An Introduction to Dynamical Systems - 2nd Edition

11.4. Invariant Measu.res 529

11.4.3. Piecewise Expanding Maps. In this section, we consider frequency
measures for piecewise nonlinear expanding maps, without assuming the Markov
property and allowing discontinuities. We de ne a Perron-Frobenius operator on
density functions that generalizes the matrix method we presented for piecewise
linear maps with a Ma.rkov partition.

We start by modifying the earlier method for piecewise linear maps with a

Markov partition to see how the densities transform. For an eventually positive

(or aperiodic) transition matrix, if mo = (m1'0,. ..,mk_0) is an initial guess and

m¢+1 = mgM are de ned inductively, then mg converges to the eigenvector m"

giving the invariant measures of the intervals. The components of the sequence of

measures are given by ,3. . L.

""'J'.e+1 = "H.1-

If we let p,-_¢ = "ii-I/L,» be the corresponding densities, then

m-‘f 1 t--L-
Pj,l+1 = = "li.l

t.. pl
= Lt az = $tij

Thus to calculate p_.,-_¢+1, we nd all intervals A, that map onto A_,- and divide
the density pm on each by its slope ,- and take the sum.

We now turn to a nonlinear map which possibly has a nite number of discon-
tinuities or points where possibly the derivative does not exist. We assume that
there are points

qo<q1 <"'<qk
such that f restricted to each open interval A_.,- = (q_,-_1,q,-) is C2, with a bound
on the rst and second derivatives. Moreover, the map is expanding, so we assume
that there is an s > 1 such that [_f’(:r)| 2 s for all qq < re < qk, with :1: 76 q,-
for any j . Finally, we assume that the interval [qmqk] is positively invariant, so
f(a:) E [qQ,q;.] for all a: in [q0,q;,].

For such a nonlinear map, analogous to the piecewise linear case, we want a
construction of a sequence of density functions that converge to a density function
of an invariant measure. Starting with p0(:v) = ‘“%%, assume that we have de ned
densities up to p¢(:c). Near any preimage y = _f“1(a:), the image is in nitesimally

stretched by I f’ (y)|, so the density p¢(y) is thinned out by %I )| and so is taken

by f to |f’(y)| Summing the terms for all the preimages of 2:,

”‘+1(’”>=P<P‘><=>= U623, @If on-

Note that we divided by the slope in the linear case and divide by the derivative
in the nonlinear case. We sum over all preimages in both cases. This operator P,
which takes one density function to another density function, is called the Perr0n—
Flrobenius operator. Lasota and Yorke [Las73] proved that the limit of the average

530 11. Invariant Sets

of the first n density functions converges to a density function p‘(:v),

1 n—1

P1 (-1=)_—nlL-1g° _n Ems)-

(=0

It is necessary to take the average of the sequence of densities because we are not
assuming any condition like aperiodicity. The construction guarantees that p'(a:)
is the density function for an inva.riant measure #p'~ We summarize these results
in the following theorem.

Theorem 11.16 (Lasota, Li, and Yorke). Let f be a possibly discontinuous piece-

wise C’ ezpanding function with expanding factor B > 1. We assume that f and

f’ have only nitely many discontinuities qq, . . . ,q;¢, and that [qq,qk] is positively

invariant. Then the following hold. '.

a. There is an invariant measure u,,- with a bounded density function p'(a:),
p‘ (zr) 5 C, for all as in [qo,qk], so up-([a', l/I) S C lb’ — a’ I for any closed interval
[a’,b’] in [q0,qk]. In fact, i_fa set S has Lebesgue measure zero, /\(S) = 0, then it
has zero measure for the invariant measure, p.,,- (S) = 0.

b. The measure u,,- can be represented as the sum of a nite number of mea-
sures /.ip- = up, + + I-‘pk such that (i) each up, is a natural measure with

w(a.-3, f) = supp(u,,,.) for a point mg and (ii) the interiors of the supports of the p,-
are disjoint; that is,

i.nt{:c:p¢(a:)>0}|"1int{a::pj(a:)>0}=(b f01"i9éj.

c. If there is a point 110 with w(a:0;f) = [qq,qk], then the density p"(:c) > 0
on [qq,qk] and u,,- is a.natural measure (i.e., k = 1 in part (b) and there is no
decomposition).

Idea of the proof. We gave the idea of the proof of part (a) before the statement
of the theorem.

(b) Li and Yorke [Li,75] showed that the density function for an inva.ria.nt
probability measure of part (a) can be split up into a nite number of parts, each
with its own density function, and the interiors of the supports of the different
density functions are disjoint. Each of the separate density functions gives an
invariant measure that is a natural measure.

(c) If there is a point 1:0 for which w(:z:0; f) equals the whole interval, then all

of the limiting measure cannot be split up, and so the limiting measure itself is a

natural measure. U

Example 11.41. Let

f(::) = 54:1: (mod 1)

be the map with one discontinuity at :1: = 3/4. It does not have a Markov partition
because (4/3)" is never an integer, so f"(1) never returns to 0. Thus, we cannot use
the matrix method for piecewise linear maps with a Markov partition. We construct
the rst few density functions by applying the Perron-Frobenius operator, which
indicates the form of the invariant density function.

11 .4. Invariant Measures 531
1

*3;

(O

gs-a

0 ___ 1
gt.- un- tes-

Figure 12. Graph of function for Example 11.41

Take p0(a:) E 1 on [0,1]. Points 0 5 :1: 5 1/3 have two preimages, while those
with 1/3 < a: < 1 have one and :1: = 1 has none, so

for05a:5§,

p1(z)= for%<z<1,

O4:-lwraw fora: =1.

To calculate the next density function p2(a:) = P(p1)(a:), we use the following

preimages: Points 0 5 :1: < 1/3 have one preimage in [0,1/4) and another in

[3/4, 1); points (1/3,4/9] have only one preimage in (1/4,1/3]; nally, points (4/9, 1)

have only one preimage in (1/3,3/4). Using the values of p1(a:) in these respective

intervals, we get '

p2(g;)= + Mm =% for05:c<§,
= fO1‘%<£E§%,

lN-IG0P-4010-‘I Mi-btN»le:uw = ma-Nu0gm: for%<a:<1,

By similar considerations,

+ %=%-(7; for05a:5§,
= for§<:c5§,

p3(:L)_ = for%<a:5§$,
for%g<a:<1.
wwmm =$aa3ss

We do not explicitly calculate more of the density functions in the sequence, but
Figure 13(a.) shows the graph of p15(a:), which was calculated using Maple. In
Figure 13(b), we plot the histogram for 500,000 iterates and 500 subintervals of
[0,1]. Notice the relative good agreement between these two plots. The spikes in
the histogram are caused by the fact that we did not use an in nite orbit; the true
limiting density is monotonically decreasing.

Example 11.42. The piecewise expanding maps with chaotic attractors given in
Examples 11.18 and 11.19 also do not have Markov partitions. In the Figures 14

532 W 11. Invariant Sets
2 ' />is(-17) -L3,/»;l"j)

1' 1

0.2 0.4 0.6 0.8 1 0.2 0.4 as 0.8 1
(a) (b)

Figure 13. Example 11.41, f(a;) = %:r (mod 1). (a) Graph of the approx-
imate density function p;5(.r) calculated by the Perron-Frobenius operator.
(b) Graph of histogram for 500,000 iterates.

1.5 p15(.'L‘) 1.5 I

0 0 _ 1.0
~1.0s 1.0 -1.05
(b)
(=1)

Figure 14. Plots for Example 11.18. (a) Graph of approximate density func-
tion p!s(I). (b) Graph oi histogram for 10,000 iterates and I: = 200 subdivi-

sions.

and 15, we plot (a) the approximate densities p1;,(:1:) calculated with the l’erron-
lfrobenius operator and (b) the histograms of a frequency for 10,000 iterates. For
the histogram, each sub.interval is w - —k (L wi.de, so we plot pj 1. k-" .

Note that each of the ineasure is supported on three intervals that correspond to
their respective attractors.

Exercises 11.4 533
1! P15(I)
1 Ll'2 Np‘!.»-Mk:IO

-0.79 1.43 -0.79 1.43
(a) (b)

Figure 15. Plots for Example 11.19. (a) Graph of approximate density func-
tion p15(a:). (b) Graph of histogram for 10,000 iterates and k = 200 subdivi-
sions.

*
Exercises 11.4

1. Explain why the rotation

Rt-,(:|:) = a: + a (mod 1)

preserves Lebesgue measure on [0,1].
2. Show that the tripling map f(a:) = 3:1: (mod 1) preserves Lebesgue measure on

l9»1l-
3. Assume that p is a. period-n point for f. Explain why

“M”up.r($) = — CM;1<p>($)-
§i—l
0_-

4. Consider the map

gs: for0$a:§§,

f(I) = 2a:—§ f0I‘%-<£B§1.

a. Draw the graph of f. Also, explain why f is an expanding map that has
a Markov partition.

b. Give the transition matrix M on measures of the subintervals, and nd
the i11V8.I‘l8.I11J measures m‘.

c. Give the densities pf that correspond to the invariant measures m‘. Sketch
the graph of the density function p‘ that takes on the values pf.

d. Show that the Perr0n—Frobenius operator for this map preserves the den-
sity function p’ found in part (c).

5. Consider the map

abH for0§.1:§;§-,

f(;1;): — rota: H 0'4: for-,]i<:c§%,

2:r— +-Hui for%<z§1.

534 11. Invariant Sets

a. Draw the graph of f. Also, explain why f is an expanding map that has
a Markov partition.

b. Give the transition matrix M on measures of the subintervals, and nd
the invariant measures m‘.

c. Give the densities pf that correspond to the invariant measures m". Sketch
the graph of the density function p‘ that takes on the values pf.

d. What is the set of all possible periods for f'?

11.5. Applications

~- 11.5.1. Capital Accumulation. Business cycles can be modeled by the amount
produced and a savings level. A discrete-time version of the Solow growth model is
given as follows. We assume a Cobb—Douglas production function Yn = A Kf,’ L}f°
with 0 < a < 1 and A > 0, where K" is the capital in period n and Ln is the labor.
Let kn = K"/1,, be the ratio of capital to labor in period n. Let s(lc,,) be the
possibly nonlinear saving function, so K,,+1 = s(k,.)Y,,. The labor is assume to
grow by L,H.1 = (1 + )\)L,,. Then

/'=n+1 = FK"+;1 = Slkn) WA K°’ L‘ :‘°‘

A
= Hi kg.

If a = s(k,,) is a constant, then we get the function

kn-I-1 = aA
= W I6:

to iterate, as we indicated in Chapter 8. This type of function cannot have any
chaotic behavior.

R. Day indicated that it is possible to get irregular or chaotic growth cycles by
using a nonlinear saving function. See [Day82]. Also, compare with Section 3.12 in
[Sho02]. Day assumes that the interest rate 1',, is given by 1',, = 5 1"-/kn = A k,‘{'1
and the per capita savings that satisfy

1-0: so
s(k,,)y,,=a(1—£)k,,=a(1—llgT)k,,,

a bk:‘_°’
kn-i-1 = 90%) = m <1 ‘“ kn

(11.1) = C (1 - B 1:15") kn,

where c = °/(1 + A) and B = b/(3,4). This function g has many of the properties of

the logistic map with one maximum and g(0) = 0 = g (B1/(1"°)). It has dynamic
behavior like the logistic map. In particular, Figure 16 shows the graphical iteration
of the single point 0.5 for the parameter values a = 0.5, B = 1, and c = 6.5. This
orbit appears to be dense in an interval approximately equal to [0.117, 0.963]. Thus»
this map appears to have a chaotic attractor for these parameter values.

