6.9. Theory and Proofs 279
The method of implicit differentiation, then says that
89 8g do _
6G (01 + 5 (01 _ 0)
g@=_%om
e 350. 0)
The implicit function theorem says, that when 3-§(0, 0) aé 0 (and implicit differen-
tiation works), it is possible to solve for a as a differentiable function of c, which
makes this equation remain equal to zero.
We show the preceding requirements on g and its partial derivatives by scaling
the variable r by e, r = ep. We show that r(21r, e,0) —e = O(c3), so g(e, 0) = O(c2).
Let p(0, e, a) = r(0, e, 0:)/6, so it is the solution for p, with p(0, e, a) = r(0, e, a)/e =
e/e = 1. The differential equation for p is given by
% = g p + ep2 C3(6,o:) — £% D_-,(9, a)]
+¥fEQ@®—%%@®%@M
— £% D4(0,a) + 2?; D3(0,a)2] + O(e3p").
Taking the linear term to the left-hand side, and using the integrating factor e‘°‘°/5,
the derivative of e“'9/5 p is
Me lzcrnmld _a
_°, 1 oz
+ :2 e“°‘a/H p3 C4 — £503 D3 — [% D4 + ‘% D§] + O(c3p4).
Integrating form 0 to 21r, we obtain
e'2"°’/5 p(2'rr, 6,01) — 1
211’ 6 1Q
= 5/ e*°‘ /B p(0,e, (1)2 C3(0,a) — F D3(0,a)] d0
0
21r
+e2-A e'°'9/5 p(0,e,oz)3 C4(0,a)
-% c,(a, 0') o=,(0, (1) - % D4(0,a) + % o=.,(o, <92] d0 + 0(£3p4)
= e h(e, a).
This last line de nes h(e,a). By the de nitions of g and h, we have
g(e, 0:) = p(21r, e,a) -1
= le2"°/'3 — 1] + ee2"°‘/B h(c,a).
Now we can calculate the quantities for g mentioned above. The value of g at
@®=mmm
¢am=e“-1=a
280 6. Periodic Orbits
is es <22)The partial derivative with respect to a is
+5 £3; (ezna/H h(£, 0,)
e=0,a=0
- W21r " °-
This veri es the assumptions previously stated, for the implicit function the-
orem, so we can solve for a as a mction 01(5) of e such that g(c,a(e) E 0.
‘These give points that are periodic with 1 = p(0,e,a(e)) = p(2-rr,e,a(e)), so
e = r(0, e,a(e)) = r(2-rr,e, a('e)). '_
We previously noted that implicit differentiation gives
E __%§(0,0) i
dew)’ §§(0.0>'
Therefore, to determine %%(0), we need to calculate %%(0, 0)..
By the previous expression for g(e, a), we have
9(¢.°) = €h(¢,0)
=¢/02? p(9,e,O)2 c,,(0,o) as
+_ E2 21' p(v.¢.o>“ ciao) - Z}; Ca(9»9) v=».<@.o>] <w
0
+ O(e3p4).
Substituting p(0,e, 0) = 1, we obtain
ag 1 ‘/-21r
—6€(0,0)=—5 0 Ca(9,0) d0=0.
The integral is zero because C3(0, 0) is a homogeneous polynomial of degree three
in sin(0) and cos(6). It follows that
0,0 El
_$8’§ =_—E(’l>§’$i§.’_/X/—\ <>>)=0.
This completes the proof of the theorem.
Although we leave the details to the reference [Rob99], we indicate the outline
of the manner in which we can determine the parameters for which the periodic
orbit appears and the stability type of the periodic orbits.
By applying implicit differentiation again to the equation
0 = am) + 2-§((@.a<@>>a'(@>.
6.9. Theory and Proofs 281
we get + 282($]0,0) C!I + w82(g0,0)(1I 2 + %69(0,0)(1'n
0 -— azg
Using the fact that o/(0) = 0, we can solve for a”(0):
M0) = _ %62§(o,o)
550.0)
_- 50 m329l“ “)-
As stated in Section 6.4, we de ne
1 2“ 1
Kn — 5;‘/0. C4(0,0t) — WC3(0, OI) D3(6, O!)
(1
It is shown in [Rob99] that
;3,27Z(0. 0) = F410r K0.
so
C!”(0) = -2K3.
This shows that the periodic orbit has a(e) — K0 :2 + O(e3) > 0, when K0 < 0
and the xed point is weakly attracting at oz = 0, and (ii) has a(c) < 0, when
K0 > 0 and the xed point is weakly repelling at a = 0.
As a check of the stability of the periodic orbit, the Poinca.ré map is
P(e,a) = r(21r, e,a).
A calculation in [Rob99] shows that '
g(t) = 1+ K06’ +0(é").
Therefore, the periodic orbit is attracting when K0 < 0 and is repelling when
K0 > 0. See the reference for details.
Change of volume by the ow
We rst prove the change of volume formula and then use it to prove the
Liouville formula.
Restatement of Theorem 6.9. Let x = F(x) be a system of di erential equa-
tions in R", with ow ¢>(t;x). Let 9 be a region in IR", with nite n-volume
and (ii) smooth boundary bd 9. Let 9(t) be the region formed by owing along
for time t, Q(t) = {¢(t;x0) : xo G 9}.
a. Let V(t) be the n-volume of Q(t). Then
dd, v(t) - /WV-Fa) dv.
where V - F(x) is the divergence of F at the point x and dV = dzl - --darn 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 = d:c1d:1:2.
282 6. Periodic Orbits
b. If the divergence of F is a constant {independent of x ), then
v(t) = v(o)e‘”*.
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 ea.ponentially; if V - F > 0, then
the volume grows exponentially.
Proof. We present the proof using the notation for dimension three, but the proof
in dimension two or higher dimensions is the same.
The time derivative of V(t) is equal to the flux of the vector eld F across the
boundary 69(t) of Q(t),
1‘/(t)=/~ (F-n)dA,
89(t) -
where n is the outward normal. Notice that it is only the component F in the
direction of n that contributes to the change of volume.‘ If this quantity is posi-
tive, then the volume is increasing, and if it is negative, the volume is decreasing.
Applying the divergence theorem, we see that the right-hand side is the integral of
the divergence, so
v(t) = / v - F(,,, av.
90)
This proves part (a).
If the divergence is a constant, then the integral gives the volume multiplied
by this constant, _
-. V(t) = (V ~ F)V(t).
This scalar equation has a solution
v(t) = v(0) ¢<“">*.
The remaining statements in part (b) follow directly from this formula for V(t),
because the divergence is constant. El
We indicated earlier how Liouville’:-1 formula can be proved by differentiating
the determinant for a time-dependent linear di 'erential'equation. It can also be
proved using the preceding formula derived from the divergence theorem, as we
indicate next.
Restatement of Theorem 6.10. Consider a system of di erential equations x =
F(x) in IR". Assume that both F(x) and are continuous forx in some open
set U in R". Then the determinant of the matrix of partial derivatives of the flow
can be written as the exponential of the integral of the divergence of the vector eld,
V - F(x), along the trajectory,
z
det (D=<¢<¢.xt>) = ‘EXP bl0 (V " F)<¢(~.x»>> <15) -
Proof. We use the general change of variables formula in multidimensions given
by / h(y) at/= / h(G(x)) |det(DG(,,))|dV.
G-(9) 9
6.9. Theory and Proofs 283
Because ¢>(—¢; ¢(¢;xo)) = X0,
DX¢(¢;d>(t:Xo)) D*¢(=;Xo) = I’
and D,,_¢(,,,°) is invertible for all t. At t = 0, it is the identity, so it has a positive
determinant. Therefore, G(x0) = ¢(t;x0) has a positive determinant of the matrix
of partial derivatives for all t,
det (D,,¢(,,,,,,)) > 0.
This fact allows us to drop the absolute value signs for our application of the change
of variables formula. By the previous theorem, we have
= / V ‘ F(x) dv =/ V ' F(¢'(l;Xo))det (D!-¢(1:xo)) dv'
xE9(t) xoE9(0)
But, we also have
-d d d
V(t) - 51;“) dV - Z/9(0)det (D,,¢(,,,,,,)) dV - gm) Edet (D,¢¢>(,,,,_o)) dV.
Setting these equal to each other, we get
~/f V - F(¢(;;xo)) det (Dxt/J(¢;,(°)) dv = / £ det (D,¢¢(¢;xo)) dV
99
for any region 9. Applying this formula to the solid balls Q, = B(x0, r) of radius
r about a point xo, and dividing by the volume of .@,., we get
_1 d
31-13% ‘/‘jar E det (D,;¢(¢;x°))
_1
— 153% /grv 'F(¢(¢=xo))d°l (D=<¢(¢;==t)) ‘W-
Because each side of the equality is the average of a continuous function over a
shrinking region, in the limit,
det (D*¢(¢:Xo)) = V ' F(¢(¢:Xo)) det (D"¢(liXo))’
which is the differential equation given in the theorem.
If we let J(t) = det (D,,¢(,,,,,,)), and T(t) = V - F(4,(,,,,,,)), we get the scalar
differential equation
iw) = Te) J<¢>.
which has the solution t
J(t)) = exp (L T(s)ds) ,
since J(0) = det(I) = 1. Substituting in the quantities for J (t) and T(s), we get
the integral form given in the theorem. U
i
Chapter 7
Chaotic Attractors
The Poincaré—Bendixson theorem implies that a system of differential equations in
the plane cannot have behavior that should be called chaotic: The most complicated
behavior for a limit set of these systems involves a limit cycle, or stable and unstable
manifolds connecting xed points. Therefore, the lowest dimension in which chaotic
behavior can occur is three, or two dimensions for periodically forced equations.
As mentioned in the Historical Prologue, Lorenz discovered a system of dif-
ferential equations with three variables that had very complicated dynamics on a
limit set. At about this same time, Ueda discovered a periodically forced nonlinear
oscillator which also had complicated dynamics on its limit set. This type of limit
sets came to be called a chaotic attractor. See [Ued73] or [Ued92] for historical
background.
In Sections 7.1 and 7.2, we prent'the main concepts involved in chaotic be-
havior, including attractors, sensitive dependence on initial conditions, and chaotic
attractors. We then discuss a number of systems which have chaotic attractors, or
at least numerical studies indicate that they do: We consider the Lorenz system
in Section 7.3, the Rossler system in Section 7.4, and periodically forced system
of equations related to those considered by Ueda in Section 7.5. Section 7.6 intro-
duces a numerical measurement related to sensitive dependence on initial condi-
tions, called the Lyapimov exponents. Then, Section 7.7 discusses a more practical
manner of testing to see whether a system has a chaotic attractor.
In general, this chapter is more qualitative and geometric and less quantitative
than the earlier material.
7.1. Attractors
Previous chapters have discussed attracting xed points and attracting periodic
orbits. In this section, we extend these concepts by considering more complicated
attracting sets. For an attracting set or attractor A, we want at least that all points
<1 near to A have w(q) C A. However, just like the de nition of an asymptotically
_
285
286 7. Chaotic Attractors
stable xed point, we require a stronger condition that a set of nearby points
collapses down onto the attracting set as the time for the flow goes to in nity.
If a system of differential equations has a strict Lyapunov function L for a
xed point p with L0 = L(p), the set U equal to a component of L_l([Lg, L0 +6])
around p is a positively invariant region, for small 6 > 0. In fact for t > 0,
¢*(U)c{xeU=L(x) <L0+6} and
¢>‘(U)={P}-
L>0
In this chapter, we consider situations where a more complicated set A is
attracting. Because we want A to have a property similar to L-stability, we give
the following de nition in terms of the intersection of the forward orbit of a whole
neighborhood rather than just using w-limit setshof points. We require that A has
a neighborhood U such that ¢‘(U) squeezes down onto A.as t goes to in nity. For
such a set, every point q in U will have its w-limit set as a subset of A.
We specify these conditions using the standard terminology from topology. Also
see Appendix A.2 for more about these concepts.
