Alligood K., Sauer T., Yorke J.A. Chaos: An Introduction to Dynamical Systems

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0 10 20

Figure 11.26 A Hopf bifurcation in the Lorenz equations.

A bifurcation diagram of the Lorenz equations for

␴

⫽ 10, b ⫽ 8 3, and 0 ⱕ r ⱕ 30,

is shown. At r ⫽ 1, the origin goes from stable to unstable, as two attracting

equilibria bifurcate. Paths of stable orbits are indicated by solid curves, while paths

of unstable orbits are represented by dashed curves and circles. A subcritical Hopf

bifurcation occurs at r ⫽ 24.74, at which point two families of unstable orbits

bifurcate simultaneously as the two attracting equilibria lose their stability. Typical

trajectories in computer simulations then move toward the chaotic attractor, ﬁrst

observed to occur at the crisis value r ⫽ 24.06.

increase and diverge to

⬁

. The behavior at r ⫽ 13.926 is quite complicated—this

is the value for the onset of chaos in the Lorenz equations.

There are chaotic trajectories but no chaotic attractor for 13.926 ⬍ r ⬍

24.06. A chaotic attractor is observed to ﬁrst occur near r ⫽ 24.06. See (Kaplan

and Yorke, 1979) for speciﬁcs of this estimation. At exactly r ⫽ 24.06, a typical

trajectory undergoes a long transient before being attracted by one of the sinks,

as shown in Figure 9.5. Therefore, throughout the parameter range 24.06 ⬍ r ⬍

24.74, the Lorenz system possesses three attractors: one apparently chaotic and

two which are not, the attracting equilibria.

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C HALLENGE 11

☞ C HALLENGE 11

Hamiltonian Systems and the Lyapunov

Center Theorem

E HAVE SEEN that the stability of an equilibrium is often determined by

the Jacobian of the vector ﬁeld. Theorem 7.18 states that when the eigenvalues of

the Jacobian all have negative real part, the equilibrium is asymptotically stable.

When a single eigenvalue has positive real part, the equilibrium is unstable. What

happens when the eigenvalues are pure imaginary (have real part equal to zero)?

In general, an equilibrium is called a center if the Jacobian of the vector ﬁeld has

only pure imaginary eigenvalues.

We want to investigate the behavior of a system near a center. We can start

to answer the question by looking at the linear case. For a two-dimensional linear

system with eigenvalues ⫾bi, the origin is an equilibrium, and all other solutions

are periodic orbits of period 2

␲

 b. For a nonlinear system, it turns out that a

center can be attracting, repelling, or neither. However, there is an important

type of nonlinear system for which we can say that the behavior is stable near a

center.

In Chapter 7, we saw examples of conservative systems such as the un-

damped pendulum. Trajectories of conservative systems are constrained to one

energy level. In other words, energy remains constant along a solution. The

set of equations that govern the motion of the pendulum is one example of a

general class of differential equations called Hamiltonian systems. Speciﬁcally,

for a smooth function H : ⺢

→ ⺢, called the Hamiltonian function or energy

function, Hamilton’s equations are:

x ⫽

⳵

y ⫽⫺

⳵

. (11.19)

Theorem 11.26 guarantees that in a Hamiltonian system, an equilibrium

with pure imaginary eigenvalues will have a neighborhood containing periodic

orbits. Your job in Challenge 11 is to construct a proof of this theorem in the

two-dimensional case.

Theorem 11.26 (Lyapunov Center Theorem.) Assume that 0 is a center

equilibrium of the Hamiltonian system (11.19) and that ⫾bi are simple eigenvalues of

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the Jacobian A of the vector ﬁeld at 0. Assume further that no other eigenvalue of A is

an integer multiple of bi. Then each neighborhood of the center contains periodic orbits,

whose periods approaches 2

␲

 b as they approach the center.

The strategy is to show that the theorem is a consequence of the Hopf Bifur-

cation Theorem. To do this, you will build a one-parameter family of differential

equations for which (11.19) is the member at a ⫽ 0. Then show that a Hopf bi-

furcation occurs at a ⫽ 0, but that no member of the family for a ⫽ 0 has periodic

orbits. The only remaining possibility is that the periodic orbits emanating from

the Hopf bifurcation point must all occur for a ⫽ 0, that is, they are periodic

orbits surrounding the center.

Step 1 To warm up, verify that (11.19) implies that

H(v) ⫽ 0, where

v ⫽ (x, y)and

H(v) ⫽

⳵

⫹

⳵

Therefore the Hamiltonian H is a constant of the motion.

These equations can be generalized to higher even dimensions in which case

H is deﬁned on ⺢

. Then for (x

,...,x

,...,y

) 僆⺢

, the two equations

of system 11.19 become the 2n equations dx

 dt ⫽

⳵

H

⳵

,i ⫽ 1,...,n, and

 dt ⫽⫺

⳵

H

⳵

,i⫽ 1,...,n.