11.5. Applications 535

Figure 16. Plot of graphical iteration for the capital accumulation model for
or = 0.5, B =1, and c = 6.5.

11.5.2. Experience dependent utility preferences. In [Ben81], Benhabib and
Day gave an example where erratic or chaotic behavior can result in a situation
where the preferences depend on experience. Assume that an agent maximizes a
Cobb-Douglas utility function U = rs“;/1"“ subject to the budget constraint pa: +
py = w. Lagrange multipliers shows that the maximum occurs at

:r=2a and y=E(l—a).
PQ

They assume that the parameter a at time period n + 1 depends on 2,, and yn
by the relationship

an-H : barn IS/11>
for a constant b. Using this dependence and the fact that qyn = w — pasn, we get
the map

w w w—pa:

sew <1~>apq‘llPll

Making the substitution g z = 2:, we get the map

(ll-2) wzbp
Zn-H = T 311 (1-_ Zn)-

The parameter A = %’

(11.3)

536 11. Inva.riant Sets

determines the type of behavior. To insure that In does not converge to 0, we need
A > 1. To insure that w Z pa;,, = wzn in the wealth constraint, we need A 3 4.
As we discussed earlier for the logistic map, the whole sequence of period doubling
bifurcations is nished by A’ z 3.57. Chaotic behavior occurs for many parameter
values with A‘ < A < 4.

11.5.3. Chaotic Blood Cell Population. As discussed in Section 9.7.3, A. La-
sota [Las77] proposed a discrete model for the population of red blood cells given
by the function

_ f°(a;) - (1 — a)a: + ba:’e'”.

On the basis of data, the parameter values taken are r = 8, s = 16, and b =
1.1 >< 10°. ~

2_

f(I)

1_

_JZ

Figure 17. Plot of graphical iteration for the blood cell model with a. = 0.81

For a = 0.81, computer simulation indicates there is a chaotic attractor. Figure
17 plots the graphical iteration of a single orbit, starting at 0.5. This plot gives
numerical evidence that there is a point with an orbit dense in the set made up of
three subintervals, approximately [0.262, 0.293] U [0.419, 0.64] U [1.353, 1.54]; hence,
numerical investigation suggests that fom has a chaotic attractor as de ned in
Section 11.2. Notice that, although the nature of the attractor is siinilar to those
for expanding maps discussed in Section 11.2.1, this map is not expanding since
fom has a critical point in the attractor.

See [Ma.r99] for further discussion and references for this model of blood cell
population.

11.6. Theory and Proofs 537

11.5.4. Mixing of Fluids. J . Ottino and his co-workers have used the ideas of
horseshoes to describe mechanisms of mixing of uids. The mixing mechanism in-
volves stretching and folding of the two-dimensional uid. To get "chaotic" mixing,
he uses a time dependent stirring processes. Through this process, he attains what
corresponds to a. homoclinic point and a horseshoe map. Further development of
his work is given in [Ott89a] and [Ott89b] by J. Ottino.

11.6. Theory and Proofs

Properties of limit sets

In part (a) of the theorem, we use the de nition of a. closed set given in Appendix
A.2 and in Section 11.2. The details of the proof of this part of the theorem are
not important and the result is used mainly to prove part (c).

Theorem 11.17. a. For any map f and for any point :20, w(a:0; f) is a closed set.
b. The set w(a:0; f) is positively invariant.
c. Ifyo is a point in w(:z0;f), then w(y0;f) C w(x0;f).

Proof. (a) Let p be a point in the closure of w(a:0; f), r > 0, and N. Since
p is in the closure, there is a point q in (p — r, p + r). Take r’ > 0 such that
(q — r',q +r') C (p — r,p + r). Because q is in the limit set, there is an n 2 N
such that f"(a:q) E (q — r’,q + r’) C (p — r,p+ 1'). Because such an n exists for
every r > 0, and N, p is in the limit set. This shows that the closure of w(a:0; f)
equals w(a:Q; f), so w(a:q; f) is closed.

(b) If f"1(a:0) converges to yo, then f"-'+1(I0) converges to _f(y0), so f(y0) is
in the limit set w(a:0; f). Thus, w(a:0; f) is positively invariant.

(c) If yo is in w(:|:0; f), then 0}’ (yo) C w(a:0; f) since it is positively invariant by
part (b). Since w(:c0; f) is closed, the w-limit points of yo must be in w(a:0; f). U

One-dimensional Lorenz map

Restatement of Theorem 11.5. Assume the following: The map f is a di “eren-
tiable function from 1R \ {c} to IR. The one-sided limits at the single discontinuity
at the point c exist and are denoted by f (c—) = b and f(c+) = a. The iterates
of the points a and b each stay on the same side of the discontinuity c for two
iterates; more speci cally,

a < f(a) < f2(a) < c < f2(b) < f(b) < b.

For as is [a, c) U (c, b], the derivative satis es f’(m) Z 6 > In particular, f is
increasing on both sides of the discontinuity.

With the above hypotheses, the following three statements follow:

a. The map f has sensitive dependence on initial conditions on [a,b].

b. For any subinterval J C [a, b] of positive length, there is an iterate n such
that f"(J) = (a, b).

c. The map f is topologically transitive on [a, b] and so [a, b] is a chaotic at-
tractor.

538 11 . Invariant Sets

Proof. (a) Take any two nearby points 1:0 < yo. If 1:0 and yo are on the same side
of c, then by the mean value theorem, there is a point Z9 between 2:0 and yo such
that

lyi — =v1| = |f(1/0) - fifvoll
= |f'(Zo)(Z/0 - Ioll
Z lyo -fl>o|-

Repeating the argument, as long as 27- = f-7'(:co) and yj = f7-(yo) are on the same
side of c for 0 § j < Ic, then

ll/i= -— Ml 2 5|?/i=-1- $k—1l

2 5" ll/0 — 10l-

Since this number grows, eventually 1:), and yk must be on opposite sides of c.
Because the map is monotone on each side, rt < c < yk. ‘

We have f(c') = b > c and f(c+) = a < c. Let r > 0 be a value such that
f(:i:) Z c+ ("/2) for c— r 5 :i: < c and f(y) 5 c— ("/2) for c < y 3 c+r. We show
that this r works in the de nition of sensitive dependence. If yk — xi, > r, then we
are done. Otherwise, both points are within distance r of c and

:1:I=+1 lJk+1>_ c+2: (c—2T)——r-

In any case, some iterates are farther apart than r, and we have proved part (a).

(b) Let J be an interval with positive length in [a, b]. We de ne a sequence of
intervals by induction: Let

J_ J if c ¢ J,

0 _ the longer subinterval ofJ \ {c} if c G J.

Then by induction, let

_,k 1 = f(Jk) if c ¢ rot).

+ the longer subinterval of f (Jk) \ {c} if c E f (Jk).

By the mean value theorem again, if c ¢ f(Jk), )\(J,,.,.1) Z /\(J';,). If c E f(-Ti),
then )\(J;,+1) Z (3/2) )\(Jk), since we cut the interval into two parts and take the
longer part. Taking the second iterate, we have

£322/\(-In) if ¢ it f(-It) U f(JI=+1).

wk”, 2 % we if c ¢ rot) n f(Ji=+1).

Iz )\(Jk) 1“ll ¢€f(-]i=)nf(J|=+1)-

The rst case assumes that c is in neither f(Jk) nor f (JH1); the second case
assumes that c is not in both f (Jk) and f(J;,+1); the last case assumes that c is in
both f(J;,) and f(Jk+1) (i.e., in two successive iterates). Since ,6 > \/5, 132/2 > 1»
and the length of every second iterate grows until two successive iterates f (J1;-4)
and f(J,,_3) contain c.

11.6. Theory and Proofs 539

Because c is in f (J,,_4), c is one of the end points of J,,_3. By assumption
(ii) on the function, _f(c‘) = b and _f(c+) = a, so f(J,,_3) limits on either a or b;
because f(J,,_3) contains c, it contains either the interval (a, c] or [c, b). The two
cases are similar, so we consider only the second case in which it contains [c, b).
Then,

f2(Jn—3) 3 f((¢.b)) 3 ((1. Cl U [¢,f(b))-
Taking the next iterate, we obtain

f3(J1l--3) 3 f((¢1»¢)) U f((¢»f(b)))

3 lob) U (11.f2(b))

vhwvwd

= (a, b),

where the last inclusion holds by assumption (iii) on the function. Since f"(J) D
f3(J,,_;;), this proves the claim.

(c) Part (b) proves that, for any two open intervals J, and J2 in [a, b], there is

an n such that f"(J1) F1 J2 74 (ll. By the theorem that follows, called the Birkho

transitivity theorem, there is a point with a dense orbit in [a, b]. El

The case in which the derivative is positive on one side of the discontinuity and
negative on the other side is more complicated. Y. Choi presents some results in
[Cho04]. Also, see [Rob00] for a summary of these results.

The next rult gives a precise statement of the Birkhoff transitivity theorem.

Theorem 11.18 (Birkhoff 'Iransitivity Theorem). Assume that F is a continuous
map from R" to itself that takes a closed subset X to itself. (In general, X can be a
complete metric space with countable basis.) Assume that, for every two open sets
U and V of X, there is an iterate lc such that F"(U) intersects V. Then, there is
a dense subset Y of X such that, for every p in Y, the forward orbit of p, 0}’ (p),
is dense in X. Moreover, if F is not necessarily continuous on all of X or R", but
is continuous on a set X’ that is dense and open in X, then the same result is true.

For more discussion of this theorem, see [Rob99].



i
Chapter 12

Periodic Points of Higher
Dimensional Maps

Chapters 9 to 11 have considered the iteration of a function of a. single variable.
Chapters 12 to 14 consider the iteration of a function of several variables. In this
context, the mction F takes a point in R" and gives back another point in the
same space IR". In Section 8.2, we gave a. few examples of such functions.

This chapter concentrates on the periodic points of such functions and the
behavior near these periodic points. We start with linear maps so we can understand
the typical behavior near xed or periodic points. Next, we discuss the stability
types and classi cation of periodic points. For those periodic points with some
contracting directions and some expanding directions (saddles), we introduce the
“stable manifold", which consists of thbse points that tend toward the periodic
orbit. We end this chapter by considering a more global situation of certain maps
on tori; we can nd many different periodic points for these particular maps.

12.1. Dynamics of Linear Maps
For a nonlinear map with a xed point, the linear terms at the xed point determine
many of the features of the phase portrait of the nonlinear map near the xed
point. Therefore, although we are primarily interested in nonlinear maps, we start
by considering the iteration of linear maps.

Let A be an n X n matrix with real entries. For a point x in IR", the matrix
product Ax gives a new point in R". This map is often called the linear map
or linear tmrwformation induced by A, but we do not worry about distinguishing
between the matrix and the linear map it induces. (The matrix depends on a choice
of basis for R"; a different basis gives a different matrix.)

i
541

542 12. Higher Dimensional Maps

The origin is always a xed point for a linear map since A0 = 0 for any n >< n
matrix A. Also, matrix multiplication has the linearity property that

A(ax + by) = aAx + bAy
for any two vectors x and y and any two scalars a and b.