De nition 7.1. For a point p in IR", the open ball or radius 1‘ about p is the\set
B(P.r) = {qE1R" = llq—pll <r}-
The boundary of a set U in R", bd(U), is the set of all the points that are
arbitrarily close to both points in U and the complement of U,
bd(U) = {p elR"__;B(p,r)nU;é0 and B(p,r) (lR" \U) aélllfor allr > 0}.
A set U in IR" is open provided that no boundary point is in U, U bd(U) = (ll.
This is equivalent to saying that for every point p in U, there is an r > 0 such that
B(p,r) C U.
A set U in 1R" is closed provided that it contains its boundary, bd(U) C U.
This is equivalent to saying that the complement of U, R" \ U, is open.
The interior of a set U is the set minus its boundary, int(U) = U \ bd(U).
The interior is the largest open set contained inside U.
The closure of a set U is the set union with its boundary, c1(U) = U U bd(U).
The closure is the smallest closed set containing U.
A set U is bounded provided that there is some (big) radius r > 0 such that
U C B(0, r).
A subset of 1R" is compact provided that it is closed and bounded.
De nition 7.2, A trapping region for a flow ¢(t;x) of a system of differential
equations is a closed and bounded set U for which ¢(t; U) C int(U) for all t > U-
This condition means that the set is positively invariant and the boundary of U is
moved a positive distance into the set for a t > 0.
We usually de ned a trapping region using a test function L : R" —> IR,
U = L_1((—<><>.C'l) = {X = L(x) S C}-
7.1. Attractors 287
Since L is continuous, a standard theorem in analysis states that U is closed and
L'1((—oo,C)) is open. If the gradient of L is nonzero on the set L‘1(C'), then
the boimdary is bd(U) = L_1(C) and int(U) = L"((—oo,C)). In this case, the
condition for the trapping region can be stated as
¢(t; U) C L"'((—oo,C)) = {x : L(x) < C} for t > 0.
De nition 7.3. A set A is called an attracting set for the trapping region U
provided that
A = Q ¢‘(U).
tZ0
Since A is invariant and any invariant set contained in U must be a subset of A,
it is the largest invariant set contained in U.
An attractor is an attracting set A which has no proper subset A’ that is also
an attracting set (i.e., if 0 96 A’ C A is an attracting set, then A’ = A). In
particular, an attracting set A for which there is a point X9 (in the ambient phase
space) such that w(x0) = A is an attractor.
An invariant set S is called topologically transitive, or just transitive, provided
that there is a point xo in S such that the orbit of xo comes arbitrarily close to
every point in S (i.e., the orbit is dense in S). We usually verify this condition by
showing that w(xq) = S.
Thus, a transitive attractor is an attracting set A for which there is a point xo
in A with a dense orbit. (This condition automatically shows there is no proper
subset that is an attracting set.)
I
Notice that an asymptotically stable xed point or an orbitally asymptotically
stable periodic orbit is a transitive attractor by the preceding de nition.
Theorem 7.1. The following properties hold for an attracting set A for a trapping
region U.
a. The set A is closed and bounded, compact.
b. The set A is invariant for both positive and negative time.
c. If X9 is in U, then w(x0) C A.
d. If X0 is a hyperbolic xed point in A, then W"(x0) is contained in A.
e. If '7 is a hyperbolic periodic orbit in A, then W“(7) is contained in A.
The proof is given in Section 7.9.
Example 7.4. Consider the system of equations given in Example 5.11:
r==y.
g=a:—2:c3+y(:c2—:c“—y2).
288 7. Chaotic Attractors
The time derivative of the test function _
_$2 + I4 yz IS
L(:c, y) - if + -2-
Z O when L < O,
L = —2y2L = 0 when L = 0,
_§ O when L > O.
On the set I‘ = L‘1(O) = {(z,y) : L(z, y) = O}, L = 0, so I‘ is invariant. This
set I‘ is a “ gure eight" and equals the origin together with its stable and unstable
manifolds. See Figure 18. At the other two xed points, L(:t2'1/2, 0) = '1/3. We
use the values *1/16, which satisfy *1/3 < ‘1/16 < 0 and 0 < 1/16, and consider the
closed set .1 _ 1 \
U1 — {(11.9) - —Ié 3 L(I.1/)5 I6}-
This set U1 is a thickened set around the I‘. For t >~ 0, ¢‘(U|) is a thinner
neighborhood about I‘. In fact, for any r > 0, there is a time t, > 0, such that
¢"(U,) C L“1(—'r,r). Since r > 0 is arbitrary, '
F = n ¢t(Ul)1
l>0
and I‘ is an attracting set. We have argued previously that there is a point (:00, yo)
outside l‘ with w(xo,yo) = I‘, so I" is an attractor. Since every point inside I‘ has
its w-limit equal to {O}, it is not a transitive attractor.
There are other attracting sets for this system of equations. For example we
could ll the holes of U1 to get a different trapping region:
' U2={(Iv.y)-L(I.1l)S E1 }-
For any t > 0, d>‘(U;) contains the set
A2={(1=.y)=L(w.y)$°}-
This set is I‘ together with the regions inside I‘. Again, as t goes to in nity, ¢‘(U;)
collapses down onto Ag, so A2 is an attracting set. Sincel‘ is a proper subset of
A2 that is also a.n attracting set, A2 is not an attractor.
There are two other attracting sets, which ll in only one hole of I‘,
A3=I‘U{(:z,y)€A2:a:Z0} and
A4=I‘U{(a:,y) €Ag:a:50}.
Again, neither of these attracting sets is an attractor.
Since the w-limit set of any point is a subset of Ag, any attracting set is a subset
of A2 and contains I‘. If any point from one of the holes of 1" is in the attracting
set then the whole side inside I‘ must be included. Therefore, the only attracting
sets are I‘, Ag, A3, and A4.
Remark 7.5. The idea of the de nition of an attracting set is that all points in a
neighborhood of A must have their w-limit set inside A. C. Conley showed that an
attracting set with no proper sub-attracting set has a type of recurrence. Every
point in such a set A is chain rec1n'rent and there is an e-chain between any two
Exercises 7.1 289
points in A. What this means is, that given two points p and q in A and any small
size of jumps e > O, there are a sequence of points, X0 = p, x1, . . ., x,, = q, and a
sequence of times, t1 Z 1, t2 Z 1, ..., t,, Z 1, such that ||¢>(t,;x.;_1)—x,-|| < e for
1 5 1' 5 n. Thus, there is an orbit with errors from p to q. There is also such an
s-chain from p back to p again. See Section 13.4 or [Rob99] for more details.
Remark 7.6. Some authors completely drop the assumption on the transitivity or
w-limit from the de nition of an attractor. Although thereis no single way to give
the requirement, we feel that the set should be minimal in some sense even though
it can only be veri ed by computer simulation is some examples.
Remark 7.7. J . Milnor has a de nition of an attractor that does not require that
it attract all points in a neighborhood of the attractor. We introduce some notation
to give his de nition. The basin of attraction of a closed invariant set A is the set
of points with w-limit set in A,
@(A;¢)={Ko=w(><o;¢)CA}-
The measure of a set .@(A;¢) is the generalized “area” for subsets of R2 and
generalized “volume” for sets in IR3 or higher dimensions.
A closed invariant set A for a ow ¢ is called a Milnor attractor provided that
(i) the basin of attraction of A has positive measure, and (ii) there is no smaller
closed invariant set A’ C A such that 93(A; ¢) \.@(A'; ¢>) has measure zero.
The idea is that it is possible to select a point in a set of positive measin-e. If
A is an attractor in Milnor’s sense, then it is possible to select a point xo such that
w(x9) C A. Since the set of initial conditions with the w-limit in a smaller set has
measure zero, w(xQ) is likely to be the whole attractor.
If we use Milnor’s de nition, then a xed point that is attracting from one side
is called an attractor; for example, 0 is a Milnor attractor for
:i: = 1:2.
Other Milnor attractors, which are not attractors in our sense, are the point (1, 0)
for Example 4.10 and the origin for Example 4.11. We prefer not to call these sets
attractors, but reserve the word for invariant sets that attract whole neighborhoods
using a trapping region.
_
Exercises 7.1
1. Consider the system of differential equations given in polar coordinates by
1‘ = r(1— 1'2)(r2 — 4),
9= 1.
a. Draw the phase portrait for r'.
b. Draw the phase portrait in the (:c,y)-plane.
c. What are the attracting sets and attractors for this system? Explain your
answer, including what properties these sets have to make them attracting
sets and attractors. Hint: Consider all attracting points and intervals for
T.
290 7. Chaotic Attractors
2. Let V(a:) = =6/6 — 5:4/4 + 22:2 and L(a:,y) = V(a:) +92/2. Notice that V’(a:) =
2:5 — 51:3 + 4:: and 0 = V’ (0) = V’ (:l:1) = V’ (:h2). Consider the systempf
differential equations
¢=a
2'1 = —V'(rv) — y [L(w»z/) — L(1.0)l-
a. Show that L = —y2 [L(a:,y) — L(1,0)].
b. Draw the phase portrait.
c. What are the attracting sets and attractors for this system of differential
equations.
3. Consider the forced damped pendulum given by
+ =1 (mod 21r),
:i: = y (mod 21r),
3) = —$in(.'B) — y + cos(1').
Here, the z variable is considered an angle variable and is taken modulo 21r.
The forcing variable T is also taken modulo 2-rr.
a. What is the divergence of the system of equations?
b. If V9 is the volume of a region 9, what is the volume of the region Q(t) =
¢(t; 9)‘? (The region Q(t) is the region formed by following the trajectories
of points starting in 9 at time 0 and following them to time t.)
c. Show that the region 3? = {(1',a:,y) : lyl 5 3} is a trapping region and
Q20 ¢(t;._?Z’)'is an attracting set. Hint: Check 3] on y = 3 and y = -3.
4 Consider the system of differential equations
w=m
1) = 3: '_ 233 _ yr
with flow-¢(t;x). Let V(:c) = §(:|:“ — 2:2) and L(:c,y)_= V(a:) + éyz.
a. Show that 9 = L"((—oo, 1]) is a trapping region and A = tzo ¢>(t; 9)
is an attracting set. Why are W“(0) and (:k\/ l/2, 0) contained in A?
b. What is the divergence? What is the area of ¢(t; 9) in terms of the area
of 9? What is the area of A?
5 Consider the system of differential equations
i = ~a: + yz,
9=—y+wa
z' = —a:y + z — 23,
with ow q5(t; x).
a. Using the test function L(:c,y,z) = %(a:2 + 3/2) + 22, show that .9? =
{ (:c,y, z) : L(rc,y,z) 5 4} is a trapping region. Hint: Find the maximum
of L on L'1(4).
b. Explain why the attracting set for the trapping region .9? has zero volume-
7.2. Chaotic Attractors 291
7.2. Chaotic Attractors
In the Historical Prologue, we discussed the butterfly effect, in which a small change
in initial conditions leads to a large change in the location at a later time. This
property is formally called sensitive dependence on initial conditions. Such a system
of differential equations would amplify small changes. E. Lorenz discovered this
property for the system of equations (7.1). We de ne a chaotic system as one with
sensitive dependence on initial conditions.
We give a mathematical de nition of a chaotic attractor in terms of sensitive
dependence. Section 13.4 de nes these same concepts for maps, and some more
aspects of the subject are developed in that section. In Section 7.7, we give an
alternative test in terms of quantities which are more easily veri ed for computer
simulations or from experimental data.
If two solutions are bounded, then the distance between the orbits cannot be
arbitrarily large but merely bigger than some predetermined constant, r. This con-
stant 1' > 0 should be large enough so the difference in the two orbits is clearly
observable, but we do not put a restriction on its value in the de nition. Also, we
do not want the distance to grow merely because the orbits are being traversed
at different speeds, as illustrated in Example 7.11. Therefore, we have to allow a
reparametrization of one of the orbits in the following de nition. A reparametriza-
tion is a strictly increasing function r : R —> R with lim,_.,,° T(t) = oo and
lim¢_._,° r(t) = —oo.
