Periodic Orbits

Proof. Fix a value of :1: = :i:‘ > a. Let P and P’ be the two points where the
trajectory starting at Y intersects the line :1; = :i:‘. For :1: > :12‘, F(:c) > F(:r') > 0.
(This uses the fact that F is monotonically increasing for a: > a.) Then,

its = - /B Fa) dy s - /P Fwy < - /P For) di = -Fa‘) ||PP'u_

As Y goes to in nity, l|PP’|| must go to in nity, so L331 < -F(a:') [|PP’|| goes to
—oo. Cl

This completes the proof of Theorem 6.5.

’Andronov~Hopf bifurcation

This section presents a partial proof of Theorem 6.6. The proof given is a com-
bination of those found in [Ca.r81] and [Cho82], and closely follows that given in
[Rob99]. This proof is more involved than most -given in this book. However, the
outline is straightforward if somewhat tedious. Using polar coordinates, we calcu-
late the manner in which the radius changes after making one revolution around
the xed point. Using the radius of the trajectory at 9 = 0 as a variable, we show
that we can solve for the parameter value u (or cu in the following proof) for which
the trajectory closes up to give a periodic orbit. This fact is shown using the im-
plicit function theorem, as explained in the proof. We do not actually show which
parameter values have the periodic orbit nor do we check their stability type: We
leave these aspects of the proof to the references given. Thus, the result proved
is actually the next (theorem, where we assume the bifurcation parameter value is
lJ0=0-

Theorem 6.21. Consider a one-parameter family of_ systems of di erential equa-
tions

(6.8) it = F,,(x),

with x in 1R2. Assume that the family has a family of red points x,,. Assume that
the eigenvalues at the xed point are a(u):ti (/.i), with o_r(;.i11) = 0, B11 = (/.40) gé 0,
and

dc:
I1

so the eigenvalues are crossing the imaginary arts at it = 0. Then, there exists
an co > O such that for 0 3 e 3 so, there are (i) differentiable functions 1u(e) and
T(c), with T(0) = 2"/50, u(0) = no, and /,1/(0) = 0, and (ii) a T(e)-periodic solution
x‘(t,e) of (6.8) for the parameter value ,u = u(e), with initial conditions in polar
coordinates given by r‘(0,e) = c and 0(0,e) = 0.

Proof. By using the change of variables y = x — x,,, we can assume that the xed
points are all at the origin. Because,

£010) 7f 0.

a(h) is a monotone function of 1.1., and given an a, we can nd a unique /J. that
corresponds to it. Therefore, we can consider that o as the independent parameter
of the system of differentia.l equations and can solve for /J. in terms of a. The

6.9. Theory and Proofs 277

eigenvalues are now functions of a, a :i: 11 (a). Note that cz is the real part of each
of the eigenvalues.

By taking a. linear change of basis, using the eigenvectors for the complex
eigenvalues, we get

D(F¢,,)0 __ (m1a1 ) — a(@t =)) =_ A(a).

The di erential equations become

(6.9) >z= A(a)x+ (BBé22(($I11,’$$g2,C’a!))) + (B‘l§$"$2’a;) +O(r").

B§ a:1,a:2,a

where B;?(a:1,:::2,a) is a homogeneous polynomial of degree j in 2:1 and 1:2, and
O(1-4) contains the higher order terms. More speci cally, we write O(r7) to mean
that there is a constant C > 0 such that

|[0(w')|] 5 cw".

Since we want to determine the effect of going once around the xed points, we
use polar coordinates. The following lemma gives the equations in polar coordinates
in terms of those in rectangular coordinates.

Lemma 6.22. The di erential equations (6.9) in polar coordinates are given by

(6.10) 1‘ = ar+r2 C3(0,oz) +r3 C4(0,a) +O(r4),

9= 5(0) + 1~o3(o, <1) + 1-2 D4(0, <1) + ow),

where C,-(l9,a) and DJ-(0,a) are homogeneous polynomials of degree j in terms of
sin(0) and cos(9). In terms of the Bf,

C,-+1(9, (1) = cos(0)B}(cos(0),sin(0), a) + sin(9)BJ3(cos(\9), sin(0), cx),
D,-+1(0, 0) = — sin(0)B; (cos(0), sin(0), a) + cos(0)B?(cos(9), sin(0), a).

Moreover,

211'
[0 C3(9, Ct) d9 = 0.

Proof. The proof is a straightforward calculation of the change of variables into
Polar coordinates. Taking the time derivatives of the equations $1 = r cos(0) and

278 6. Periodic Orbits

$2 = r sin(0), we get

(:3) = (:*:£3; ;";i;?§€’) <2)»

(5) = § (’.";‘-.’f.‘<?> 11'???) (ii)

_ < cos(0).'i:1 + sin(0):i:2 )
_ —r"1 si11(9):i:1 + r‘1 cos(9):b2

_ (Bar > + < cos(0) B1} + sin(0) B3 )

— (a) —r‘1 sin(0) BQ + F1 cos(0)B§

+ , cos(9) B5‘ + sin(6) B§ + O(1-4)

—r‘1 sin(0) Bé + 1"‘ cos(0)B§ O(1-3) ’

where the B;-‘ are functions of rcos(0), rsin(0), and cf, B§(rcos(0),rsin(0),a).

Since the Bf are homogeneous of degrees j, we can factor out r from the Bf, and
get the results stated in the lemma.

The term C3 (0, 0:) is a homogeneous cubic polynomial in sin(0) and cos(0) and,

therefore, has an integral equal to zero. U

Since 9 96 0, instead of considering 1‘, we use g5 and nd out how r changes
when 0 increase by 21r. Using equation (6.10), we get

dl '_ L ~_ ar + r2 C3(0, a) + 1'3 C4(0, a) + O(1-4)
d0 %-'1 5(0) + rD3(0, oz) + r2 D4(0, cr) + O(r3)

= - 1' + 1'2 B C3(9,a) _ ~;'7 D3(0, 01)]
B9

+ r3 C4(9, Cr) -— Eli C3(0, (I) D;r(9, C1)

- 1% D4(0, C!) + 2% D3(0, a)2] + 0(1-4).

The idea is to nd a periodic orbit whose radius is e for some 0:. So, we switch
parameters from a to the radius e of the periodic orbit. Let r(0, e, 0:) be the solution
for the parameter value Q, with r(0,e, cu) = c. We want to solve r(21r,c,a) = c,
which gives a periodic orbit. The solution with e = 0 has r(0,0, a) = 0, since the
origin is a xed point. Therefore, e is a factor of r(0, e, a), and we can de ne

g(6,a)r=2l'%,e a__—_c

If g(e, oz) = 0 for e > 0, then this gives a periodic orbit as desired.
To show that we can solve for a as a function of c, 0(6), such that g(c, 01(6)) = 0»

we use the impl.icit function theorem. We show that g(0,0) = 0 and 3-§(O, 0) 94 0-