Example 12.1. Consider the linear map

L(r1)=(§ 0 1'1

:22 -- L}

If'we start with a point (0) , then the iterates are

Zn1 _ _- _ (rxigm0> ) _ 29,50%) +=§°g’ ap) I

_ _

= -- 0t A” : )e@r&a°>s+e>%-a°> '

_ 0 3(2) (%)3z(0)

“" ' - — ' s)<=£°>+e>”»-9" 'g+"/‘?/H7HNNHre-Ia»la-H'6:NH;L__)H/L1 "is/"W/“WK-5 MtnhMlkUl Ul @UH7-IU1:L-0/LH/L) HNH/'g7/H\7/ ;_/;_/ /"_$Z-5 /0-1\:-/-‘&\I'"t,‘il+l’:_+°'~°<.1l*=6»"-5|-=

Notice that forward iterates involve powers of 4/5 and 1/5, which are the entries on
the diagonal (the eigenvalues).

Since the determinant of the matrix of the linear 1nap is 4/25 76 0, the matrix
has an inverse

E4 ' 540 = ( fl 0 ) -
—— -5 5
( Oil-l

Therefore, it is possible to take backward iterates

a*> 4 Ia 2mg-1) =L xgo) = _5 0 ma5 Igo) -

Again, notice that backward iterates involve powers of 5/4 and 5, which are the
inverses of the entries on the diagonal (and the eigenvalues). We return to a more
complete analysis of this linear map in Example 12.2.

We now give notation for a general linear map A that we used in the preceding
example. If we start with a point (or vector) x(0) in R", then we can repeatedly
apply the matrix A to obtain new points that we label as xu), xlzl, :

xm = Ax(°) and for k Z 1.
xv‘) = Ax("‘1) = Akxw)

12.1. 7Linear Maps 543

If A has a nonzero determinant, then it has an inverse A", and we can form the
backward iterates

x(“l) = A_1x(°) and
x(_") = A_1x(“"+1) = A_'°x(°) for - k 5 —1.

The points with the simplest dynamics are those that are taken into a scalar
multiple of themselves; that is, points which satisfy

Av=/\v for v7é0.

A number /\ is an eigenvalue of a matrix A if there is a nonzero vector v such that

Av = /\v.

The vector v is called an eigenvector of A, corresponding to the eigenvalue A.
These vectors and numbers satisfy

(A-/\I)v=0

for a nonzero vector v, so

det (A — /\I) = 0.

This preceding equation is called the characteristic equation of the matrix A.

Eigenvalues and Eigenvectors

To nd the eigenvalues and eigenvectors, the rst step is to solve the

characteristic equation '

det (A — /\I) = 0,

for the roots /\1, , ,\,,. The /\_.; are the eigenvalues, and can be real
, or complex. For each root /\_,-, row reduce A — ,\,-I to nd a nonzero

solution v7 of the equation

(A — /\J' V = 0.

The vector v5 is called an eigenvector for the eigenvalue /\,-.

The characteristic equation is especially easy to write down for a two-by-two
matrix.

Theorem 12.1. For a two-by-two 1nat1~i.1: A = (lg with trace -r = tr(A) =

a + d and determinant A = det(A) = ad — bc, the the characteristic equation is
O = /\2 — 'r)\ + A.

Proof. The characteristic equation is
0_-det(G -C A d_ll /\)-(_a—/\)(d—,\) _ bc__/\2 _ (a+d)/\+(ad _ bc). U

544 12. Higher Dimensional Maps

If the eigenvalues are distinct (i.e., /\, 9’: /\_,- when i 94 j), then it can be shown
that the n eigenvectors v1, . . ., v" are linearly independent; that is, if

c1v1+---+c,,v"=O

then all the c_,- = 0. This property can be expressed by saying that the only solution
of the matrix equation

61
(vl,...,v") ( j=0

an
is_c, = -- - = en = 0 (i.e., the homogeneous equation has only the trivial solution).
The homogeneous equation has only the trivial solution if and only if

det(v1,...,v") 76 0.

Also, for any column vector x = ($1, . . . ,a:,,)T, it is possible to solve

311 $1
(v1,...,v") ( j = (

yn $n
for the coef cients yr, so any x can be written as a linear combination of the
eigenvectors, or

x=y1v1 +---+y,,v".
Because any vector can be written uniquely as a linear combination, any set of n
vectors in IR" for which det(v1, . . . ,v") 96 0 is called a basis of lR".

If vi is an eigenvector for the eigenvalue A]-, then A vj = ,\,- vi , and by induction

A’ vi = A (Avi) = A (,\,- vi) = ,\,» (Avi) = ,\,- (,\,-vi) = A? vi,
A3 vi = A(A’ vi) = A (>\§ vi) = ,\§ (Avi) = i§ (A,-vi) = ,\§ vi, and
A'° vi = A (A"-1 vi) = A ()\f“ vi) = ,\§-1(Avi) = A;-1(,\,~vi) = /\f vi

for ls Z 1.

If A-1 exists, then all the eigenvalues )\_,- 76 0 and for negative powers,

A_1vj = /\J71v7 and
A_" vj = A17" v-ll for — k 5 —1.

Combining, we have

Ah vj = A? v-l for all integers Io.

Note that, if l)\,-| < 1, then ||A"vj|] = |)\j|"||vj|| goes to zero as Ic goes to
in nity and goes to in nity as k goes to minus in nity. If |)\,-| > 1, then ||A" vj|| =
|/\j|" Ilvi-|| goes to in nity as ls: goes to in nity and goes to zero as k goes to minus
in nity.

12. 1. Linear Maps 545

Real distinct eigenvalues

Assume that n = 2 and that there are two real eigenvalues /\1 76 /\2, and
l/\1| < |)\2| < 1. Let v1 and v2 be the corresponding eigenvectors. Then, any point
x can be written as a combination of the eigenvectors:

x = y1v1+y2v2.

Then,

A"x=y1A'°v1 +y2A"v2

= yl )\'fv1+ yg A’; v2.

Taking absolute values and applying the triangle inequality yields

l|A'° XII = l|y1>\'fv1+ 1/2 »\'£ V2"
S ly1l~ |i\1l" - ||v‘I| +li/zl -I/\2l" - llvzll.

which goes to zero as lc goes to in nity.

For a linear map with distinct eigenvalues that are each real and of absolute
value less than one, the origin is called a linear stable node (or just stable node).

Example 12.2 (Linear Stable Node). Consider the linear map

<>.(D HI-I

$2 - é $2
/P? UUlullh V

As stated for the general situation, the origin 0 is a xed point of this linear map.

The characteristic equation, '

, , - ,) <. > <. >o=da2_
/\ 0 = -4 -,\ -1 -,\,
5

has roots /\ = 1/5, 4/5, both of which are real and less than one in absolute value.
For,\2=4/5,

40 4 -3

A_§I_(- --woe» )"‘(0 0)’
(BB

where we write ~ when one matrix can be row reduced to the other. Thus,

the components of an eigenvector satisfy 41:1 — 3112 = 0, v1 = %'Ug, which has a

solution

vi = (3).

The reader can either repeat the preceding process for A1 = § or just check that
an eigenvector is

v1= (Q).

Since both of these eigenvalues are less than one in absolute value, Ak'x goes to
zero as lc goes to in nity for any point x, so 0 is an attracting xed point for the

546 12. Higher Dimensional Maps

linear map. However,

2

A(ag)= in and
E131
“=1
I1 _ ‘Z
K/Aeventually converge to zero.
\\A(@)1I-W 5 >l (@)Ii-Thus, the rst iterate of
is farther from the origin even though higher iterates

7/

Figure 1. Phase portrait for a. linear stable node

If x = ylvl + ygV2 is written as a linear combination of the eigenvectors, then
A(y1v1 + ygvz) = y1Avl + 1,/2 Av2

— 1/1 l5vl + 92 i5v’ -
Taking powers, we get

1 '< 4 '=
Ak(U1V1+ I/2V2) = U1 (E) V1 + 1/2 (3) V2-

Notice that (1/5)‘: goes to zero faster than (4/5),‘, so if yg 76 0, then the iterates
approach the origin in a direction that tends to the line generated by the eigenvector
for 4/5:

Ak (S11v1 + I/2V2) = G)’: [U1 GYV1 + Z/2 V2] ,

and the part inside the square brackets approaches the vector y2v2. Thus, the
orbits approach the origin in a direction asymptotic to the eigendirection of the
weaker contraction. Figure 1 shows the phase portrait of this rnap using several
representative orbits.

Table 1 gives the general procedure for drawing the phase portrait for a stable
node.

12.1. Linear Maps 547

Phase portrait for a linear stable node

Consider the case where a 2 x 2 matrix A has eigenvalues that are real
and distinct with |/\1| < I/\;;| < 1 and that they have eigenvectors v1
and v2 respectively.

(1) Draw two straight lines through the origin in the directions of the
two eigenvectors, vl and v2. Mark representative orbits on the
four half-lines by arrows between points on various orbits.

(2) In each of the four regions between the eigenvector lines, draw in
representative orbits which are linear combinations of orbits that
start at v1 and v2:

A"(y1v1+ 3/;v2) = y1 /\'f V1 + yg A; V2.

If yg 94 0, then as k goes to in nity, the orbit approaches the
origin in a direction asymptotic to the line for the eigenvector V2
corresponding to the eigenvalue whose absolute value is closer to
one.

Table 1

If n = 2, both eigenvalues are real, and |)\1| > 1/\g| > 1, then the iterates of
points go to in nity as lc goes to in nity. Backward iterates go to the origin as k goes
to minus in nity. The origin for a linear map of this type with all the eigenvalues
real, of absolute value greater than one, and distinct, is called a linear unstable
node. The p1'OC6ClU.1"6 to draw the phase portrait of an unstable node is similar to
that for a stable node, with obvious changes between k going to in nity and minus
in nity. Exercise 1 in this section contains an example of a linear unstable node.

Finally, assume that |/\1| < 1 < I/\2| for n = 2. The origin for this situation is
called a linear saddle. Let vl and V2 be the corresponding eigenvectors. Any point
can be written as a linear combination of the eigenvectors, x = y1v' + ygV2. Then,
the iterates

Akx = y1)\fv1+yg/\§v2

have the rst term go to zero and the second term go to in nity (if yg 95 0), so

A'°x—y;/\§v2 =y1/\'1‘v1

goes to zero. Thus, iterates are asymptotic to the line generated by v2. Backward
iterates are asymptotic to the line generated by v1:

A_'°x—y1/\['°vl =y;/\§'°v2

goes to zero as -k goes to minus in. nity. If both yl and yg are nonzero, then
||A" x|| goes to in nity as k goes to both plus and minus in nity.

Example 12.3 (Saddle). Consider

L $1 _ — —— I1
1132 _ _OJ“-7*‘ p-nt0(-J 272 '

548 12. Higher Dimensional Maps

The characteristic equation,

0 = ,\2 - Z/\ + 1,

has roots A = 1/2, 2. A direct calculation shows that the corresponding eigenvectors
are

v1 = and v2 = .

See Figure 2 for the phase portrait of the iterates.