De nition 7.8. The ow ¢(t; x) for a system of differential equations has sensitive
dependence on initial conditions at xq provided that there is an r > 0 such that, for
any 6 > 0, there is some yo with ||y0 — xg|| < 6 for which the orbits of xo a.nd yo
move apart by a distance of at least 1'. More speci cally, for any reparametrization
r : R —> R of the orbit of yo, there is some time t1 > 0 such that
||¢>(T(¢1);yo) — ¢(t1;Xo)|| 2 1'-
De nition 7.9. The system is said to have semitive dependence on initial condi-
tions on a set S provided that it has sensitive dependence on initial conditions for
any initial condition X0 in S. More precisely, there is one r > 0 (that works for all
points) such that, for any xo in S a.nd any 6 > 0, there is some yo in the phase
space such that the orbits of X0 and yo move apart by a distance of at least 1' in
the sense given in the previous de nition. In this de nition, the second point yo is
allowed to be outside of the set S in the ambient phase space.
If S is an invariant set, we say that the system has sensitive dependence on
initial conditions when restricted to S, provided that there is an r > 0 such that,
for any X4] in S and any 6 > 0, there is some yo in S such that the orbits of xo and yo
move apart by a distance of at least r in the sense given in the previous de nition.
In this de nition, the second point yo must be selected within the invariant set S.
Example 7.15 gives an example that shows the di erence in these two ways of
considering sensitive dependence on initial conditions on an invariant set.
292 7. Chaotic Attractors
Example 7.10. For a simple example with sensitive dependence, consider the
system in polar coordinates given by
1" = 'r(—1 + 1'2),
9= 1.
The set 'r = 1 is an unstable periodic orbit and is invariant. Any initial con-
dition ($0,319) on the periodic orbit has sensitive dependence, because an initial
condition (:z:1,y1) with r1 76 1 eventually has |r(t) — 1| > 1/2, and ||¢(t; (:v1,y1)) —
¢(t;(:z:Q,y0))|| > 1/2. This system does not have sensitive dependence when re-
stricted to the invariant set 1' = 1, since points on the periodic orbit stay the same
distance apart as they ow along.
Example 7.11. This example illustrates why -it is necessary to allow reparametri-
zations in the de nition of sensitive dependence.
Consider the system of differential equations given by
9 = 1+ %sin(1/1) (mod 21r),
11-1 = 0 (mod 21r ),
y = —y-
The set
U={(9.1/1.1/)=059s21r.051/152*.I1/Isl}
is clearly a trapping region,
¢‘(U) 5 {(0.¢.y) = 0 s 0 s 21:, 0 s w s 21r.|1/ls e-‘}-
Its attracting set is
A={(6,'4/1,0):0§0§21r, 0$1,b§21r}.
Since ti > O, any invariant set must contain arbitrary 0. Also, 1,/z(t) is constant.
The values of 1b also cannot be split up by a trapping region. Therefore, A has no
proper subsets that are attracting sets and it is an attractor. However, since ¢(t)
is constant, there is no point whose w-limit equals all of-A. Therefore, A is not a
transitive attractor.
Next, we explain why this example does not have sensitive dependence on
initial conditions when restricted to A. Consider a point (00,1,b,0) in A and a
second point (91,1/1,0) in A that is nearby. Let ¢(t; (I9,-,1l1_,-,0)) = (0,-(t),1,b,-,0). If
sin(¢0) 75 sin(1/11), then
|91('5) - 90(¢)| = |91 - 90 + 1/2($i11(¢1) — $in(¢'o))|-
This will grow until it is as far apart as possible modulo 2-rr, which is a distance 1r.
However, we can reparametrize the second orbit so that 01(¢(t)) = 01+t+‘/2 sin(1/10).
Then,
|91("'(t)) —_9o(t)| = |91 — 90],
and the distance between the two orbits with reparametrization is a constant. Thus,
the ow does not have sensitive dependence on initial conditions when restricted
to the attractor.
7.2. Chaotic Attractors 293
De nition 7.12. A chaotic attractor is a transitive attractor A for which the ow
has sensitive dependence on initial conditions when restricted to A.
Remark 7.13. For a chaotic attractor, we require that the system have sensitive
dependence when restricted to the attractor and not just sensitive dependence on
initial conditions at all points in the attractor. Example 7.15 shows that these
two conditions are not the same: Our stronger assmnption eliminates the type of
nonchaotic dynamic behavior this example exhibits.
Remark 7.14. As indicated in the Historical Prologue, T.Y. Li and J . Yorke in-
troduced the word “chaos” in [Li,75]. Although their paper contains many math-
ematical properties of the system studied, which ones were necessary to call the
system chaotic was not precisely stated.
Devaney was the rst to give a precise mathematical definition in [Dev89]. His
de nition concerned a chaotic set (or a chaotic system, after it was restricted to a.n
invariant set). The three properties he required were that the system have sensitive
dependence on initial conditions, the periodic points be dense in the set, and there
exist a point with a dense orbit in the set. Having a dense orbit is equivalent to
being able to take the xo in A with w(xq) - A, i.e., it is a transitive attractor. See
Appendix B for a discussion of the reason why a “generic” system has the periodic
orbits dense in any attractor.
Martelli in [Mar99] states the following condition, which combines transitivity
and the condition that the system has sensitive dependence on initial conditions
when restricted to the attractor A.
There is an initial condition xo in the attractor A such that w(x,-,) = A and
the forward orbit of xo is unstable in A (i.e., there is an r > 0 such that for
any 6 > 0, there is a point yo in A with "Yo - X0|| < 6 and a t1 > 0 such that
li¢(t1§x¢°) - ¢(i1;¥o)|| 2 1')-
Ruelle and Takens introduced the related concept in 1971 of a strange attmctor.
See [Rue71] or [Guc83]. This concept emphasizes the fact that the attractor has
a complicated appearance, geometry, or topology. In Section 14.1, we introduce the
box dimension which is a measurement of some aspects of the complicated geometry
of a chaotic attractor.
Our or Devaney's de nition of chaos does not directly require that orbits exhibit
random behavior as a function of time. However, transitivity together with sensitive
dependence implies that is is true.
Example 7.15 (Nonchaotic attractor). We consider the same system of differ-
ential equations considered in Examples 5.11 and 7.4,
ri= = 1/.
Q= :c—2a:3—y(—a:2+:z:4+y2).
In Example 7.4, we showed that I‘ = L_1(0) is an attractor by using the test
function 242
L(.,-, mly) = _
2
The attractor P equals the union of the stable and unstable manifolds of the xed
point at 0.
294 7. Chaotic Attractors
If we take the initial point pg in F and take the nearby point qo outside I‘,
then ¢(t; pg) tends toward the origin 0 while ¢(t; qo) repeatedly passes near (i:1,0),
even with a reparametrization. These orbits are a distance approximately 1 apart,
and the system does have sensitive dependence on initial conditions at all points
on I‘ within the ambient space.
However, if we take nearby points po and qo both in I‘, then ¢f. (pg) and 411 (qo)
both tend to the xed point 0 as t goes to in nity and so they get closer together:
therefore, the system does not have sensitive dependence on initial conditions when
restricted to I‘. Thus, this systems is not a. chaotic attractor.
There are several aspects of the dynamics of this system that make it reasonable
not to call it chaotic. For any point po in I‘, the w-limit set w(p0) is just the xed
point 0 and not all of F. So, any point p1 with w(p1) = I‘ is not in I‘ but pl
is outside 1". The trajectory for such a pl spends periods of time near the xed
point, then makes an excursion near one branch of the homoclinic connection, and
then spends even a longer period of time near the xed point before making an
excursion near the other branch of the homoclinic connection. See Figure 1 for the
plot of a: as a function of t for a point starting slightly outside of A. This trajectory
in phase space and the plot of one coordinate versus time do not seem “chaotic” in
the normal sense of the term; they seem almost periodic, with bursts of transition
of a more or less xed duration. The length of time spent near the xed point 0
increases each time the trajectory passes by 0.
Z
1
t
0
200
4
Figure 1. Plot of 1: as a function of t for initial conditions (0,0.05,0) for
Example 7.15
Example 7.16 (Nonchaotic attractor with sensitive dependence). We mod-
ify Example 7.11 to make it have sensitive dependence on initial conditions when
restricted to the attractor. Consider the system of differential equations given by
+= 1 (mod 21r),
9= 1 + %sin(1/1) (mod 21.),
1,li=0 (mod21r),
Q=—u
7.2. Chaotic Attractors 295
This system still has an attractor with {y = 0 For two points ('r_.,, 0,-,1,b,-,0) in
the attractor, in order for the -r variables to stay close, we cannot reparametrize
the trajectories. If sin(1,b0) qé sin(1,b1), then |00(t) —01(t)| will become large until it
reaches a value of 1r. Thus, the system has sensitive dependence on initial conditions
when restricted to the attractor. However, the attractor is not transitive since 111
is a constant. Therefore, this is not a chaotic attractor.
Example 7.17 (Quasiperiodic attractor). This example illustrates the fact that
a quasiperiodic systems does not have sensitive dependence on initial conditions
when restricted to its attractor. Consider the system of di erential equations
511 =1/1 +1I1(1 --Ti-1/i)1
?J1= -911 +111 (1 -"Ii — Iii)»
512 = \/5112 + 3$2 (1 — 59% ‘ 9;).
3); = —\/20:2 + 3y; (1 — —
If we introduce polar coordinates 1-? = 2:? + yf, ta.n(01) = yl/a:1, 'r§ = 1;; + yg, and
tan(6;) = 92/1:2, then the differential equations become
1', = 1-, (1 - 1}),
d1= 1 (mod 211'),
1, = 31, (1 - 1-3),
dz = \/2 (mod 21r).
The set 1'1 = 1 = 1-2 is easily determined to be an attracting set. Since the motion
on this set is quasiperiodic, there is a dense orbit and the set is an attractor. Two
trajectories, with initial conditions (0}'0,'0-)0) and (0%)), 9&0) in the angles, satisfy
(0; (1), 0; (1)) = (01, + 1,05,, +1 \/5) and
(9i(t)1 9% (15)) = (9i,o + $193.0 + t V5)-
Therefore, the distance between the trajectories stays a constant, and there is not
sensitive dependence on initial conditions,
||(9i(i)19i(¢)) - (9i(t).9§(¢))I|
= I (91.11 + t.@%,1. +1»/5) — (vi... +¢.@%,11 + ¢ ~/§)||
= |l(9i,0 _ 9i,o19i,0 ‘ 93,0)"-
Since the system does not have sensitive dependence on initial conditions, the
set is not a chaotic attractor. Since quasiperiodic motion is not “chaotic” in ap-
pearance, this classi cation makes sense for this example.
The system considered in Example 7.11 is another example of an attractor that
does not have sensitive dependence and also is not transitive. Therefore, there are
& couple of reason it does not qualify as a chaotic attractor.
So far, we have not given very complicated examples with sensitive dependence
and no examples with sensitive dependence when restricted to an invariant set, so
no chaotic attractors. In Section 7.3, we indicate why the Lorenz system has a
296 7. Chaotic Attractors
chaotic attractor, both (1) by a computer-generated plot of trajectories and (2)
by reducing the Poincaré map to a one-dimensional function. In Sections 7.4 and
7.5, computer-generated plots indicate that both the Rossler system of differential
equations and a periodically forced system possess a chaotic attractor.
Exercise 7.2
1. Consider the system of differential equations given in polar coordinates by
1‘ = 'r(1— 'r2)(1'2 — 4),
0= 1.
a. Does the system have sensitive dependence on initial conditions at points
in either of the sets {r = 1 } or {r = 2 Explain your answer.
b. Does the system have sensitive dependence on initial conditions when re-
stricted to either of the sets {'r = 1 } or {r = 2 Explain your answer.
c. Is either of the sets {r = 1 } or {r = 2} a chaotic attractor?