Notice that ||A"v1|] goes to zero as It goes to in nity and goes to in nity
as lc goes to m.inus in nity. On the other hand, ]|A“v2|| goes to in nity as k
goes to in nity and goes to zero as k goes to minus in nity. For any other point
x = y; v1 +y2v2 with 111 76 0 .and yg 96 0, ||Akx|| goes to in nity as k goes to either
in nity or minus in nity.

v1 v2

Figure 2. Phase portrait for a linear saddle

Phase portrait for a linear saddle
(1) Draw two straight lines through the origin in the directions of

the two eigenvectors. Mark representative orbits on the four half-
lines by arrows between points on various orbits, with the orbits
along one line approaching the origin as k goes to in nity, and
the orbits along the other line going to in nity.
(2) In each of the four regions between the straight line solutions,
draw in representative orbits which are linear combinations of
orbits that start at v1 and v2.

Table 2

The eigenvalues can be negative, as the next example illustrates.

12.1. Linear Maps 549

Example 12.4 (Negative Eigenvalues). Consider

L I1 = i

The characteristic equation, $2 1 D—l\7ln-7 L} /I?H!-IMy-I L)

o=,\2_§2,\_ 1,

has roots A = ‘1/2, 2. A direct calculation shows that the corresponding eigenvec-

tors are

_3 and
/_\ to \_/ /_\ >-»- ;/

The image,

L(-3)=-%(~3)»
is on the opposite side of the origin. The second iterate returns to the original side

but has been multiplied by (—1/2)2 = 1/4:

”(»3)=%(-3)-

Thus, the linear map is a ip on the line spanned by the eigenvector for the eigen-
value ‘1/2. See Figure 3.

V2

V1
I

Figure 3. Phase portrait for a linear saddle with a ip

Repeated eigenvalues
If there is a repeated eigenvalue, there are not always as many independent

eigenvectors as the multiplicity of the eigenvalue as a root of the characteristic
polynomial. For n = 2, if there are two independent eigenvectors for a multiple
eigenvalue, then the matrix has to be a diagonal matrix; for example,

0.5 0
0 05 '

550 12. Higher Dimensional Maps

However, a matrix such as

0.5 1
O 0.5

has only one independent eigenvector v = (1,0)T. To nd a second independent
solution, the theory of linear algebra shows that the nonhomogeneous equation

(A-/\I)w=v

has a solution for a vector w. The vector w is called a generalized eigenvector for
/\. (See Appendix A.3 for further discussion of generalized eigenvectors.) Then,

" Aw = /\w+v,

Azw = XAW + Av
= /\(/\w+v)+/\v

= )\2w+2)\v,

Aaw = AZAW

= /\A2w+A2v

=,\(>\’w+2,\v) +,\*v

= /\3w+3/\2v, and

Akw = A"'1Aw

= /\A"“1w+A"'1v
._= ,\ (,\'=-‘w + (k - 1) ,\"‘2v) + ,\'=-Iv

= /\kw+k/\"'1v.

Notice that, if |/\| < 1, then these iterates still go to the origin, since k)\"'1 goes
to zero. However, the rst iterate can get longer. For example, if

he :>
with 0 < a < 1, then

||A<r>r|=|<:>|=m»r

For n = 2, with a multiple eigenvalue /\ where |/\| < 1 but only one inde-
pendent eigenvector, the origin is called a degenerate stable node. II the multiple
eigenvalue has [/\| > 1 and only one independent eigenvector, then the origin is
called a degenerate unstable node.

Example 12.5 (Degenerate stable node). Consider

—2
A 3,

' 5NI!-5IQ?"

12.1. Linear Maps 551

which has the single eigenvalue 1/2 with multiplicity two. To nd an eigenvector,
we row reduce A — %I:

1 -1 2 -1 2
A-El-(_%1)~( 0 0).

Thus, we need —'v; + 2112 = 0 and v1 = 2112. So an eigenvector is

v = 21 .

To nd the vector w, we row reduce the augmented matrix:

-1 2 2 -1 2 2
-1 1 1 ”< 0 0'0)"

Thus, we need —w1 +2wg=2; one solution is w1=0 and 'l.U2=1,01'

w=(‘f).

Then,

Aw=—w+v,

A’ w =-(-21 w+v)+-21 v =2-1 2w+2-2v, and
NI!-IN»-I

A3w=—21’ —21 w+v +—22’ v=—21°w+—232v,

Ak wi- 2ik1 _1_I(2w+v)+ ak2—k_ll v 1_ .2?1 kw+2ik_I0 1v.

I

The rst few iterates of w move farther from the origin, but eventually they tend

into the origin. Since A" = [v + 217 w] approaches the line generated by v,

every orbit approaches the origin in a direction asymptotic to the line generated by

the eigenvector. See Figure 4 for the phase portrait.

is-> V

L-%—>

L

Li
Figure 4. Phase portrait for a degenerate stable node

552 12. Higher Dimensional Maps

The general procedure for drawing the phase portrait for two equal real eigen-
values is given in Table 3.

Phase portrait for two equal real eigenvalues

We consider the stable case; the case of an imstable system is similar,
with obvious changes between k going to in nity and minus in nity.
First, assume that there are two independent eigenvectors (and the
matrix is diagonal).

If there are two independent eigenvectors, then all solutions go
straight in toward the origin. The origin of this system is called
a stable star, all points go toward the. origin along straight lines.
Next assume that there is only one independent eigenvector v, and a
second generalized eigenvector w, where (A — /\I)w‘= v.
(1) Draw the two orbits that move in straight lines toward the origin
along the line generated by v, connecting points on the orbits by
arrows.
(2) Next, draw the orbit that has initial condition w and then comes
in toward the origin, with limiting displacement vector from the
origin being a multiple of the vector v (i.e., the orbit /\"w +
k /\'°‘1 v).
(3) Draw the orbit with initial condition —w which should be just
the re ection through the origin of the previous orbit.

‘Table 3

Complex eigenvalues

For n = 2, the matrices that are in simplest form with a pair of complex
eigenvalues are given by

A _ a. —b _ |)\|cosl9 —|/\|sin0 _ W cos0 —sin0

_ b a _ |»\|sin0 |/\|cos0 _ sin0 cos0 ’

where |/\| = x/a2 + bi, a = |/\| cos 0, and b = |).| sin 0. A direct check shows that the
eigenvalues are

a;tz'b= |/\| (cos0:hisin0).

The matrix cos0 —sin6
Rm) _ ( sin0 cos0 >

corresponds to a rotation through an angle of 9, and the scalar I/\1 gives the expan-
sion or contraction factor. Taking powers, we see that

A" = |/\|"R(k0)

is a rotation through an angle of k6, and an expansion or contraction by I/\|". If
|/\| < 1, then the map is a contraction, and the origin is called a stable focus; if
I/\| > 1, then the map is an expansion, and the origin is called an unstable focus;
and if |/\| = 1, then the map is a rotation, and the origin is called a linear center.

12. 1. Linear Maps

Finally, if every eigenvalue of A has IAI < 1 (whether they are real or complex),
then the origin is called a linear sink. If every eigenvalue of A has |/\| > I (whether
it is real or complex), then the origin is called a linear source.

Example 12.6. Let

A=(;;?)-

Then, the characteristic equation is 0 = A2 — 6 /\ + 25, which has roots /\ = 3 zt 2
So, |).| = \/32 + 42 = 5 > 1, and the map is an expansion. See Figure 5.

Figure 5. Phase portrait for a linear unstable focus

I

Phase portrait for a pair of complex eigenvalues
Assume that the eigenvalues are /\ = |/\|e*"9 with 0 76 0, rr, so the
eigenvalues are not real.
1 (1) If |/\f = 1, then the origin is a linear center with each orbit lying

on an ellipse around the origin. The direction of motion can be
either clockwise or counterclockwise, depending on the entries in
the matrix. To determine the direction, check the image of ( 1, 0)T
(or any other vector).

(2) If |)\| < 1, then the origin is a stable focus, and the orbits rotate

either clockwise or counterclockwise around the origin as they
come in toward it.

(3) If |A| > 1, then the origin is an unstable focus, and the orbits

rotate either clockwise or counterclockwise around the origin as
the iterates get larger and larger in absolute value.

Table 4

554 12. Higher Dimensional Maps

Norm of a matrix

We often want to know the largest amount by which a linear map stretches
a vector: The ratio |lA""/||v|| is the amount the vector v is stretched, and the
maximum of this ratio over all nonzero vectors v gives the maximum amount of
stretching. Because "/||v]| is a unit vector, and

|IAv|l = Al
Ilvll Ilvll

it follows that rs{%}=.sr:."A""-

~

We de ne the norm |]A|| of the matrix A by '-

ll A ll = Imglll A“ll .

Notice that ]|A|| Z 0. In fact, the norm IIAII is the square root of the largest
eigenvalue of the symmetric matrix ATA. See Appendix A.3.

Usually, we do not need to know the exact value of the norm but only that
such a number exists. We use the norm, at least implicitly, to see that the linear
part of a map determines many of the properties of the phase portrait near a xed
point.

Example 12.7. Consider the matrix

A st 1)

with multiple eigenvalue a, where 0 < a < 1. The norm of powers of A goes to
zero, as the following argument shows. The power

M = k “.Zk,-1 ) and

T 21¢ k 21¢-1
(Ah) Ak = (k$k—1 a2k +ak2a2k—2)-

The eigenvalues of this matrix are

2a2k + k2a2k—2 i /T€2a4k——2 _,__k4a4F_-4

2
= an + a2k—2%2 5: an-2%2 , /4e’/k2 +1_

The largest eigenvalue is found by ta.king the positive square root, which is close to
0.2" + a2'°‘2l<:2 for large Ic. The norm ]|A'°||, which is the square root of the largest
eigenvalue, is close to Va“ + a2'°“2k2. Both a2" and a2"'2k2 go to zero as lc goes
to in nity, so the norm of A“ goes to zero, and the iterates of any point converge
to the origin.

12.2. Classi cation of Periodic Points 555

Exercises 12.1

1. Find the eigenvalues and draw the phase portraits for the linear maps that the

a <05following matrices represent: b <3 1>
P
0.25 0.25
9
./-\ PPvior c»n- O»-O-J ;/
-3
e 0.4 0.2 1'"Cl-
/_\/-\ c‘J._‘l—l—l—-I I--l \_/
' /'3 —0.2 0.4 ;/L/

12.2. Classification of Periodic Points

The preceding section presented the phase portraits of various types of linear maps.
In this section, we turn to the different concepts of stability for periodic points. We
gave them earlier for one-dimensional maps, but we repeat them with the notation
for higher dimensional maps. When reading the de nitions, the reader should think
of the linear examples given in the preceding section, but the de nitions apply also
to nonlinear maps. After giving the de nitions, we indicate conditions on the
eigenvalues of a linear map that imply the di 'erent types of stability.

For a point pg in IR", the open ball of radius 1‘ (without the boundary) is the
set

B(Po/F) = {X 5 R" 5 "X - Poll < T}-
Sometimes, people call this the open disk, especially when n = 2.

De nition 12.8. A period-k point pg for a map F is called Lyapunov stable pro-
vided that, for each 1' > 0, there exists a 6 > 0 such that, if x is in B(pg,6), then
F1-(x) is in B(F1(pg),'r) for all j Z 0. We often write L-stable for Lyapunov stable.

A period-lc point pg for a map F is called unstable provided that it is not
Lyapunov stable. Thus, pg is unstable provided that there exists an rg > 0 such
that, for each 6 > O, there exists a point X5 in B(pg,6) and an jg 2 0 such that

||Fj‘(Xa) — Fj‘(P0)|| 2 1'0-

The linear stable node in Example 12.2 shows that it is often necessary to take
6 < r. In two dimensions, linear maps for which the origin is L-stable include linear
stable nodes, degenerate stable nodes, linear stable foci, and linear centers. Theo-
rem 12.2 gives precise conditions for a linear map to be L-stable in any dimension.
Linear maps for which the origin is U.I'1SlL3.bl6 include linear saddles, unstable nodes,
and unstable foci.

De nition 12.9. A period-Ic point pg for a map F is called attracting (or asymp-
totically stable, or a periodic sink) provided that

(i) it is Lyapunov stable and

556 12. Higher Dimensional Maps

(ii) there is a 61 > 0 such that, if x is in B(pg,61), then ||F5(x) — F7(pg)|[ goes
to zero as j goes to in nity.

It follows that, if pg is a periodic sink and xg is a point in the set B(pg,61) with
61 as in the de nition, then w(xg,F) = Of.’ (pg) is the orbit of pg.

The origin is an attracting xed point for all linear maps for which all the
eigenvalues have absolute value less than one, which includes linear stable nodes,
degenerate stable nodes, and linear stable foci. For a linear map on [R2 with a pair
of complex eigenvalues of absolute value equal to one (the origin is a linear center),
the origin is L-stable but not attracting.
De nition 12.10. A period-k point pg for a map F is called repelling (or a periodic
source) provided that there is an r1 > 0 such that, if x 96 pg is in B(pg,r1), then
there exists a j = j,‘ such that

||Fj‘(X) — Fj"(P0)|| 2 ri-

We really should add the condition to the de nition of a repelling periodic point
that pg is L-stable for F4, but we wait until the end of the section when we discuss
the inverse of a map.

A periodic point is unstable if some points move away, and it is repelling if
all points move away. The origin is repelling for all linear maps for which all
the eigenvalues have absolute value greater than one; this includes linear unstable
nodes, degenerate unstable nodes, and unstable foci. Note that a saddle xed point
is unstable but not repelling.

Figure 6. (a) Attracting. (b) Riepelling. (c) Unstable but not repelling (saddle).

Theorem 12.2. The stability type of a linear map A.x is classi cation as follows.
a. If all the eigenvalues /\_,- of A have |/\_.,-[ < 1, then the origin is attracting.
b. If all the eigenvalues /\,- of A have I/\_,-| 3 1, and each eigenvalue with

I/\j| = 1 has multiplicity one, then the origin is L-stable.
c. If one eigenvalue /\,-,, of A has I/\_,-0| > 1, then the origin is unstable.
d. If all the eigenvalues /\j of A have I/\_,-| > 1, then the origin is repelling.

12.2. Classi cation of Periodic Points 557

Example 12.11. Consider the map in polar coordinates given by

r+1

FT = 2

(0) (02 (21r - 0)? (mod 21r)

with xed point

P (01) - (01 )-
We show that this xed point is not Lyapunov stable but satis es condition (ii)

of De nition 12.9 for an attracting xed point. Because 02 (21r — 0)2 > 0 for

0 < 0 < 21r, the orbit of every point on r = 1 goes around the circle and has

w((1,6g)) = (0,21r) = (1,0) (mod 2rr). Similarly, for iterates of (rg,0g) with

rg 96 0, Tj goes to 1 and 01- goes to 21r, so w(rg,6|g) = (1,0). Thus, the xed

point satis es condition (ii) for asymptotic stability. However, the xed point is

not L-stable, since points with small 0g > 0 go around near 0 = 1r before returning

to near 0g = O. See Figure 7.

4-—

Figure 7. Unstable xed point for Example 12.11

For nonlinear maps, we consider the matrix of partial derivatives to determine
the linearized stability of a periodic point. Then, Theorem 12.3 asserts the extent
to which the periodic point has the same stability type for the nonlinear map as
the linearized map.

De nition 12.12. Let F be a nonlinear map from IR" to IR" with coordinate

functions Fr. The matrix of partial derivatives at a point p, or the derivative, is

the n x n matrix

8F. 56F31(1)) a3—F;%(P)

DF<p> = (f(1p' )) = an,E or-1,E (P)
Tl (P) ' ' '

The row is determined by the coordinate function and the column is determined
by the variable used to calculate the partial derivative.

558 12. Higher Dimensional Maps

Appendix A.1 indicates the manner in which the Taylor expansion extends to
multiple variables to show that

F(x) = F(P) + DF(p)(X - P) + 0(l|X — PIIZ).

where |O(]]x — P||2)i S CIIX - p||2 for some constant C > 0, so it is a higher
order term. Since the term O(||x — p||2) is smaller than DF(p)(x — p) for small
displacements, this expansion can be used to show that the linear terms dominate
near a xed point.

For an iterate of pg, if we let p_,- = F7 (pg), then by the chain rule for partial
derivatives applied to the matrices, we obtain

_ D(F'°)(p.) = DF(p._.>DF(p.._n ' ' ' DF<p.>DF(p..>.

which is a product of the matrices. In one variable, the order of multiplication of
the derivatives does not matter, but here in higher dimensions, the order of matrix
multiplication does matter: the matrix to the extreme right is evaluated at the
starting point and is the rst to act on the vector; the matrix to the extreme left
is evaluated at the last point p;,_1 and is the last to act on the vector.

Example 12.13. Consider the map 1:

Fe - toy
Its derivative is
DF(=v.v) = (11 23')-

1r pg = then p1 .= and

D(F")<.,.> = DF(a,2)DF(2,1)

=6 3)(i 3)=(? 3)"

We often want to specify that a function has a certain number of partial deriva-
tives, we introduce the following terminology.

De nition 12.14. For an integer r 2 1, a map F(x) from IR" to IR" is said to be

C’, provided that F is continuous and all partial derivatives up to order r exist and

are continuous; that is,

6"F
ina 11:1 . . . 6 nzn

exists for all (i1,...,i,,) with all ij Z 0 and i1 + + in § r, and these partial

derivatives are continuous as a function of x. I.f partial derivatives of all orders exist

(i.e., F is C’ for all r), F is said to be C°°.

The next theorem gives the stability of periodic points for nonlinear maps.

Theorem 12.3. Let F from IR" to IR" be C2. Assume pg is a period-lc point. Let
A1, ..., /\,, be the eigenvalues of D(F")(,,,,).

a. If all the eigenvalues /\_,- of D(F")(,,°) have |/\j| < 1, then the periodic orbit
O; (pg) is attracting.

12.2. Classi cation of Periodic Points 559

b. If one eigenvalue A,-O of D(F")(p°) has |)\_,-0| > 1, then the periodic orbit
O§I(p0) is unstable.

c. If all the eigenvalues /\j of D(Fk)(p°) have I/\,-I > 1, then the periodic orbit
(9;(po) is repelling.

Remark 12.15. If all the eigenvalues Aj of D(F")(p°) have |/\,-| 5 1, at least one
eigenvalue has |/\,-0| = 1, and each eigenvalue with IA,-| = 1 has multiplicity one,
then the periodic point could be attracting, L-stable, or 11I1Sl58.bl8Z The linear terms
are not su icient to determine the stability type.

De nition 12.16. A period-lc point pg is called hyperbolic provided that I/\,-| qé 1
for each eigenvalue /\j of D(F'°)(po).

A period-k point po is called a saddle provided that it is hyperbolic, I/\,-1| < 1
for some jl, and |/\_,-,| > 1 for some other 3'2 (i.e., there are both an attracting
direction and a repelling direction).

Example 12.17. We consider the stability types of the xed points for the non-
linear map given by

F( =, y) = (11, a$ir1(I) ~ tr)-

The matrix of partial derivatives is

01
DF(=.v) = ( ac0s(I) _1 )'

The xed points satisfy

:1: = y and
y = asin(a:) — y.

Substituting :2: for y, we nd that the second equation becomes

2:1: = a sin(a:) or

sin(a:) = E.
G

There is always a xed point with a: = y = 0. For 0 < a < 2, this is the only xed
point, since the slope of 2"’/0, is then greater than one. For a > 2, there is at least a
second xed point (:1:,,, ma) with 0 < ma < 1r, since the slope of 2”/a is less than one.

At the origin,

01

DF(D,0)= < a _1)v

which has characteristic equation /\2 + /\ — a = 0 and eigenvalues

—1:l:\/1+4‘:

At: -

We consider various ranges of the parameter a where the eigenvalues are real or
complex.

For 0. < '1/4, the eigenvalues are not real, and

uh = i/11 + %-4 — 1 = F = \/|a'|-

560 12. Higher Dimensional Maps

Thus, for a < -1, the origin is a repelling xed point with complex eigenvalues; for
-1 < a < *1/4., the origin is an attracting xed point with complex eigenvalues.

For a 2 '1/4, the eigenvalues are real. The eigenvalue )\_ = -1 for a = 0;
therefore, for "1/4 5 a < 0, we have -1 < A- 5 '1/2 and '1/2 5 »\+ < 0, and the
origin is attracting. The eigenvalue /\.,. = 1 for a = 2; therefore, for 0 < a < 2, we
have /\_ < -1 and 0 < /\+ < 1 and the origin is a saddle. Finally, for 2 < a, we
have that /\_ < -2 and I < /\+ and the origin is repelling.

Summarizing the stability of the origin, we have the following properties:

a < - |_| source with complex eigenvalues,
sink with complex eigenvalues,
-1 < a < -—
sink with rea.l eigenvalues,
J=~|- saddle with real eigenvalues,
source with real eigenvalues.
1
-Z < a < 0

O<a<2
2<a

Tiuning to the xed point (:1:a,:1:,,) for a > 2, the matrix of partial derivatives
is

0 1
_1 ) '
DF($-ll) _ (

It has characteristic equation /\2 + /\ - a cos(a:,) = O and has eigenvalues

-1 :h ,/1 +4a cos(a:,,)

Ag; = .

The eigenvalue /\_ = -1 when cos(:|:,,) = 0, 2:“ = "/2, asin("/2) = 1r, or
a = 1r. For this parameter value of a = 1r, )\_ = -1 and /\.,. = 0. For 2 < a. < 1r,
0 < $4 < "/2. Since cos(:i:,,) > 0, /\_ < -1 and /\+ > O. For this range of
parameters, we want to see that A4. < 1, or ,/1 + 4a cos(a:,,) < 3. We would need

3 > \/1 +4a cos(a:,,),

cos(a:,,) < 2 _ sin(a:,.,) or
a 21,, '

:c,, < tan(:i:,,).

This last inequality is true when 0 < 2,, < "/2, so the earlier inequalities are true.
Therefore, for 2 < a < 1r, 0 < )\.,. < 1, and (a:,,,:r,,) is a saddle point. In particular,
for a = 3, 0:3 z 1.50, A- w -1.189, and ,\+ z 0.189.

For a > 1r, the eigenvalues can be real or complex. Since a:,, is an increasing
function of a, there is a bifurcation value 0.2 such that the eigenvalues are real for
1r < a 5 (12 and complex for a2 < a. For 1r < a 5 a2, the eigenvalues are real and
cos(a:,,) < 0, so 0 5 ,/1+4acos(:1:a) < 1, -1< /\_ 5 A1, < 0, and (:i:,,,a:,,) is B
sink.

12.2. Classi cation of Periodic Points 561

For a2 < a,

‘Ail = ‘/1 “ (1 +24acos(za.))