2. Consider the system of differential equations given in exercise 2 in section 7.1.
L li C0 = I/(1,0).
a. Does the system have sensitive dependence on initial conditions at points
in the set L“(C0)? Explain your answer.
b. Does the system have sensitive dependence on initial conditions when re-
stricted to the set L‘1(C0)'? Explain your answer.
c. Is the set L"1(Co) a chaotic attractor?
3. Consider the system of di erential equations given by
+ = 1 + %sin(9) (mod 21r),
9=0 (mod21r),
a':=—:1:.
a. Show that the system has sensitive dependence on initial conditions when
restricted to A = {(1',0,0) :0 3 1- 3 21r, 0 3 0 3 21r
Hint: Note that + > 0 at all points but the value changes with 0.
b. Discuss why A is not a chaotic attractor. Does it seems chaotic?
4. Let v(.1.-) = —2a:6 + 15¢‘ - 24.12, for which v'(@) = -129:5 + 605* - 48¢,
v'(0) = v’(¢1) = v'(s=2) = 0, v(0) = 0, v(s=1) = -11, a.nd v(a=2) = 16.
Also, V(:r) goes to —oo as :1: goes to dzoo. Let L(:r,y) = V(a:) +142/2.
a. Plot the potential function V(:c) and sketch the phase portrait for the
system of differential equations
a:=y,
y=-Way
Notice that L(:|:2,0) > L(0,0).
7.3. Lorenz System 297
b. Sketch the phase portrait for the system of differential equations
:i: = y
19 = -V'(='1l— 1/ L($. 1/)-
Pay special attention to the stable and unstable manifolds of saddle xed
points.
c. Let A = {(:r,y) G L“(0) : -2 < :r < 2}. Does the system of part (b)
have sensitive dependence on initial conditions at points of the set A?
Explain why. Does it have have sensitive dependence on initial conditions
when restricted to A? Explain why.
d. What are the attracting sets and attractors for the system of part (b)?
7.3. Lorenz System
In this section we discuss the Lorenz system of differential equations, which is given
by
1': = —aa:+ay,
(7.1) ;i)='ra:—y—a:z,
i = -—bz+:cy.
The quantities 0, r, and b are three positive parameters. The values of the param-
eters that Lorenz considered are a = 10, b = 3/3. and 1- = 28. We will x a = 10
and b = 3/3, but consider the dynamics for various values of 1', including Lorenz’s
original value of 1' = 28.
In Section 7.8.1, we discuss some of the history of E. Lorenz’s development of
this system a.nd how it models convection rolls in the atmosphere. In this section,
we discuss various properties of this system, including why it has a chaotic attractor.
This system gives a good example of a chaotic attractor, as de ned in Section
7.2. It has only linear and quadratic terms in three variables and is deterministic,
because the equations and parameters are xed and there are no external “sto-
chastic” inputs. Nevertheless, this simple deterministic system of equations has
complicated dynamics, with apparently random behavior. The chaotic behavior re-
veals itself in at least two ways. First, the plot of an individual trajectory can seem
random. See Figure 2. Second, Figure 3 shows the results of numerical integration
that indicate that it has sensitive dependence on initial conditions restricted to its
invariant set, because nearby initial conditions lead to solutions that are eventu-
ally on opposite sides of the invariant set. The initial conditions (1.4, 1.4, 28) and
(1.5,1.5,28) are followed to time t = 1.5 in Figure 3a. and t = 2.3 in Figure 3b.
Notice that the solution with smaller initial conditions switches sides on the second
time around, while the other solution stays on the same side. After time 2.3, the
rst solution has crossed back over to the right side, and the second solution is
now on the left side. The trajectories of these two solutions have very little to do
with ea/ch other, even though they have relatively similar initial conditions. The
fact that systems with so little nonlinear-ity can behave so di erently from linear
systems demonstrates that even a small amount of nonlinearity is enough to cause
chaotic outcomes.
298 7. Chaotic Attractors
0% . E
A\ . wt/» IV [ll ’"‘<\_
\/
l.‘
(8) (b)
20 I
10 l
|lL-‘it'll:-L t
‘I Fri I‘ H'F[1o0
-10
I
-20
(C)
Figure 2. Lorenz attractor for 0' = 10, b = 8/3, and 1- = 28: Two views of the
phase portrait and the plot of :2: as a function of t
Warwick 'I\1cker [Tuc99] has proved that the Lorenz system has sensitive de-
pendence on initial conditions and a chaotic attractor for the parameter values
0' = 10, b = 3/3, and r = 28. His proof is computer assisted in the sense that it used
the computer to numerically integrate the differential equations with estimates to
verify the conditions.
7.3.1. Fixed Points. The xed points of the the Lorenz system (7.1) satisfy
y = rt, :r(1- — 1 — z) = 0, and bz = $2. The origin is always a xed point, and if
1- > 1, then there are two other xed points,
Pl‘ = \/H’/Tfllyr — 1) and
P- = (wit. 1).
7.3. Lorenz System z
40
z
40
|I O $ '' ' it
-20 |I -20 O l
20
20 (bl
(3)
20 .l I
‘ZWlulWM1l lllhllllll1ll12°.
_2Q-I
'(<=)
Figure 3. Sensitive dependence for the Lorenz system of equations: Initial
conditions (1.4,1.4,28) and (1.5,1.5,28) followed up to t = 1.5 in (a) and
L = 2.3 in the (b); gure (c) gives the plot oft versus 1: up to time 20
The matrix of partial derivatives of the system of differential equations is
DF(,,_v_1.) = -0 0 O .
T — z -1 —:r
y zr -b
At the origin, this reduces to
DF(o) = —a 0
7‘ —1
00 Q"©©
The characteristic equation for the origin is
(/\+b)().2+(a+ 1)/\+cr(1—1')) =0
300 7. Chaotic Attractors
The eigenvalues are
A, = —b < 0,
A" : -—(a+1)— \/(0;-1)2+4a(r—1) <0, and
A _ —(o'+1)+\/(a+l)2+4a('r— 1)
tl— 2'
The rst two eigenvalues are always negative, and A, is labeled with s for “weak
stable” and A“ is labeled with ss for strong stable. For r > 1 and 0 > b — 1,
" A“ < A, < 0._
The last eigenvalue Au is negative for 0 < r < 1 and positive for r > 1. We label it
with a u for “unstable” since it is positive for most of the parameter values. Thus,
the origin is attracting for 0 < r < 1; for 1' > 1, the origin is a saddle with all real
eigenvalues, two negative (attracting) and one positive (repelling).
For 1' > 1, the two other xed points P+ and P‘ have the same eigenvalues,
so we consider only P"'. At the xed point, 1‘ — z = 1 and y = rs, so the matrix of
partial derivatives is
—cr 0 O ,
1 —l —a:
z a: —b
and has characteristii: equation
.~ (.
0=p,(A) = A3+A2(a+b+ 1) +A(ab+b+a:2)+2a::.-2
= A3 + A2(a+ b+ 1) + Ab(1' +a)+2ba(1' - 1).
All the coe cients are positive, so it has a negative real root A1: One of the
eigenvalues is always a negative real number. (This is true because the characteristic
polynomial p,(A) is positive at A = O and goes to —oo as A goes to —oo: Since p,(A)
varies continuously as A varies from 0 to —oo, it must be zero for some negative
value of A.)
We now discuss the manner in which the other two eigenvalues of Pi vary as
1' varies and a = 10 and b = 3/3 stay xed. The eigenvalues are all real for r near
1. Plotting the graph of p,.(A) as 1' increases indicates that for about r Z 1.3456,
there is only one real root and so another pair of complex eigenvalues. See Figure
4.
For small 1f > 1.3456, the real part of the complex eigenvalues is negative-
It is possible to solve for the parameter values, which results in purely imagirl -l'.Y
eigenvalues by substituting A = iw in p,.(A) = 0 and equating the real and imagi!181'Y
parts of p,(iw) to zero. For A = iw, A2 = —w2 and A3 = -1203, so
0=—iw3—w2(a+b+1)+iwb('r+a)+2ba(r—1)
= 'iw(—w2 + b('r + 0)) — w2(a + b + 1) + 2ba(r -1).
7.3. Lorenz System 301
PW P(/\)
1 100
100 T
A A
-10 -5 0 -10 -5 0
(a) (b)
z>(»\) P0,)
100 - I 2-10-’
AA 0
-1.2892
-1° -5 0 -1.2895
(C) (dl
Figure 4. Plot of the characteristic polynomial for (a) 1' = 1.2, (b) 1' = 1.5,
(c) 1- = 13455171795, and (d) r = 13455171795.
So, equating the real and imaginary parts to zero, we get
I 2 -_
(1‘+o'](o'+b+ 1) = 2a(1‘— 1), or
a(0+b+3)=1'(a—b—1),
T1: a(0+b+3).
a—b—1
For 0 = 10 a.nd b = 3/3, this gives
1'1 = % z 24.74.
At this parameter value 1'1, there are a pair of purely imaginary eigenvalues and an-
other negative real eigenvalue. For values of 1' near 1'1, there is a “center manifold”,
a two-dimensional surface, toward which the system attracts. It has been veri ed
that there is a subcritical Andronov—Hopf bifurcation of a.n periodic orbit within
the center manifold. The orbit appears for values of 1' < 1'1 and is unstable within
the center manifold. Since there is still one contracting direction, the periodic orbit
is a saddle in the whole phase space. The xed point goes from stable to unstable
as 1' increases past 1-1, and there is a saddle periodic orbit for r < 1-1.
Besides the Andronov—Hopf bifurcation, there is a homoclinic bifurcation as r
varies. For small values of 1', the unstable manifold of the origin stays on the same
302 7. Chaotic Attractors
W’(°) W“(0)
('1)
W0) W"<0>
(bl
W” (0) Wu (0)
(C)
Figure 5. Unstable manifold of the origin for (a) 1' < r0, (b) 1' = 1-0, and
(¢) T > To
side of the stable manifold of the origin a.nd spirals into the xed points Pi. See
Figure 5(a). For r = 1'0 re 13.926, the unstable manifold is homoclinic and goes
directly back into the origin. See Figure 5(b). For 1' > 1'0, the unstable manifold
crosses over to the other side of the stable manifold. See Figure 5(c). Thus, there is a
homoclinic bifurcation at 1' = 1'0. For 1' = r0, the strong contraction e"" << 1 causes
the solution to contract toward a surface containing the homoclinic connections.
The sum of the eigenvalues A0 + A, > 0 so the homoclinic bifurcation spawns an
unstable periodic orbit in this surface for r > 1'0. Numerical studies indicate that
this periodic orbit continues up to 1' = 1-1, when it dies in the subcritical Andronov-
Hopf bifurcation at the xed points Pi. See Sparrow [Spa82] for more details.
7.3. Lorenz System 303
7.3.2. Attracting Sets.
Example 7.18 (Fixed point sink for Lorenz system). Assume that 0 < 1' < 1.
For these parameter values, the Lorenz systems has only one xed point. By using
a Lyapunov function, it is possible to show that the origin is globally attracting for
these values. Let
2 2 Z2
L1(a:11/12) = 5; + y? +
This function is positive and measures the square of the distance from the origin
with a scaling in the :1: coordinate. Then,
L- 1: ;1:1::. z:+y.y+zz.
= a:(y—:z:) +y(1'a:—y —:z:z) +z(—bz+a:y)
= —a:2+(1'+1)a:y—y2—bz2
2
where we completed the square in the last equality. If _1' < 1, then 1' + 1 < 2,
(1' + 1)/2 < 1, and the coe icient of y2 is negative. If L; = 0, then y = z = 0
and 0 = :1: — #15 it follows that a: = Egl y = 0. Therefore, L1 = 0 only when
(a:,y, z) = (0, 0, 0), and L1 is strictly negative on all of 1R3 \ This shows that
L1 is a global Lyapunov function and that the basin of attraction of the origin is
all of 1R3.