= V al COs($a)l

= \/E (1 - sin2(:r,,))1/2

4 Z l/2

= J5 _ >
(1

which go to in nity as a goes to in nity. Thus, this xed point becomes a source
for large a, and there is a bifurcation value a3 such that I/\i| < 1 for 0,2 < a < a3,
and |/\i| > 1 for a3 < a.

The following list summarizw the stability of the xed points (a:,,, :|:,,):

2 < a < 1r saddle with real eigenvalu,
1r < a 5 a2 sink with real eigenvalues,
a2 < a < a3 sink with complex eigenvalues,
source with complex eigenvalues.
a3 < a

We do not consider the xed points with :1: > 1r, which occur for sufficiently
large parameter values.

The Hénon map

M. Hénon introduced the map that bears his name as an example with a speci c
quadratic formula that could be iterated on a computer. This map also illustrates
the ideas of stability for xed points and period-2 points. Note its similarity to the
logistic map.

The Hénon map is given by

F(:c,y) = (a - 1:2 — by,:r:).

The two constants a and b are parameters. Some authors, including Hénon, write
the map di erently and the parameter b has the opposite sign.

Fixed Points of the Hénon Map
The xed points satisfy

(11 - I2 — blue) = (1.21)

or

y = a: and

2: = a - 11:2 — by.

Substituting z for y, we obtain the equation

0=I2+(b+1):z:—a

with roots

xi _ -(b+1)¢ ,/(1+b)2 +4a

562 12. Higher Dimensional Maps

Thus, the two xed points are

(a:+,:c+) and (a:_,:r:_).

The xed points are real if

(1 +b)2 +4.12 0.

For general parameter values, the matrix of partial derivatives is

1JF(,_,,,=( -2a1: -b0).

The determinant, det (DF(,,,,,)) = b is the factor by which area changes. If b
is‘ negative, then the map re ects in one direction and is orientation reversing.
(Again, in the original paper, b < 0 corresponds to orientation preserving and b > O
corresponds to orientation reversing.)

For example, if a = 0 and b = -0.3, then

,,,: i_ 2M_2 =0, _,,_.,

and the two xed points are
(0,0) and (-0.7, -0.7).

At the xed point (0, 0), for a = O and b = -0.3,

0 0.3
DF(0-U) = (1 0 )>

which has characteristic equation

0= ,\’ -0.3,

and eigenvalues /\ = :l;\/0.3 w :t0.5477. Since I/\| = \/0.3 < 1 for both eigenvalues,
the xed point (0, 0) is attracting.

At the other xed point (-0.7, -0.7),

1.4 0.3
DF(-0.1,-0.7) = ( 1 O ) »

which has characteristic equation

0 = A2 - 1.4x-0.3,

and eigenvalues

A __ m1.4 1 ,/(1.4)? +i1.2

__ _14¢T\/3_.16

z 1.5888, -0.1888.

Since | - 0.1888| < 1 and |1.5888| > 1, the xed point (-0.7, -0.7) is a saddle and
unstable, but not repelling.

12.2. Classi cation of Periodic Points 563

Period-2 Points of the Hénon Map and
A period-2 point satis es

(11,141) =F(f¢0-1/0)= (a-155* bl/0.10)
($0.1/0l= F(I1,!/1) = (fl — Ii " 591.1111)-
It follows that yl = :0, yo = 21,

mo = a - - by;

= 0 — 1/3 — bro.

0=y§+(1+b)a:0-a,
and

a:1 = a. - 23 - byo

= fl - Hi — 5911,

0=yf+(1+b)a:1-a.

We leave it as an exercise to show that the points on a period-2 orbit satisfy

as, + y_-,- = 1 + b.

Substituting 1:0 = 1 + b - yo into the equation 0 = yg + (1 + b):co — a, we obtain

0=y§ -(1+b)y0+(1+b)2—a.

We now take parameter values for which this equation has roots with simple
expressions, b = -0.3 and a = 0.57. We leave it to the reader to check that both
xed points are saddles for these parameter values. The value yo for the period-2
point satis es

0 = 1/3 - (0.7)y0 + (0.7)? - 0.57
= 113- O.7y0 - 0.08,

with roots yo = 0.8, - Hhus, the period-2 orbit is

0.8 -0.1 .

,_»_‘_Q/3?- -0.1 < 0.8 ) “"th
and
0.8 -0.1

F(-0.1)"( 0.8)

-0.1 0.8

F< 0.8)‘(-0.1)"

The matrix of partial derivatives is

DFl0.s,-0.1) = DF(-0.1,0.s) DF(0.s,-0.1)

_ 0.2 0.8 -1.0 0.3 _ -0.02 0.06

_ 1O 10 — -1.6 0.3 '

The characteristic equation is

0 = ,\’ - 0.28 ,\ + 0.09.

564 12. Higher Dimensional Maps

with eigenvalues

0.28 1 ,/(0.28)? - 0.30
’\ = W

= 0.14 :h i \/0.0704

z 0.14 :1: iO.2653.

The absolute value of the eigenvalues satis es

I/\| = \/(0.14)2 + 0.0704

= V0.09
= 0.3 < 1.

Therefore, the period-2 orbit ‘is attracting.

Inverse and or-limit set

We de ned the w-limit set in Section 11.1. We now proceed to de ne the 0:-
limit set if the map has an inverse and it is possible to follow the orbit for backward
iterates. We have used inverses before when de ning conjugacy for one-dimensional
maps and for linear maps. However, we want to solidify the idea of a.n inverse for
a nonlinear map with more variables. Therefore, we start by de ning an inverse of
a function and calculating the inverse of the Hénon map.

A map is one to one provided that if x1 96 x; then F(x1) 96 F(x;). For a
continuous function on the line, this is the same as saying that the function is
monotone, either increasing everywhere or decreasing everywhere.

If a function is one to one, it is possible to de ne an inverse function de ned on
the image of the function. The image of a function F is the set of all y for which
there is some x such that F(x) = y,

image(F) = {y : F(x) = y for some x in the domain of F

The map F from X to Y is called onto provided image(F) = Y. Sometimes we say
the map F is onto Y to emphasize the image.

Assume that F is a one-to-one function from a space X to a space Y with
image J. Then, the inverse F“ is de ned from J onto X by F“1(y0) = xo if and
only if F(x0) = yo. The compositions F‘1 0 F is the identity on X and F 0 F“1 is
the identity on J .

De nition 12.18. Let U and V be two open sets in R" (or metric spaces). A
map h from U to V is called a homeomorphism provided that (i) h is continuous,
(ii) h is one to one on U, (iii) h is onto V, that is h(U) = V, and (iv) the inverse
h‘1 is a continuous map from V to U.

De nition 12.19. Let U and V be two open sets in IR". For an integer r Z 1, a
C’ di eomorphism F from U to V is a homeomorphism from U to V such that F
and its inverse F_1 are C’.

For a diffeomorphism, it follows that det(DF(,,)) 74 0 at all points x and that

D(F")<y) = (DF00) -1 -

where y = F(x).

12.2. Classi cation of Periodic Points 565

Example 12.20. We calculate the inverse of the Hénon map. If

($1,111) = F($0»!l0) = ('1 — $3 * bl/0,110),
then

a:1=a—a:g—by0,
1/1 = I0-

Substituting yl for 2:0 and solving for yo, we get

$1 =<1—y§—byo.

bi/0 =0"?/i-$1,

yo=a—— v;bi ——$1-

Therefore, we have expressions for 2:0 and yo in terms of 1:1 and 111» and the inverse
is given by

_ 2_
($0,?/0) = F_1($1,!/1) = (Z/1, 9—'$%i)-

Since both the map and its inverse have continuous partial derivatives of all orders,
the Hénon map is a C°° diffeomorphisrn from R2 onto IR2.

De nition 12.21. If the map F has an inverse, then the orbit of a. point xo is the
set of all both forward and backward iterates of xo and denote it by

0F(X0l={Fj(X0)1—°°<j<°°}-

De nition 12.22. Let F be a map on a space X, and let xo be an initial condition
in X. Assume that F has an inverse F‘1. A Point q is an a-limit point of xo for
the map F provided that there is a. sequence of iterates —lc,- going to minus in nity
such that F"'°1(x0) converges to q. This means that the orbit of xo keeps coming
back close to q under backward iteration. A more precise way of saying this is that,
for any e > 0 and for any N > 0, there exists a k Z N such that

|lF“"(Xo) — qll < @-

The a-limit set of x0 for F is the set of all Oz-limit points of X0 for F:

a(xo; F) = {q : q is an a-limit point of xq for F

The property of a periodic point being repelling can now be expressed in terms
of the inverse of a map. A period-k point pg for a homeomorphism F on IR" is
repelling, provided that it is attracting for F_1; that is, the following conditions
are satis ed:

(i) For every 1' > 0, there is a 6 > 0 such that

F_j(B(P0=5l) C B(F'j(Po),1‘)
for all j Z 0 (so —j 5 0).
(ii) There is an 1'1 > O such that, if xo is in B(po,r1), then a(x0;F) = OF(PQ).

12. Higher Dimensional Maps

Exercises 12.2

Determine the stability type of the xed point at the origin for each of the
linear maps in Exercise 1 of Section 12.1.

Show that the linear map with matrix

11
O1

is unstable. F .1; _ ':v+y+:;’
Let
y _ 2z+3y '

Find the xed points and classify them as source, saddle, sink, or none of these.

Let
F : <2:|:y+y)_
y 3 y — :1:

Find the xed points and classify them as source, saddle, sink, or none of these.
Let

F :1: = Ii "3 + 33/2 .
y Z 1! + § y

Find the xed p__oin-ts,_and classify them as source, saddle, sink, or none of these.
Consider the Hénon map.

a. Show that, if (:c+,:z:+) and (a:_,:c_) are the two xed points, then

2+ + a:_ = -1 — b.

b. Show_that, if { (:r:0,y0), (:c1,y1) } is a period-2 orbit, then

1+b=Io+1lo=I1+'y1-

Consider the Hénon map with b = -0.2.
a. Show that, for a Z -0.16 = 0.0, there are xed points. Find the eigenvalues
of the single xed point for a = -0.16.
b. Show that, for a > —0.16, a:_ < -0.4, /\+ > 1, and )\_ = ‘O-2/,\+ satis es
-1 < /\_ < 0, so (a:_,a:_) is a saddle point.
c. Show that, for the xed point (a:+,a:+), the eigenvalue }\_ = —1 for a =
0.48 = a1. Using the continuity of the eigenvalues, conclude that the xed
point is attracting for -0.16 < a < 0.48.
d. Show that there is a period-2 orbit for a > 0.48, and that the product of
the two values of :1: on the orbit is 0.8’ — 0., a:0:r1 = 0.82 — a.. Hint: Use the
results of the previous exercise about period-2 orbits for the Hénon map.
e. Show that the characteristic equation for this period-2 orbit is

,\’ - (4¢.,¢1 + 0.4),\ + 0.04 = 0.

12.3. Stable Manifolds 567

Letting —p = :t0:c1, show that one of the eigenvalues is -l when

0 = 3p. + 0.4/J. — 0.84 or
__2.8

#— 6-

Show that this occurs for a = 0.64 + 2'8/6 = a2. Also, for a1 < 0. < G2, the
period-2 orbit is attracting.
8. Let 0. > 0, and de ne Fa from R2 to itself by

F,,(a:,y) = (1 — a:t2 + y, 1:).

a. Find all period-2 points for Fa.
b. Find all values of a for which the period-2 cycle is of saddle type.

12.3. Stable Manifolds

A linear map for which the origin is a saddle has contracting and expanding direc-
tions. If a nonlinear map has a xed point for which the matrix of partial derivatives
is a saddle, then the nonlinear map has curves (or SlJ.l.'f&C6S), for which the points
on these curves have a similar type of behavior with respect to the xed point.

Considering a nonlinear map in two dimensions, with a. saddle xed, there is a
curve of points, called the stable manifold, that converge to the xed point under
forward iteration. There is another curve of points, called the unstable manifold,
that converge to the xed point under backward iteration. These curves are impor-
tant in separating the points that pass the xed point on one side from those that
pass the xed point on the other side, so in two dimensions they are sometimes
called sepamtrices.

These stable and unstable manifolds are important for a variety of reasons. A
stable manifold of one saddle xed point can form the boundary of the basin of
attraction of an attracting xed point as illustrated in Section 12.3.2. In Section
13.3, we explain how an intersection of stable and unstable manifolds of a xed
point implies the existence of an invariant set on which the map has complicated
dynamics.

In higher dimensions, these curves are replaced by surfaces or higher dimen-
sional “manifolds.” In mathematics, the term manifold refers to curves, surfaces,
and higher dimensional objects. (In common usage, manifold means many times or
an object with many openings.) Appendix A.2 provides a more thorough discussion
of the term.

If the origin is a saddle xed point for a linear map on the plane, then there is
a line on which the map is a contraction and another line on which the map is an
expansion. For example, for

A= i 0

02

568 12. Higher Dimensional Maps

andlcZO,

Ah = as and

l> H" “?=l-*r- o'1

~< \_/‘/ <:.>.mh$30;/;/
HHMonaco»-I?
The points on the eigenspace for A = 1/2 are the set of all points x such that Akx
goes to 0 as k goes to in nity. This set of these points is called the stable manifold
of the origin, denoted by

W’(0,A)={x:A“xgoesto0askgoestooo}

The points on the eigenspace for /\ = 2 are the set of all points x such that A"x
goes to 0 as k goes to minus in nity. This collection of point is called the unstable
manifold of the origin, denoted by

W"(0,A)={x:A'°xgoesto0aslcgoesto —o0}={(£2)}.

Now, consider a nonlinear map F in R2 with a xed point p, such that the
linearization A = DF(p) has a saddle at the origin. The main theorem of this
section says that there are invariant curves tangent to the lines p + W°(0, A) and
p + W“(0, A) that consist of points whose iterates F"(x) tend to p as k goes to
in nity or minus in nity,’ respectively:

W“’(p, F) = {x : ||F"(x) — p|| goes to 0 as k goes to 00} and

W“(p,F) = {xz |lF'°(x) — p|| goes to 0 as k goes to — oo

We start with an example in which we can calculate these curves explicitly.

Example 12.23. Consider the map

:1: = éa:

F (11) (211 + I’) ’

which has a xed point at the origin. The derivative is

DF(0) = (012 0 .
2)

The map preserves the y-axis on which it is an expansion, so that

F 0 = 0 .

(11) (211)

<:>=<s.>»Thus, the iterates of the points by the inverse map,

Stable Manifolds 569

converge to the xed point at the origin. Therefore, the unstable manifold of the
xed point is the y-axis:
W“(01F) = {(O|y)T

A

W"(0,F)

W’(0. F)

V

Figure 8. Stable and unstable manifolds for Example 12.23

The stable manifold is not as easy to nd. Assume that it is the graph over the
:1:-axis of a function y = 1,/1(:z:). Since the curve goes through the origin, 112(0) = 0;
since the cm've is tangent to the eigendirection (1,0)-r, qb'(0) = 0. Express 1,!) in
terms of a power series,

2/=¢(I) =¢1zw“+a3a:“+--- ,

with no constant or linear term. Then, the iterate is expressed as

$1 l _ F £13
23:2 “'1' 97; _ “.1 -'5].

l2:1:

:22 + 2a,- as’

%:c
(20-2 +1)a:2 + E;32a,~ asj

$1 2-'l+1 0.1- '
_ (4 (2:11 + 1):1:§ +

where we have used 22:1 = as. Equating the coefficient of ref, a2 = 4(2 (Z2 + 1). So,
-4 = 7a2, and 02 = *4/7. Equating the coe icient of xi for j > 2, a,_,- = 25"‘Ha,-.
So, (25'*" — 1)aJ- = 0 and aj = 0. Thus, the inva.ria.nt ciuve is given by

1/ = we-) = -31’-

570 12. Higher Dimensional Maps

Since this curve is invariant by F, the iterates of points on the curve satisfy

($21)) = F (1/$30)) Z and

($31)): F (viii-'1>l = (llllll = '

Thus, ark = :1:02_" goes to zero as k goes to in nity. Also, since 1/1 is continuous,
yk = 1l»(:z:k) must also go to zero. Thus, the pair of points (a:;,,y;,) goes to the xed
point at the origin, and

" w"(0, F) = —§¢f)T }.

See Figure 8.

We next give the general de nition of the stable and unstable manifold of a
periodic point.

De nition 12.24. Let F be a diffeomorphism and let p be a period-q point of
F. The stable manifold of a period-q point p is the set of points whose iterates are
asymptotic to the iterates of p. More speci cally,

W"(p, F) = {x : ||F'°(x) — F"(p)|| goes to 0 as lc goes to oo}.

The stable manifold of the orbit of a period-q point p is the set of points whose
iterates are asymptotic, to the iterates of p, or

. q——1

W‘(0F(p>.F) = U W*<F1'(p),F>={><=~»(><)= ops») }-

:i=°

The unstable manifold of a period-q point p is the set of points whose iterates
are backwardly asymptotic to the iterates of p:

W“(p, F) = {x : ||F"(x) — F"(p)|| goes to 0 as k goes to — oo}.

The unstable manifold of the orbit of a period-q point p is the set of points that are
backwardly asymptotic to the orbit of p:

q—1

W“(0F(P)»F) = U W"(Fj(Pl»Fl={X1°'(Xl= @i-(p) }-

J'=0

We also want to de ne the local stable and unstable manifold. For 6 > 0, the
local stable manifold of p of size 6 is the set

W§(P,F) = {X E W’(P,F) = |lF'°(X) — F"(P)|| S 5 for all k Z 0}-

Similarly, for 6 > 0, the local unstable manifold of p of size 6 is the set

W.;‘(p, F) = {X E W"(p,F) = l|F'°(>=) — F"(r>)|| s 6 for an k s 0}

= {x 2 [|F'°(x) — F"(p)]| 3 6 for all k 3 0}.

12. 3. Stable Manifolds 571

Remark 12.25. If p is an attracting xed point, then the stable manifold of p is
what we earlier called the basin of attractiom

W’(P.F) = @(P; F)-
From this point on, we use the notation for the stable manifold for this basin of
attraction.

The next theorem characterizes the local stable and unstable ma.nifolds of a.
periodic point.

Theorem 12.4 (Stable Manifold Theorem). Let F be a C’ diffeomorphism on lR2
for 1" Z 1, and let p be a saddle period-q point with eigenvalues /\, and /\,,, with
|)\,| < 1 and |/\,,| > 1. For sufficiently small 6 > 0, the local stable and unstable
manifolds of p of size 6, W5’(p,F) and W§‘(p, F), are C’ curves tangent to the
directions given by the eigenvectors for the eigenvalues /\, and Au, respectively.
Moreover, they are characterized by

W;(p, F) = {X = ||F"(x) _ F'=(p)[| 5 6 for all k Z 0} and

Wl‘(P.F) = {X= l|F“(X) - F'°(P)l| S 5 for all k S 0}-
These curves are locally invariant in the sense that

F(W§(p,F)) C W§(F(P).F) and
F(Wl‘(P.F)) D W5‘(F(P).F)-

The (global) stable and unstable manifolds are equal to the union of the iterates
of the local stable and unstable manifolds, respectively:

Wmn=UrwmW@ml

k=l '

Wmn=@WmuWmm.

k=l

The global stable manifold cannot cross itself, but can wind around in a com-
plicated fashion. See Figure 9 for an example of this phenomenon for the Hénon
map with a = 1.6 and b = -0.3. The stable manifold of the xed saddle point
p = (z_,z:_) contains the curves in the gure that are more vertical than horizon-
tal and also points outside the frame of the gure; the unstable manifold contains
the curves that are more horizontal than vertical, which stay completely in the win-
dow. (The unstable manifold looks like it has sharp corners, but it really has just
very sharp smooth bends.) The gure indicates the second xed point q = (a:.,., :1:+),
but not its stable and unstable manifolds.

Remark 12.26. Poincaré discovered the importance of the stable and unstable
manifolds in the last quarter of the nineteenth century. In particular, if the unstable
manifold of a periodic orbit crosses the stable manifold of the same periodic orbit,
then each manifold is embedded in a complicated manner and the map has an
invariant set with complicated dynamics modeled by symbolic dynamics. He used
these ideas to explain why an N-body problem governed by Newtonian laws of
attraction, such as om solar system, can be unstable. His ideas relate to the
construction of a horseshoe for a transverse homoclinic point given in Section 13.3.

572 12. Higher Dimensional Maps

3-

cl
1|

_~,4~ 5

pl

l, at

Figure 9. Stable and unstable manifolds of the xed point p for the Hénon
map with a = 1.6 and b = -0.3. Note that the stable manifold run out of the
window shown.

12.3.1. Numerical Calculation of the Stable Manifold. Theorem 12.4 asserts
the existence of stable and unstable manifolds. Using computer simulation, there
are various ways to estimate stable and unstable manifolds for diffeomorphisms
in two dimensions. Thesimplest technique is to pretend that the local unstable
manifold is a short line segment through the xed point p in the direction of the
unstable eigenvector v“:

w,;*(p) ={p+tv“ 1 -65:56}.

Iterating points on Wj‘(p) determines the global unstable manifold. Assume that
we place N evenly spaced points on one side of this local unstable manifold; then,

x’ = p + 6v“,

for 1 5 j 5 N. Taking a few iterates gives a fairly long unstable manifold, namely,

{F"(><1') = 1 sj s N}.

for a xed k > 0. For example, we could take N = 1000 and k = 5.
The stable manifold could be found by a. similar method using backward iter-

ates. Let

y’ = p + 6v‘,

for 1 59' 3 N. Then,

{F‘“(y’> = 1 s 1 s N},

for a xed k > 0, &pp1'O)CllI'l8.l36S the stable manifold.

12.3. Stable Manifolds 573

This primitive method is often used to plot the stable and unstable manifolds of
a xed point in the plane. There are a couple of problems that cause the locations
of these numerically calculated curves to be imprecise, especially far out on the
curves. First, the local stable and unstable manifolds are approximated with line
segments. Second, the numerical iteration of points has round-off errors. Finally,
the points on the curve can spread apart as higher iterates are taken. However, the
contracting and expanding nature of the map itself often causes the general shape
of the manifolds to be correct even if one cannot verify their numerical accuracy.

The book [Pa.r89] by Parker and Chua gives a better algorithm that varies the
number of points along the unstable manifold to keep them evenly spaced farther
out on the manifold. Their algorithm also increases the number of points used when
the curve bends more sharply.