Example 7.19 (Attracting set for Lorenz system). We now consider the
Lorenz system for 1' > 1, where the origin is no longer attracting. For this range of
1', we use the test function
L,(,,2y2,,)_=_ 2
to show that all the solutions enter a large sphere, but we cannot use it to determine
the nature of their w-limit sets. The time derivative is
L2 = a:(—a:z: + ay) + y(1'a: — y — :1:z)+(z — 1' — a)(—bz + xy)
= —a:1:2 + azy + my — y2 — :1:yz — bzz + rnyz + rbz — my + abz — axy
= -022 — ya — b (.22 ~ (1' + a)z)
22
= —o:c2—y2—b(z— ———T_;U) + —L(T:U).
The sum of the rst three terms is negative, and L2 is negative for
22 _1'+a 2 b(r+o)2
011 +1; +b(z T ) >~T .
304 7. Chaotic Attractors
The level set formed by replacing the inequality by an equality is an ellipsoid with
the ends of the axes at the following six points,
:l:\/b(1'+J) OE Oix/b(r+a)1'+o
2,/5 " 2 ' ’ 2 ‘T ’
(0,0,0), (0,0,1'+o).
The values of L2 at these points a.re
(igen), ("—':,3l2(b+1), and 0.
2
For 0 > 1 and b < 3, the largest of these is Qi2i)—. Let C2 be any number slightly
greater than this value, '_
Let 2
Cg > ——+2i-—U)(T .
U2 = LT1((_°o1 = {($11/:2) 3 L2(z:l/iz) $ C2
The interior of U2 is the set with a strict inequality,
il1t(U2)={(a:11/13) 1L2($13/11) < C2
If ¢(t; x) is the ow for the Lorenz system, then the calculation of the time derivative
of Lg shows that ¢(t;U2) C lI1lJ(U2) for t > 0. Thus, Ug is a trapping region,
and
A = n ¢(t; U2)
130
is an attracting set. ,
For 1' = 28, the xed points P* are saddles and computer simulation shows
that they are not part of the attractor. Therefore, to get a trapping region for an
attractor, we need to remove tubes around the stable manifolds W’ (P*) UW‘ (P‘)
from the open set U2. At the end of this section, we discuss how the resulting
attracting set can be shown to be a chaotic attractor.
In fact, the trajectories for the Lorenz system tend to a set of zero volume. To
see this, we need to consider how volume changes as we let a region be transported
by the ow. The next theorem shows how the volume of any region goes to zero as
it is owed along by the solutions of the differential equation.
Theorem 7.2. Consider the Lorenz system of differential equations.
a. Let 9 be a region in IR3 with smooth boundary 89. Let Q(t) be the region
formed by owing along for time t,
Q(t) = {¢>(t;X0) 1x0 G 9}.
Finally, let V(t) be the volume of @(t). Then
v(t) = v(o) 8-<='+1+">*.
Thus, the volume decreases exponentially fast.
b. If A is an invariant set with nite volume, it follows that it must have zero
volume. Thus, the invariant set for the Lorenz equations has zero volume.
7.3. Lorenz System 305
Proof. By Theorem 6.9, we have
v(t) = I v - F dV.
@(¢)
For the Lorenz equations, V - F = —a —- 1 - b is a negative constant. Therefore, for
the Lorenz equations,
1'/(¢)= -(0 + 1+ b)V(t),
and the volume is decreasing. This equation has a solution
v(t) : V(O) e—(a+l+b)¢’
as claimed in the theorem. The rest of the results of the theorem follow directly.
The invariant set A is contained in the regions Q(t) of nite volume. Therefore,
vol(A) 5 vol(9(t)) = vol(9(0)) e_("+1+°)‘,
and it must be zero. El
Table 1 summarizes the information about the bifurcations and eigenvalues of
the xed points for the Lorenz system of differential equations for 0 = 10 and
b = 3/3, treating r as the parameter which is varying.
1 < 1' Bifurcations for the Lorenz system
1 < 1' < 1'1
1- = 1'0 The origin is unstable, with two negative eigenvalues (two
r = 1'1 contracting directions) and one positive eigenvalue (one ex-
r1 < 1' panding direction).
1' = 28
The xed points Pi are_sta.ble. For r larger than about 1.35,
there is one negative real eigenvalue and a pair of complex
eigenvalues with negative real parts.
There is a homoclinic bifurcation of a periodic orbit which
continues up to the subcritical Andronov—Hopf bifurcation at
T = 7'1.
There is a subcritical Andronov—Hopf bifurcation. The peri-
odic orbit can be continued numerically back to the homoclinic
bifurcation at 1- = 1-0.
The xed points Pi are unstable. There is one negative real
eigenvalue and a pair of complex eigenvalues with positive real
parts.
A chaotic attractor is observed.
Table 1. ro '-=1 13.926, and 1-1 z 24.74
When the Lorenz system became more widely analyzed in the 1970s, mathe-
maticians recognized that computer simulation for 1' = 28 indicates its attracting
Bet possesses certain properties that would imply that it is transitive and has sen-
sitive dependence on initial conditions and so a chaotic attractor. (1) There is a
strong contraction toward an attracting set that is almost a surface. (2) Another
306 7. Chaotic Attractors
direction “within the surface-like object” is stretched or expanded. (3) This expan-
sion can occur and and still have the orbits bounded because the attracting set is
cut into two pieces as orbits ow on different sides of the stable manifold of the
origin. (4) Finally, the two sheets of the surface are piled back on top of each other.
See Figure 2. Because conditions (1) and (2) cannot be veri ed point-by-point but
are global, no one has been able to prove the existence of a chaotic attractor by
purely human reasoning and calculation. The mathematical theory does provide
a basis for a geometric model of the dynamics introduced by J. Guckenheimer in
[Guc76], which does possess a chaotic attractor. The geometric model is based
on assuming certain properties of the Poincaré map for the Lorenz system as dis-
cussed in Section 7.3.3. These properties of Poincaré map in turn imply that there
is a chaotic attractor. As mentioned earlier, Warwick 'I\1cker ['Iiuc99] has given
a computer-assisted proof that the actual Lorenz system for r = 28, 0 = 10, and
b = 3/3 has a chaotic attractor; hence, the name Lorenz attractor has been justi ed
by a computer-assisted proof. .
7.3.3. Poincaré Map. Most of the analysis of the Lorenz system has used the
Poincaré map, which we discuss in this section.
By looking at successive maximum 1,, of the z-coordinate along an orbit, Lorenz
developed an argument that the orbits within the attracting set were diverging. He
noticed that, if a plot is made of successive pairs of maxima (z,,,z,,+1), the result
has very little thickness and can be modeled by the graph of a real-valued function
h(z,,). See FigLu'e 6. The function h.(z,,) appears to have a derivative with absolute
value greater than one on each side. So, if z,, and zj, are on the same side, then
[h(z,,) — h(z§,)| = |h'(§_.)| - Izn — z§,| > |2,, -z{,|, where Q is some point between 2,, and
z§,. This argimient can be used to show that the system has sensitive dependence
on initial conditions, provided that the reduction to the one-dimensional map h can
be justi ed.
50 - Z-n+1
40 -
so - I,- - \\
I Z-n
30 II
40 50
Fig-ure 6. Plot of zn versus z,,+1
Instead of using Lorenz’s approach, we connect the argument back to Poincaré
maps. It is difficult to prove the actual results about the Poinca.ré map, but Warwick
7. 3. Lorenz System 307
Tucker ['Ihc99] has published a computer-assisted proof of these facts. He uses
the computer to calculate the Poincaré map with estimates that verify that his
analysis is valid for the actual equations. Rather than try to develop these ideas,
we connect the observed behavior to the geometric model of the Lorenz equations
that was introduced by Guckenheimer. The geometric model was further analyzed
by Guckenheimer and Williams. See [Guc76] and [Guc80].
The idea of the geometric model is to take the Poincaré rnap from z = 20 to
itself, only considering intersections between the two xed points P1’ and P“. A
numerical plot of the return points of one orbit is shown in Figure 7.
10- y
/.
./
-10-
Flgure 7. Plot of the orbit of one point. by the two-dimensional Poincaré map P
The Lorenz system near the origin is nearly linear, so we approximate the
dynamics by
it = au,
1'1 = —cv,
2 = —bz,
where a = ,\,, ~ 11.83, C = ->._.,, z 22.33, and b = -,\, = 8/3. The variable u is
in the expanding direction and 11 is in the strong contracting direction. For these
values of a, b, and c, we have 0 < b < a < c, so 0 < b/a. < 1 and °/e > 1. We consider
the ow from {z = 20} to {u = 5:141}. We restrict the transversal E by
E = {('u,v,zo) : |u|, Iv] 5 a},
and de ne
$1 = {(iu1.v.Z) |vl»l-1| S 5}-
We take a equal to the value of u for the rst time W"(0) crosses z = 20 between
P+ and P‘. Thus, a is small enough that the cross section is between the two
xed points P+ and P“. The points on E O W’(0) tend directly toward the origin
and never reach 21, so we must look at
E'=ZI\{(u,'u,z0):u7é0} =E+UE_,
308 7. Chaotic Attractors
where 2+ has positive u and E‘ has negative u.
The solution of the linearized equations is given by u(t) = e°‘u(O), v(t) =
e“"*v(0), and z(t) = e""z(0). We take an initial condition (u,v,z0) in E’ (u yé O)
and follow it until it reaches E1. We use u > 0, or a point in 2+, to simplify the
notation. The time r to reach E; satis es
e‘"u = 1/.1,
CT = u 1/a .
Substituting in this time,
-- 'u(-r) = (e"']'°v
- (-> 1‘u. -=/"e
u
u‘/“v
ui/“ ,
2(1) = (eT)_b Z0
Zoltb/G
ug/° l
Thus of the linear system, the Poincaré map P1 : Z’ —> 21 is given by
Pl (uvv) = (z(T)r'U(7'))
= (zo u1_b/°|u|"/“, ufc/° |u|°/° '0).
The image of E’ is given in two pieces, P1(E*) = 21* and P1(E‘) = Ef. See Figure
8.
The map Pg from S1 back to E takes a nite length of time, so it is differentiable.
For the actual equations, this is di icult to control and has only been done by a
computer-assisted proof. The geometric model makes the simplifying assumption
that the ow'ta.kes ('u.1,z(r),v('r)) in 21+ back to (u,'v,z) = (—a + z('r),v(r),z0).
What is most important is that P2 takes horizontal lines ('u.1,v,z1) back into lines
in E with the same value of u; that is, the new u value depends only on 2 and
not on v. The Poincaré map for u negative is similar, but the u coordinate is
a — ufb/“z0]u|"/". Therefore, we get the Poincaré map, P = P2 o P1, of the
geometric model
P (U 1 U) = n v)) = l((— a + zqufb/°u"/“,u1_°/° u.‘/“ v) for u > 0,
a a _ zo U1_ b/Oluib/a,u1_C/Glulc/av) for u < O.
Notice that the - rst coordinate flmction depends only on u and is independent of
11. The second coordinate function is linear in v. The image of E’ by P is as in
Figure 9. Compare it with the numerically computed map in Figure 7, where the
wedges are very very thin.
The derivative of the rst coordinate function f(u) is
f'(u) = zo 11.-1b/“ |u| _ 1+”/“ for u e.ither pos.iti.ve or negati.ve.
7.3. Lorenz System 309
' I I
/I E_ I' Z1+
If ’ ’ I
I
1 ,’
1I
I» » »’/
1 |I
I
I
_1
E1 ll
|
I
I
I
I
Figure B. Flow of E past the origin
2
P(2—) P(2+)
Figure 9. Image of‘E’ by Poincaré map
L‘:1»)
'U.
Figure 10. Graph off
Since b/Q < 1, it has a negative exponent, and f’ (u) is greater than one if ‘u. is
small enough. This gives the expansion in the u-direction of the Poincaré map.
The partial derivative of the second coordinate function is
5<9—g=u_1°/°|ul°/“<1 for |u|<u1.
310 7. Chaotic Attractors
Therefore, the Poincaré map is a (strong) contraction in this direction,
U'1_ g)(u9»v2)l <lv1_ U2l'
The contraction in the v-direction and the fact that the rst coordinate depends
only on u, allows the reduction to the one-dimensional map, which takes u to f(u).
The map f has a graph as shown in Figure 10 and replaces the one-dimensional
map h.(z) (approximating the plot of (z,,, z,,+1) given in Figure 6) used by Lorenz.
The fact that the derivative of the one-dimensional Poincaré map is greater than
one implies that the ow has stretching in a direction within the attractor; this
stretching causes the system to have sensitive dependence on initial conditions as
is discussed in the next section. The cutting by the stable manifold of the origin
is reflected in the discontinuity of the one-dimensional Poincaré map; it allows the
attractor to be inside a bounded trapping region even though there is stretching.
Thus, a combination of cutting and stretching results in a chaotic attractor.
l/
Figure 11. Flow on the branched manifold
The fact that there is a consistent contracting direction over the attracting set
is the most important assumption made about the geometric model for the Lorenz
system. This condition allows the reduction to a one-dimensional Poincaré map.
This conditions is di iculty to verify for the actual system for r = 28, and has only
been done with computer assistance.
7.3.4. One-dimensional Maps. We saw that the Lorenz system of differential
equations has a trapping region for an attracting set. We verify this directly for the
one-dimensional maps we are considering. We also verify that they have a chaotic
attractor, and so the geometric model for the Lorenz system has a chaotic attractor.
We start with stating the de nition of a chaotic attractor in this context.
De nition 7.20. A closed interval [p,q] is a trapping interval for f : IR —+ R
provided that
¢1(f(l1>. q])) C (P. =1)-
We add the closure to the earlier de nition, because the image does not have to be
closed since the map can have a discontinuity.
7.3. Lorenz System 311
We denote the composition of f with itself k times by _f"(:r). For a trapping
interval [p,q], f']k>0 cl(f"([p, q])) is called an attracting set for [p,q]. An attractor
is an attracting set A for which there is a point 2:0 with w(:z:0) = A. An attractor
A is transitive provided that there is a point we in A for which w(:r0) = A. An
attractor is called chaotic provided that it is transitive and has sensitive dependence
on initial conditions when restricted to the attractor.
The one-dimensional Poincaré map has one point of discontinuity and derivative
greater than one at all points in the interval. The following theorem of R. Williams
gives sufficient conditions for such maps to be transitive and have sensitive depen-
dence on initial conditions.
Theorem 7.3. Assume that 0. < c < b and that f is a function de ned at all
points of the intervals [a, c) and (c, b], taking values in [a, b] and the following three
conditions hold. (i) The map f is di erentiable at all points of [a, c) U (c, b], with a
derivative that satis es f’ Z '1 > \/2 at all these points. The point c is the
single discontinuity, and
lim f(:r) = b and lim f(a:) = a..
:|:<c,:|:—»c z>c,:\:—~c
(iii) The iterates of the end points stay on the same side of the discontinuity c for
two iterates,
G S f(¢1)S f2(@) < 6 < f’(b) S f(b) S b-
Then, the following two results are true:
a. The map f has sensitive dependence on initial conditions at all points of [a, b].
b. There is a point 2:‘ in [a, b] such th_at w(:z:") = [a., b].
c. If 0. < f (a) and f(b) < b, then the interval [a, b] is a chaotic attractor for f.
The proof of this general result in given in Chapter 11; see Theorem 11.5. In
this chapter, rather than prove this theorem about nonlinear maps, we simplify to
a map that is linear on each of the two sides. For a constant 1 < '7 3 2, let
__ 'y:c+l fora:<0,
f7(x)—{'y:r—1 for:r:Z0.
The only discontinuity is 2: = 0, and the interval corresponding to the one in
Theorem 7.3 is [-1, 1]. The images f.,([—l,0)) = [1—'y, 1) and f.,([0,1]) = [-1, —1+
7] together cover [-1, 1), f [-1, 1] = [-1, 1). (The discontinuity causes the image
not to be closed.) For 1 < 1 < 2, f.,([—1,0)) = [1 — '7, 1) gé [-1, 1) and f-,([0,1]) =
[-1, -1 + 7] 96 [-1, 1). Each subinterval separately does not go all the way across
the interval but together cover (—1,1), just as is the case for the Lorenz map f.
However, for 7 = 2, f2 takes both [-1, 0) and [0,1] all the way across, _f2([—l,O)) =
l~1i1) and f2(l01: l_1i1l'
The following theorem gives the results for the map f.,.
Theorem 7.4. Consider the map f.,(a:) for \/2 < 'y 5 2 de ned previously.
8- The map f., has sensitive dependence on initial conditions when restricted to
l_1¢
312 7. Chaotic Attractors
1
-f'r(3)
0
‘ ZB
1.
0 .1
Figure 12. Graph of f-,(.1:)
b. There is a point z‘ in [-1, 1] such that w(z‘,f~,) = [-1, 1].
c. For \/5 < 'y < 2, the interval [-1, 1] -is a chaotic attractor for _f.,.
The proof is given in Section 7.9.
Exercises 7.3
1 . (Elliptical trapping region for the Lorenz system) This exercise considers an
alternative bounding function given by
L3(a:,y, z) = r:z:2+ay2+2a(z—2r)2 .
a. Calculate L3. _
b. Find a value C3 > 0 such that, if L;;(a:,y,z) > C3, then L3 < 0.
c. Conclude that all solutions enter and remain in the region
U3 = {(93.3/,1) I L3($,1/ll) S C3}
(i.e., that U3 is a. trapping region).
2. Consider the tripling map T(:c) = 3:1: (mod 1) on [0, 1] de ned by
3a.: for0§a:<§,
T06): 3:c—1for§5a:<§,
3a:—2 for3_§a:<1,
0 for1=:1:.
Show that this map has sensitive dependence on initial conditions. Hint: If
yn = :|:,, + 63" is on a. different side of a discontinuity from :2", then yn+1 =
:c,,+1 + 3(63") — 1. Try using 1' = 1/4.
7.4. Rossler Attractor 313
3. Consider the extension of f2 to all of lR de ned by
1 :2 — % for zr < -1,
F(x): fO1'—1§$<O,
- for 0 5 :1: < 1,
—IO»-l\Jl\DN HHH++ ,-g,__r->- for a: > 1.
a. Show that [—2, 2] is a trapping interval.
b. Show that [-1, 1] is the attracting set for the trapping interval [-2, 2].
Hint: In this case, you must calculate F'°([—2, 2]) for k 2 1.
c. Explain why [-1, 1] is a chaotic attractor for F.
7.4. Riissler Attractor
The chaos for the Lorenz system is caused by stretching and cutting. Rather than
cutting, a more common way to pile the two sets of orbits on top of each other is
by folding a region on top of itself. Stretching, folding, and placing back on top of
itself can create an attracting set that is chaotic and transitive, a chaotic attractor.
Rossler developed the system of differential equations given in this section as a.
simple example of a system of di erential equations that has this property, and
computer simulation indicates that it has a chaotic attractor. However, verifying
that a system has a chaotic attractor with folding rather than cutting is even more
delicate. The Smale horseshoe map discussed in Section 13.1 was one of the rst
cases with folding that was thoroughly analyzed, but it i.s not an attractor. The
Hénon map, considered in Section 13.4, also has stretching and folding. Much work
has been done to show rigorously that it has a chaotic attractor.
The Rossler system of differential eqdations is given by
1: = —y — z,
2) = r + y.
2 = b + z(a: — c).
Computer simulation indicates that this systems has a chaotic attractor for a =
b = 0.2 and c = 5.7.
If we set z = O, the (:2, y) equations become
¢=—%
3) = a: + ay.
The eigenvalues of the origin in this restricted system are
i _ a :t'2\/4 — a7
A - —i——2 ,
which have positive real parts, and a solution spirals outward. In the total three-
dimensional system, as long as z stays near z = b/(c _ 3) where xi = 0, the solution
continues to spiral outward. When :2: — c becomes positive, z increases. For these
larger z, :1: decreases and becomes less than c, so z decreases. To understand the
entire attracting set, we take a line segment :21 3 :1: 3 (132, y = 0, and z = 0, and
follow it once aroimd the z-axis until it returns to y = 0. Since the orbits on the
314 7. Chaotic Attractors
outside of the band of orbits lift up and oome into the inside of the band while those
on the inside merely spiral out slightly, the result is a fold when the system makes
a revolution around the z-axis. All the orbits come back with z z 0; the inside and
the outside orbits come back with the smallest values of 1:; and the middle orbits
return with the largest values of 2:. See Figure 13.
’ 30y
40 Z /R
A
"\’. Y -ad
(bl
0 ‘i
'30 I 30 40 40 z
(*1)
40 z
y l av
~30 0 30 -30 0 30
(C) (<1)
- Figure 13. Rbssler attractor for a. = 0.1, b = 0.1, and c = 14
To completely analyze this system theoretically is dif cult or impossible. How-
ever, its two-dimensional Poincaré map is very thin and it is like a one-dimensional
map. (Because of the folding, this reduction is not really valid but gives some indi-
cation of why the system appears to have a chaotic attractor.) The one-dimensional
map is nonlinear but is similar to the following linear map which we call the shed
map:
{2—a,+a.a: ifa:5c,
go = a — aa: i. f 2: Z c.
where c = 1 — 1/Q, so ga(c) = 1 for both de nitions. We take \/§ < a 5 2. This
map has slope ia everywhere, so its absolute value is greater than \/§. Also,
O<ga(O)=2—a.<l—1/a,=c.
For 0. = 2, we get the map called the tent map. See Section 10.3. By a proof
very similar to that for the Lorenz one-dimensional map, we show that this shed
map has sensitive dependence and is transitive.
7.4. Rossler Attractor 315
1
90(3)
C ___ -- - - - - - - - - - - - - - - - - - __-
0c ._.$
1
Figure 14. Shed map
Theorem 7.5. Consider the shed map g,, for \/§ < a 3 2.
a. The shed map g, has sensitive dependence on initial conditions when restricted
to [0, 1].
b. The shed map has a point z"‘ in [0,1] such that w(z") = [0, 1].
c. For \/5 < a. < 2, the interval [0,1] is a chaotic attractor.
The proof is similar to that for the one-dimensional Lorenz map and is given
in Section 7.9.
7.4.1. Cantor Sets and Attractors. -The Poincaré map for the Rossler attrac-
tor stretches within the attractor, contracts in toward the attractor, and then folds
the trapping region over into itself. The result is that one strip is replaced by two
thirmer strips. If the process is repeated, then the two strips are stretched and
folded to become four strips. These strips are connected within the attractor at
the fold points, but we concentrate on the part of the attractor away from the fold
points and look in the contracting direction only. Thus, we idealize the situation
in what follows, and seek to understand a much-simpli ed model situation.
in ii L i—K2
Figure 16. The sets Kg, ...K3 for the Cantor set
The situation in the Lorenz system is similar, except that the sheet is cut rather
than folded. In Figure 7, the sheets are so thin that the gaps between the sheets
are not visible. However, this system also has the structure of a “Cantor set” in
the direction coming out of the attractor.
316 7. Chaotic Attractors
The Cantor-like structure of the attractor in the contracting direction is pa.rt
of the topological nature of these chaotic attractors, which motivated Ruelle and
Takens to call them strange attractors.
We introduce Cantor sets in Section 10.5 in terms of ternary expansion of
numbers. The middle-third Cantor set K consists of all points in the interval [0, 1]
that can be expressed using only 0’s and 2’s in their ternary expansion. It is the
intersection of sets Kn, which are shown for 0 5 n 5 3 in Figure 15. The Cantor
set has empty interior, is closed and bounded, and is uncountable.
__ Exercises 7.4
l. Prove that the tent map T given in this chapter has a point z‘ with w(z') =
[0, 1]. Hint: Compare with Theorem 7.4(b).