To obtain greater accuracy, a shooting method can be used. In this approach,
one checks to see on which side of p the iterates of points x pass. If one point
x1 passes on one side and another point x; passm on the other side, then x3 =
(1/2)(x1 + X2) is a better guess. This new point x3 passes on the opposite side of
p than does one of the original points. Using these two points we can continue to
re ne which point is on the stable manifold.

12.3.2. Basin Boundaries. As we mentioned at the beginning of Section 12.3,
the stable manifold of a saddle xed point can form the boundary of the basin of
attraction of an attracting xed point. We use the Hénon family of maps with
b = 0.3 to illustrate this principle:

Fa,0.3($| y) = (a _ I2 " yum)‘

This family has xed points for a Z '1-32/4 = —0.4225. When the xed points rst
appear (a. z —0.4225), one of the xed points p = (:c_,a:_) is a saddle point and
the other q = (a:.,., a:+) is an attracting xed point. The stable manifold of q is an
open set in the plane, previously called the basin of attraction of the xed point.
For values of a near —0.4225, computer studies show that the basin of attraction
W‘(q;F,,,0_3) has the stable manifold W“’(p;Fa,0_3) as its boundary. See Figure
10. This is true not only for the Hénon family but also for any family of maps of
the plane that creates a pair of xed points as a parameter is varied in the same
manner as the Hénon family. (See [Pat87].)

12.3.3. Stable Manifolds in Higher Dimension. There is a version of the sta-
ble manifold theorem for a saddle periodic point in IR“. Assume that p is a period-q
point for a dilfeomorphism F in IR". Let A = D(F‘7)(p), with eigenvalues Aj for
lgjgn. Assume]/\,-|<1forlgj§'n.,<nand|/\j|>lforn_.,+1$j§_n.
Let vi be a corresponding eigenvector or generalized eigenvector for /\,-. Then the
stable eigenspace at p is the set of vectors

IE"=span{v-"zlgj5n,}={y1v1+---+y.,,,v"’:y1,...,y,,,6lR}.

Translating this subspace to p gives the hyperplane p + IE’. Similarly, the unstable
eigenspace at p is the set of vectors

lE“=span{v-" :n,+l gjgn}

:' {y'n,+1vn‘+1 +ynvn:yn,+1)'-“I!/H GR}:

574 12. Higher Dimensional Maps

1

I I I1 I I I
-3 p 1 3

1-"ii

Tiiii
-3

.n
‘#1

Figure 10. Stable manifold of the saddle xed point p as the boundary of the

basin of attraction of a second xed point q for the Hénon map with a = -0.4

and b = 0.3 '

and the corresponding hyperplane through p is p + IE“.

For a. saddle periodic point, both IE’ and IE" contain nonzero vectors. In R3,
one is a. line and the other is a. plane.

Theorem 12.5. Let F be a C’ di eomorphism on IR" for r 2 1, and let p be
a saddle period-q point. For su iciently small 6 > 0, the local stable and unstable
manifolds of p of size 6, W; (P. F) and W_§‘(p,F), are C’ manifolds given as graphs
over the hyper-planes p -+ IE’ and p + E", respectively. The manifold W§(p,F) is
tangent to the hyperplanep + IE’ at p, and W§‘(p,F) is tangent to p + E“ at p.
These manifolds are locally in'ua1~iant; that is,

F(W§(p.F)) C W§(F(P)=F)
F(Wi‘(P.F)) 3 W3‘(F(P),F)-

The global stable and unstable manifolds are the unions of the iterates of the
local stable and unstable manifolds, respectively:

W"(p,F) = U; F-'=(w;(F'=(p),F)) and

Wm». F) = UXh, F*<Ws'<F-'=<p>. F»-

Example 12.27. Consider the map
F(a:, y, z) = (E1 x, E1 y, 22 + 2:2 + 312),

which has the origin as a. xed point. The matrix of partial derivatives at the origin
is

DF(Q) = ©
_ Q) .

2QQNH-‘ QNH-“Q

12.4. Hyperbolic Toral Automorphisms 575

The map preserves the z-axis, and

00
F0 = 0 .

z 2z

So, F expands the z-axis, and

W“(0,F) = {(0,0,z)T = Z e 112}.

Just as for the map in 1R2 given in Example 12.23, the stable manifold is given by
4

W8(0|F) = {($iy)z) : z = _;(I2 + yz) } '

Exercises 12.3

1. Consider the map

(11) = F (an) = (0.5:: — 43,/3)

in :1 2 1/ '

a. Find the inverse of F.
b. Find the stable and unstable manifolds of the xed point at the origin.

2. Consider the map

2:1 1: 0.5:|:—4y3+8z2

2/1 =F 2; = 22/

Z1 Z 42

a. Find the inverse of F.
b. Find the stable and unstable manifolds of the xed point at the origin.

12.4. Hyperbolic Toral Automorphisms

This section considers a particular type of map on a torus. These maps have in n-
itely many periodic points that are dense in the tota.l space. They can be thought
of as higher dimensional maps related to the doubling map in one dimension, but
they are now have an inverse.

S. Smale popularized these maps as examples with in nitely many periodic
points. These are concrete examples of systems studied extensively by D.V. Anosov,
hence they are often called Anosov diffeomorphisms. They are de ned on a torus
by means of a hyperbolic linear map on a Euclidean space, so they are also called
hyperbolic toral automorphisms.

The torus or two torus T2 is the set of points (:12, y) for which each coordinate
is taken modulo one. You can think of each variable as being an angular variable
modulo 1. This space can be considered as the surface of a donut or bagel by the
following process. Any number 2: taken modulo 1 can be represented by a number
as’ with 0 5 2' < 1. Thus, the set of points in T2 is related to the unit square:

S={(:c,y):0§a:§1, 031/$1}.

576 12. Higher Dimensional Maps

If S is rolled in the a:-direction and the line segment {0} x [0,1] is glued to the line
segment {1} x [0, 1], with each point (0, y) glued to the point (1, y), then the result
is a cylinder. See Figure 11(b). Subsequently, the cylinder is bent in the y-direction
so that the “circle” [0, 1] x {0} is glued to the “circle” [0, 1] x {1}, then the ends
of the cylinder are brought together to form a surface like the surface of a donut.
See Figure 11(c). Because :1: = 0 is glued to a: = 1, a point whose a:-coordinate is
near 0 is close to a. point whose a:-coordinate is near 1. The same thing is true for
points with y near 0 and 1.

i» i__;
i»
I‘

<3) 75' (C)

Figure 11. Construction of a torus: (a) Unit square. (b) Cylinder fonned by
attaching vertical edge together. (c) Torus formed from cylinder by attaching
horizontal edges together.

The hyperbolic toral automorphisms are de ned by means of a. linear map on
1R2. Therefore, we introduce a notation for the “quotient map” 1r that ta.kes a point
in R2 and gives a point in the torus. This quotient map is essential the same as
taking points modulo 1 in each coordinate. If

a:'=:z:(mod1) and y'=y(mod1), then
:z:'=a:+m and y'=y+n,

for two integers m and n (positive, zero, or negative). If such integers exist, then
the points (:i:, y) and (:c', y’) in R2 represent the same point in the torus. Let 1r be
the map from [R2 to T2 such that 1r(:r,y) = 1r(:r’,y’) if and only if there are two
integers m and n such that a:' = :1: +m and y’ = y+n. This projection accomplishes
the gluing we described previously.

We introduce hyperbolic toral automorphisms by means of a speci c example,

(12.1) A=

This matrix has integer entries and a determinant equal to 1. (Examples with a
determinant equal to -1 also work.) Because the determinant is 1, the inverse also
has integer entries:

A_1 __ ( _,1 -12 ) .

The matrix A induces a linear map from IR? to itself:

A = IR’ -» 1122.

12.4. Hyperbolic Toral Automorphisms 577

If 1r(:r:,y) =1r(:1:',y'), with 2:’ =:1:+m and y'=y+n, then

2:’ _ a:+m _ :1: m _ zr 2m+n

Ab’) 'A(v+"l _A(@/l +A("> “Ab/l + (m+")’

and therefore,

"(A(Z/ll ="(A(Z))-
Thus, two points that are identi ed by 1r are taken to two points identi ed by 1r,

so A induces a map FA from T2 to T2. In fact, if p is a point in T2, and (a:,y) is

any point with 1r(a:,y) = p, then cs»

makes the map well de ned. Thus, if :1: and y are taken as modulo one variables,

then to nd the image by FA, we nd A and take each new coordinate modulo
one.

Because A-1 also has integer entries, it also induces a map

FRI : T2 —> T2,

which is the inverse of FA; so, FA is one to one and onto T2.

Figure 12 shows the image of the unit square S by the matrix A. The regions
that are identi ed are labeled with the same letters, so it can be seen that, with
the identi cations, the image A(S) covers the unit square exactly once.

A(S)'
s

D

c
B ,»’ B

nI” A

C

Fig-ure 12. The image of the unit square by the toral automorphism FA

The eigenvalues of A are
>.,=E2 >1 and 0</\2=—3i-_2 —5 <1,

so the matrix is hyperbolic.
For any 2 x 2 matrix A which (i) has integer entries, (ii) has a determinant

equal to i1, and (iii) has eigenvalues with 0 < [}\2| < 1 < |/\||, the induced map

578 12. Higher Dimensional Maps

FA is called a hyperbolic toral automorphism or an Anosov di 'eomorphi.sm. For an
n x n matrix A, the assumptions are basically the same, excepts for (iii') all the
eigenvalues have |/\;| 76 1, at least one eigenvalue has I/\,-| > 1, and at least one
eigenvalue has I/\k| < 1.

We next discuss the periodic points of the map FA. The following theorem
concerns the periodic points of a. hyperbolic toral automorphism.

Theorem 12.6. Assume that FA is a hyperbolic toral automorphism on 'lI‘2. Then
the periodic points of FA are exactly the points 1r(a;,y) where both zr and y are
rational numbers. Therefore, there are in nitely many periodic points, and they are
dense in 'll‘2.
M Flurthermore, all the periodic points are hyperbolic." if p is a period-k point,
then the eigenvalues of D(F§_')(p) are hf and /\'§, -where

,\1=¥>1 and 0<A,=%/5<1.

Proof. Consider the set of rational points that all have the same denominator q:

°"=l(?‘ITL ;TlL’°£’”<*"°§"<"l"

These numbers are not necessarily written in lowest terms, so there can be a com-
mon factor of m and q or n and q. Since

Q 2mj;n
Q
> _j :
/'7 -Q2-Q m+nj’
Q

the map FA takes 1r(Qq) to itself. Since FRI also preserves 1r(Qq), FA restricted
to 1r(Qq) must be one to one and onto (i.e., a permutation of the nite set Qq with
qz elements). Since this set is nite, each point must be periodic, with period less
than or equal to qz.

The union of these sets,

X

UQq={(;) :0§_:r<1, 0$y<1, :r:a.ndya.rerational },
1q=

is dense in the unit square. Therefore, the projection of this union by 1r is dense in
T2. This proves that the periodic points are dense in T2.

The eigenvalues of A are /\1 and /\2, as given in the statement of the theorem.
The derivative of the map FA at all points is A. So, if p is a period-k point, then

0 (Ff,)(p) = A“.

The eigenvalues of A'° are the km power of those of A, and so they are /\'f and /\§
as stated in the theorem.