2. Consider the saw toothed map
éaz for 1: < O,
32 for 0 5 :1: 5 3%,
S(:c)= 2-3:: for§5:c5§,
3a:—2 for%5a:51,
§a.:+% fora:> 1.
a. Show that S restricted to [0,1] has sensitive dependence on initial condi-
tions. .,
b. Let J = [.—1,2]. Calculate J1 = S(J), J2 = S(J1), ..., Jn = S(J,,_1).
Argue why J is a trapping region for the attracting set [0,1].
c. Explain why [0,1] is a chaotic attractor for S.
3. Consider the tent map with slope i4,
41: for0<a:<0.5
T(I) =TM) = {4-4:; for0._55:c-'.51.’
Let
K,,(4) = {$0 : T-"(:1:0) G [0,1] for 0 5 j 5 n}
and
K(4) = Q K,,(4) = {$0 . rm.-0) e [0,1] for o 5; < 00}.
1:20
a. Describe the sets K1(4) and K2(4).
b. How many intervals does K,,(4) contain, what is the length of each interval.
and what is the total length of K,,(4)'? Why does K (4) have “length” or
measure zero‘?
c. Using an expansion of numbers base 4, which expansions are in the Cantor
set K (4)?
d. Give two numbers which are in K (4) but are not end points of any of the
sets K,,(4).
e. Show that the set K(4) is nondenumerable.
7.5. Forced Oscillator 317
4. Consider the tent map with slope :t,6, where ,6 > 2,
Ha: forO<:c<0.5,
Th) = T'6(z) : { - Ba: for 0.g5 2:5 1.
Let
Kn(5)=l$01Tj(Io)5lO»1lf°1‘0 sj s n}
and
K( ) = Q K,,( ) = {$0 = Tj(:c0) E [0,1] for 0 5 j < O0}.
n>0
a. Describe the sets K1( ) and K2 ([3). Let a = 1 -2 ii. Why is it reasonable
to call the Cantor set K (H) the middle a Cantor set?
b. How many intervals does K,,( ) contain, what is the length of each inter-
val, and what is the total length of K,,( )? Why does K(,6) have “length”
or measure zero?
7.5. Forced Oscillator
Ueda observed attractors for periodically forced nonlinear oscillators. See [Ued73]
and [Ued92]. In order to get an attracting set, we include damping in the system.
Example 7.21. We add forcing to the double-welled Du ing equation considered
in Example 5.1. Thus, the system of equations we consider is :i5 + 6 :i: — :1: + 1:3 =
Fcos(w t), or
(7.2) T = 1 (ITlOd'21vp) ,
:i:=y,
Q =a:—:v3 — 6y+Fcos(wr).
The forcing term has period 2'”/o in r, so we consider the Poincaré map from r = 0 to
r = 2"/W. The divergence is -6, which is negative for 6 > 0. Therefore, the system
decreases volume in the total space, and any invariant sets have zero volume, or
zero area in the two-dimensional cross section r = 0.
For F = 0 and 6 > 0, the system has two xed point sinks at (a:,y) = (:t1,0),
which attract all the orbits except the stable manifolds of the saddle point at the
origin. In terms of the potential function, the orbits tend to one of the two minima,
or the bottom of one of the two “wells”. For small F, the orbits continue to stay
near the bottom of one of the two wells (e.g., F = 0.18, 6 = 0.25, and w = 1). See
Figure 16 for a time plot of a: as a function of t.
For a sufficiently large value of F, the system is shaken hard enough that a
trajectory in one well is pulled over to the other side. Therefore, orbits can move
back and forth between the two sides. For F = 0.40, the plot of x(t) is erratic. See
Figure 17 for a time plot of at as a function of t. The plot of (z(t), y(t)) crosses itself
since the actual equations take place in three dimensions; therefore, these plots do
not reveal much. The Poincaré map shows only where the solution is at certain
times. It is a stroboscopic image, plotting the location every 21r/w units of time
when the strobe goes off. The munerical plot of the Poincaré map for F = 0.4,
318 7. Chaotic Attractors
3!
'I t
_1_ 100
Figure 16. Plot a: as a function oft for system (7.2), with F = 0.18, 6 = 0.25,
and w = 1
6 = 0.25, and w = 1 indicates that the system has a chaotic attractor. See Figure
18.
The stretching within the attractor is caused by what remains of the saddle
periodic orbit rs = 0 = y for the unforced system. The periodic forcing breaks the
homoclinic orbits and causes the folding within the resulting attractor. The chaotic
attractor is observed via numerical calculation, but it is extremely difficult to give
a rigorous mathematical ‘proof.
a:
§1-
Figure 17. Plot a; as a function of t for system (7.2), with F = 0.4, 6 = 0.25,
and w = 1
Exercises 7.5 319
1.50 y
Iv /I /
1’ , .‘ 1’ ='=
lI .
|.
=.
-0.75
Figure 18. Plot of one orbit of Poincaré map for system (7.2), with F‘ = 0.4,
6 = 0.25, and w =1
Example 7.22. In this example, we add forcing to Example 7.15 and consider the
system of equations
+ = 1 (mod 21r),
it = yr
1) = sc — 22:3 + 1/(:2 — 2:‘ — yz) + F cos('r).
For these equations, an arbitrarily small forcing term makes the stable and the
unstable manifolds of the saddle periodic orbit (which was formerly at the origin)
become transverse rather than coincide. When this happens, the numerical result
is a chaotic attractor with a dense orbit. See Figure 19. This indicates that a
“generic” pertiubation of the system of Example 7.15 results in a system with a
chaotic attractor.
This system has stretching and folding like the Rossler system. The stretching
comes from orbits coming near the xed point and being stretched apart by the
expansion in the unstable direction. The folding is caused by the different locations
of orbits when the periodic forcing changes its direction.
Exercises 7.5
1. Consider the forced Du ing system of differential equations
(H1Od21v,;/),
:i: = y,
y=:z:—-1:3 —6y+F cos(w1').
Using the divergence, show that any attractor for 6 > 0 must have zero volume,
or zero area for the Poincaré map.
320 7. Chaotic Attractors
1/
0.5
-1.2 I
1.2
\
\ \
\/ -0.5
Figure 19. Plot of one orbit of Poincaré map for System (7.3), with F = 0.01
2. Show that the attractor for the system of equations
'i' = 1 (mod 21r),
i = 2/,
1) = a: — 22:3 + y(a:2 — 0:‘ — 112) + F cos(1')
must have zero volume, or zero area for the Poincaré map.
3. Consider the forced system of equations _
'i' = 1 (mod 21r ),
i = :1.
1) = 2:3 -6;/+ F cos(w1').
Using a computer, show numerically that the system has a chaotic attractor
for 6 = 0.1 and F =12.
7.6. Lyapunov Exponents
We have discussed sensitive dependence on initial conditions and how it relates to
chaos. It is well de ned mathematically, can be veri ed in a few model cases (see
Theorems 7.4 and 7.5), and it can be observed in numerical studies. In order to
quantify the rate at which nearby points move apart, we introduce the related con-
cept of Lyapunov exponents. They are more computable with oomputer simulations
than sensitive dependence and arises from considering in nitesimal displacements.
They measure the average divergence rate per unit time of in nitesimal displace-
ments and generalize the concept of eigenvalues of a xed point and characteristic
multiplier of a periodic orbit.
By using the linear approximation for xed t, we get
¢(t;y0) _' ¢(t;x0) z DX¢(t;xo)(y0 " X0)-
7.6. Lyapunov Exponents 321
for small displacements yo — xo. If "yo — xQ|| is “in nitesirnally small”, then the
“'n nitesimal dis lacement" at time t is v t = D ¢ . y — x . To make the
“in nitesimal disi])Jlacements” rigorous, we(chnside,f fg ggnto vect:>)rs to curves of
initial conditions, x, of initial conditions. Letting
V0 = Q2 3-Dd
as 8:0
v(t) = §0¢(t;X@) 8:0 = D=¢¢<¢.».8,>xg,' 8:0 = D=<¢<¢=>=.W<>»
then v(t) satis es the rst variation equation, Theorem 3.4,
(7-4) EdV0) = DF(¢(¢;xo))v(t)'
The growth rate of ||v(t)|| is a number E such that
||\'(¢)|| ~ Ce“-
Taking logarithms, we wa.nt
hi "v(t)" as ln(C) + ft,
1nu:<¢>|| g 1n<t0> + ,,
,3 = Hm 111l”_(*E_
t—>oo If
We use this last equation to de ne the Lyapunov exponent for initial condition xo
and initial in nitesimal displacement vo. There can be more than one Lyapunov
exponent for a given initial condition xo; in fact, there are usually n Lyapunov
exponents counting multiplicity.
I
De nition 7.23. Let v(t) be the solution of the rst variation equation (7.4),
starting at xo with v(0) = v0. The Lyapunov exponent for initial condition xo and
initial in nitesimal displacement vo is de ned to be
e(Xo;V0) = t—lim~oo t,
whenever this limit exists.
For most initial conditions X9 for which the forward orbit is bounded, the
Lyapunov exponents exist for all vectors v. For a system in n dimensions a.nd an
initial condition xo, there are at most n distinct values for €(x0; v) as v varies. If
we count multiplicities, then there are exactly n values, lZ1(x0) = £(x0; vi), E2(x0) =
£(x0; v2), ..., (!,,(x0) = £’(x0; v,,). We can order these so that
@1(X0) Z @2(X0) Z Z 3n(X0)-
The Oseledec multiplicative ergodic theorem 13.17, which we discuss in term of
iteration of functions in Section 13.5, is the mathematical basis of the Lyapunov
exponents.
First, we consider the Lyapunov exponents at a xed point.
Theorem 7.6. Assume that xq is a a:ed point of the di erential equation. Then,
the Lyapunov ercponents at the ased point are the real parts of the eigenvalues of
the xed point.
322 _ 7. Chaotic Attractors
Proof. We donot give a formal proof, but consider the different possibilities. First
assume that v7 is an eigenvector for the real eigenvalue /\,-. Then, the solution is
v(t) = e’\1‘v7.
Taking the limit of the quantity involved, we get
- . ln n
: hm ln(e’\1")+ln|]v5]|
t—~oo t
. in llvj ll
— A-i + 21320 if
= '_
The third equality holds because ln ||v-7' || is constant and the denominator t goes to
in nity.
If /\_,- = 01,- + i j is a complex eigenvalue, then there is a solution of the form
v(t) = e"-*‘(cos( _,-t)u" + sin([iJ-t)wj).
Taking the limit, we have
eat. =11) = ,1g;;, TIn l|v(¢)|l
= hm ln(e°=") + ln || cos( ,-t)u-7' + sin( ,-t)w7|]
= bi +Hm JRt 1n||ws<a~¢>u1I t+ smtw.-¢)w1~||
=01".
The last equality holds because the numerator in the limit oscillates but is bounded
so the limit is zero.
Finally, in- the case in which there is a repeated real eigenvalue, there is a
solution of the form
w(t) = e’\1‘wj + te’\"vj.
Taking the limit, we get
- -. J _= glnll“/(1)1|
e(x°’w ) ¢1l»n<f== t
= lim 1 n(e»\1t )+ 1 n||wj +tvj ||
t—voo t 1)
1 1' t 1
t—wO
= /\_-".
The last equality holds because the numerator of the last term grows as ln(t) and
the denominator grows as t, so the limit is 0.
If we have the sum of two solutions of the preceding form , the rate of growth
is always the larger value. For example, if v(t) = C'1e*"v1 + Cge*°‘v2, with both
7. 6. Lyapunov Exponents 323
C1,C2 95 O and /\1> /\2, then
.1 t
£(XQ;C1V1 + CQV2) = 33.120
= lim l.n(e’\“) ln||C1v1 + Cge(’\"’\1)‘v2]|
t—>oo t + 3
= Al + hm ln |IC'1v1 + C'ge('\’_’\')‘v2||
t—~o|o t
= A1.
The last equality holds because /\2 — ,\1 is negative, so the term e(’\’_"*)' goes to
zero, the numerator is bounded, a.nd the limit is zero. D
When we consider solutions which are not xed points, the vector in the direc-
tion of the orbit usually corresponds to a Lyapunov exponent zero.
Theorem 7.7. Let xo be an initial condition such that r/>(t;x°) is bounded and
w(xq) does not contain any red points. Then,
“X0; F(X0)) = 0-
This says that there is no growth or decay in the direction of the vector eld, v =
F(xq).
Proof. The idea is that
F<¢<¢.=<.>> = §;¢<s.¢<¢;==0>> 3:0 = git + tum) 8:0
= w3 e. ¢<s;=<@>> ‘$20 = v.¢(...., §¢<@.==0>
.9=U
= Dx¢(t;x°)
This says that the linearization of the ow at time t takes the vector eld at the
initial condition to the vector eld at the position of the ow at time t. If we let
x_, = ¢(s;x0), then
v(t) = §6¢<tx.> 8:, = F<¢<tx0>>
is the solution of the rst variation equation; that is,
di,F<¢(¢.=<o>> = vF<..<....»F<¢<¢.x@>>-
If the orbit is bounded and does not have a xed point in its w-limit set, then the
quantity l|F(¢(t;x0))|] is bounded and bounded away from 0. Therefore,
g(xo;F(xo)) = ‘lim l D . F = lim £"£(2(tl-)_)_“ = [)_
—>oo t t—~oo t U
De nition 7.24. Let xo be in initial condition such that ¢(t; Kg) is bounded and
the Lyapunov exponent in the direction F(x0) is zero. If the other n — 1 Lyapunov
exponents exist, then they are called the principal Lyapunov exponents.
324 7. Chaotic Attractors
Theorem 7.8. Let xo be an initial condition on a periodic orbit of period T.
Then, the (n — 1) principal Lyapunov exponents are given by the (ln|,\,~|)/T, where
,\,- are the characteristic multipliers of the periodic orbit and the eigenvalues of the
Poincaré map.
We calculate the Lyapunov exponents for a periodic orbit in a simple example.
Example 7.25. Consider
:t = —y+1:(1 -1:2 -312),
s2==v—v(1—w’*y2)-
This has a periodic orbit (a:(t),y(t)) =_ (cost,s_in t) of radius one. Along this orbit
the matrix of partial derivatives is given by -
_ 1—r2—-2a:2 —1—2y:r .
A(t)'( 1—2:cy 1—r2—2y2
_ -—2COS2t -1 - 2costsin(t)
' 1—2costsint -251112: '
In this form, the solutions of the equation v(t) = A(t)v(t) are not obvious. How-
ever, a direct check shows that
<::.::> <-er)
satisfy 3d5v(t) = A(-t)v(t) (i.e., they are solutions of the time-dependent linear
differential equation). "(The second expression is the vector along the orbit.) Since
(==(0).e/(0)) = (1.0).
a%sms%$ M
shnet)
one Lyapunov exponent is
f((1,0);(1,o))=1..m —1 ln 6_1=(°°St ,)
t—~00 t Slll
1 —2t
= lim ————n()e
t—~oo t
= _2,
and the other Lyapunov exponent is
@<(1.<>>.(o.1>>=,1;-;_._, § in ('jj§§‘>)
=.*1a11§L’
=0.
7.6. Lyap unov Exponents 325
The Lyapunov exponents are easier to calculate using polar coordinates, where
' 1" = r(1 -— 1'2),
9= 1.
At r = 1, the partial derivative in the r variable is
§5r(1—r2 )r:l -1—3r2 r=1_—2.
Therefore, the linearized equation of the component in the r direction is
Ed v, = —2v,,
so, v,-(t) = e'2‘v,(0). Again, the growth rate is -2. The angular component of a
vector ‘U9 satis es the equation,
—ddtvo =0'
and the growth rate in the angular direction is 0. Thus, the two Lyapimov exponents
are -2 and 0.
The following theorem states that two orbits that are exponentially asymptotic
have the same Lyapunov exponents.
Theorem 7.9. a. Assume that q‘>(t;x0) and ¢(t;yq) are two orbits for the some
di erential equation, which are bounded. and whose separation converges exponen-
tially to zero (i.e., there are constants —a < 0 and C Z 1 such that
|I¢>(¢;Xo) - ¢(t;Y0)l| 3 C e‘“‘
fort Z O). Then, the Lyapunov exponents for xo and yo are the same. So, if
the limits de ning the Lyapunov exponents ezrist for one of the points, they earist
for the other point. The vectors which give the various Lyapunov exponents can be
different at the two points.
b. In particular, if an orbit is asymptotic to (i) a fired point with all the eigen-
values having nonzero real parts or (ii) a. periodic orbit with all the characteristic
multipliers of absolute value not equal to zero, then the Lyapunov exponents are
determined by those at the red point or periodic orbit.
The Liouville formula given in Theorem 6.10 implies that the sum of the Lya-
punov exponents gives the growth rate of the volume. This leads to the next
theorem, which gives the connection between the sum of the Lyapunov exponents
and the divergence.
Theorem 7.10. Consider the system of di erential equations x = F(x) in IR".
Assume that xo is a point such that the Lyapunov exponents £1(xO), . . . £,,(x@) exist,
a. Then, the sum of the Lyapunov exponents is the limit of the average of the
divergence along the trajectory,
fl 1T
J2 ejlxol = T A V ' F¢(¢;Xo) ‘it-
b. In particular, if the system has constant divergence 6, then the sum of the
Lyapunov exponents at any point must equal 6.
326 7. Chaotic Attractors
c. In three dimensions, assume that the divergence is a constant 6 and that xo
is a point for which the positive orbit is bounded and w(x0) does not contain any
red points. If £1(x0) is a nonzero Lyapunov exponent at xo, then the other two
Lyapunov exponents are 0 and 6 — E1.
Applying the preceding theorem to the Lorenz system, the divergence is —a —
1 — b, which is negative, so the sum of the Lyapunov exponents must be negative.
For the parameter values 0 = 10 and b = 9/3, the sum of the exponents is -13.67.
Numerical calculations have found Z1 = 0.90 to be positive (an expanding direction),
E2 = 0 along the orbit, so E3 = —13.67—£1 = -14.57 must be negative (a contracting
direction). Since Z1 + £3 is equal to the divergence, to determine all the Lyapunov
-exponents, it is necessary only to calculate one of these values numerically.
7.6.1. Numerical Calculation. For most points xo for which the forward orbit
is bounded, the Lyapunov exponents exist. Take such a point. Then for most vo,
E(x0,vo) is the largest Lyapunov exponent, because the vector D¢>(,_,,°)v@ tends
toward the direction of greatest expansion as t increases-. (Look at the ow near a
linear saddle. Unless the initial condition vo is on the stable manifold, emvo tends
toward the unstable manifold as t goes to in nity.) Thus, the largest Lyapunov
exponent can be calculated numerically by integrating the differential equation and
the rst variation equation at the same time,
=2 = F(x),
\'r = DF(,.) v.
The growth rate of the resulting vector v(t) gives the largest Lyapunov exponent,
em) = @(=<@.v@> = ,1g;-O
It is more dif cult to calculate the second or other Lyapunov exponents. We
indicate the process to get all n exponents for a differential equation in IR".
First, we explain another way of understanding the smaller Lyapunov expo-
nents. Let Jr; = D¢(r.,,.o) be the matrix of the derivative of the ow at time k. Let
U be the set of all vectors of length one,
U={x: = 1}.
The set U is often called the “sphere” in IR". The image of U by Jk Jk(U) is an
ellipsoid with axis of lengths rs), ..., if‘). (If the matrix J), were symmetric, it
would be fairly clear that the image was an ellipse from topics usually covered in a
rst course in linear algebra. For nonsymmetric matrices, it is still true. It can be
shown by considering J1-Jk which is syrmnetric, where JI is the transpose.) The
vectors which go onto the axes of the ellipsoid change with k. However, for most
points,
(1')
ej (X0) = klivlgo
Rather than considering the ellipsoids in all of lR", we consider only the ellipsoid
determined by j longest directions within Jk(U), which we label by BE). The
volume of ES) is a constant involving rr times r ) - - -1-,9), which is approximately
7. 6. Lyapunov Exponents 327
e'°("+"'+‘1). Thus, the growth rate of the volume of the ellipsoids E?) as k goes
to in nity is the sum of the rst j Lyapunov exponents, given by
, vol(E(j))
=8] +...+ej_
Thus, we can determine the growth rate of the area of ellipses de ned by two
vectors; this gives the sum of the rst two Lyapunov exponents.
If we take n vectors vé ...vf,‘, then each vector D¢(,_x°)vg most likely tends
toward the direction of greatest increase, so E(xq,vf,) = E;(x0) are all the same
value. However, the area of the parallelogram spanned by D¢(¢_,,,,)v$ and D¢(,,,,°)v§
grows at a rate equal to the sum of the two largest Lyapunov exponents, and the
volume of the parallelepiped spanned by {D¢(¢_,,o)v§,, .. ., Dq‘>(,_,,°)vf,} grows at a
rate equal to the sum of the rst j largest Lyapunov exponents. To calculate this
growth, the set of vectors in converted into a set of perpendicular vectors by means
of the Gram—Schmidt orthogonalization process. Thus, we take n initial vectors V6
...vg, a point xo, and initial time to = 0. We calculate
X1 = ¢(1;X0) and
wi = 1)¢(,,,,,,v5 for 1 gj g n
by solving the system of equations
2': = F(x),
W1 = DF(x) W1 ,
W" = DF(x) Wn,
for 0 = to 3 t g to + 1 = 1, with initial conditions
x(0) = xo,
wj(O)=v'8 forl jgn.
Then, we make this into a set of perpendicular vectors by means of the Gram-
Schmidt orthogonalization process, setting
v11__w1,
"? “"2 " (WW2- ) "i
(W)wG (W)W3-V2
"i ‘"3 ‘ ~§. "i " “i'
"n‘”_“" _ (lWIvn'lVIl12I )v‘1_____ (|W|v11.1‘V'-11‘1|-|12l"?-1 '
The area of the parallelogram spanned by vi and vf is the same as the area for w‘
and W2; for 3 5 j 3 n, the volume of the parallelepiped spanned by V} . . . vi is the
328 7. Chaotic Attractors
same as the volume for W‘ . . .w7. Therefore, we can use these vectors to calculate
the growth rate of the volumes. We set t1 = to + 1 = 1.
We repeat the steps by induction. Assume that we have found the point xk,
the n vectors v}, . . .v,’§, and the time tk = k. We calculate
x;,.,_1 = q‘>(1;x|,) and
wj = D<i>(1,,,,,)\r'}; for 1 3j 3 n
by solving the system of equations
1': = F(x),
wl = DF(,,) w‘,
W" = DF(,,) w",
for 0 = t;, 3 t 3 t;, + 1 = k: + 1, with initial conditions
Xlkl = Xx.
w7(k:)=v;i for13j3n.
Then, we make this into a set of perpendicular vectors by setting
"I1=+1 = W1,
vk2 +l 21
=w2 ( i“"—H' vl"b—’-k1+";-1) 1
— v k+lv
"2 + 1 = w3 _. (iw||"si+-v1'lll+2 1)Vi+ 1 — (w|'|;".Zv+z1+||21> vi-+ 1 1
'=+‘ nvt... ll" *+1vn = wn _( W“ - vll=+1) V1 _ _ _ _ _ W| ~"/2-.2v%"kl+'l11’ "1vn—1_
We set tk+1 = ti. + 1 = k + 1. This completes the description of the process of
constructing the vectors.
The growth rate of the length of the vectors vi usually gives the largest Lya-
punov exponent, and the growth rate of the length of vectors vi give the j"‘ largest
Lyapunov exponent,
E]-(X0) = vj
LIN for 1 3j 3 n.
Exercises 7.6
1. For a system of differential equations with constant coef cients, the sum of the
Lyapunov exponents equals the divergence. What is the sum of the exponents
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