Now we assume the Hamiltonian satisﬁes the hypotheses of the two-

dimensional Lyapunov Center Theorem; that the origin 0 is an equilibrium,

and that ⫾bi are eigenvalues of the Jacobian at 0. Embed the Hamiltonian system

in a one-parameter family of differential equations:

x ⫽ a

⳵

⫹

⳵

y ⫽ a

⳵

⫺

⳵

, (11.20)

where a is a scalar parameter. Notice that the original Hamiltonian system occurs

at a ⫽ 0.

Step 2 Show that 兵(a, 0):a 僆⺢其 is a path of equilibria.

Using the notation gradH for the gradient of H, and letting J ⫽



⫺10



, we can rewrite the system (11.20) in the following matrix form:

˙v ⫽ (aI ⫹ J)gradH(v).

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C HALLENGE 11

Step 3 Show there are no periodic (nonequilibrium) orbits of (11.20) for

a ⫽ 0. [Hint: Let (x

(t),y

(t)) be a solution of (11.20). Verify that (dH dt)(x

(t),

(t)) ⫽ a|gradH|

The goal now is to prove that (11.20) has a Hopf bifurcation at a ⫽ 0,

in which case all bifurcating periodic orbits must occur at a ⫽ 0 (the original

Hamiltonian). To that end, we need to show that a path of eigenvalues of the

linearized system crosses the imaginary axis at ib when a ⫽ 0.

Let f

(v) ⫽ (a ⫹ J)gradH(v). Then D

(0) ⫽ (aI ⫹ J)M, where M is

the 2 ⫻ 2 matrix of second partial derivatives of H (with respect to the phase

variables) evaluated at v ⫽ 0.

Step 4 Show that M is a nonsingular symmetric matrix, and that its in-

verse M

⫺1

is also.

Step 5 Show that

˙v ⫽ JMv (11.21)

is a Hamiltonian system with Hamiltonian function E(v) ⫽

Mv.

Step 6 Let c(a) ⫹ d(a)i be a path of eigenvalues of D

(0) ⫽ (aI ⫹ J)M.

Conclude from Steps 3 and 4 that if a ⫽ 0thenc(a) ⫽ 0.

Step 7 Show that if

␭

is an eigenvalue of (aI ⫹ J)M, then ⫺

␭

is an

eigenvalue of (⫺aI ⫹ J)M. [Hint: If (aI ⫹ J)M ⫺

␭

I is singular, so is M

⫺1

[(aI ⫹

J)M ⫺

␭

M.]

Step 8 Use Steps 6 and 7 to conclude that the eigenvalues of D

(0)

cross the imaginary axis at ⫾bi when a ⫽ 0.

Step 9 Use the weak version of the Hopf Bifurcation Theorem in Remark

11.22 to show that there are periodic orbits of period approaching 2

␲

 b in every

neighborhood of the origin.

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E XERCISES

11.1. Find a saddle-node bifurcation and a period-doubling bifurcation for f

(x) ⫽

a sin x.

11.2. Find a period-doubling bifurcation for g

(x) ⫽⫺x(a ⫹ x). For what parameter

value a does the period-two orbit itself go through a period-doubling?

11.3. Find a period-doubling bifurcation for f

(x) ⫽ x

⫺ ax.

11.4. Let

T (a, x) ⫽



ax if x ⱕ

a(1 ⫺ x)ifx ⬎

Draw the tent map bifurcation diagram for (a, x)in[0, 2] ⫻ [0, 1].

11.5. Find all saddle-node and period-doubling bifurcations of ﬁxed points for the fol-

lowing maps.

(a) f

(x) ⫽ a ⫹ ln x, for x ⬎ 0.

(b) f

(x) ⫽ a ⫺ ln x, for x ⬎ 0.

(x) ⫽ a ln x, for x ⬎ 0.

(d) f

(x) ⫽ a ⫹ x ⫺ e

11.6. Let f

) ⫽ (ax

⫺ x

⫹ ax

). Show that the family f

has two paths of

ﬁxed points (paths in a). Find parameter values at which ⫹1 is an eigenvalue of

(f) evaluated along these paths. Explain what happens to the stability of the

ﬁxed points at these parameter values.

11.7. Find the saddle-node and period-doubling bifurcations of ﬁxed points for the

enon map f

(x, y) ⫽ (a ⫺ x

⫹ 0.3y, x).

11.8. Give an example of a one-parameter family of scalar maps f and a ﬁxed point

x of

such that f



(x) ⫽⫺1 and there is not a period-doubling bifurcation at (a, x).

11.9. (Period-doubling bifurcation.) In this exercise, a speciﬁc condition on derivatives

is shown to guarantee that period-doubling bifurcations are isolated. Assume that

(x) ⫽ f(a, x) is smooth (partial derivatives of order at least 2 exist and are

continuous). Let (

a, x) be a ﬁxed point of f with

(a, x) ⫽⫺1. Since this value

is not ⫹1, there is a path of ﬁxed points (a, x(a)) for a near

a with x(a) ⫽ x.

(a) Show that if

dadx

(

a, x) ⫹

(a, x)

(

a, x) ⫽ 0, (11.22)

then

dadx

(a, x(a)) ⫽ 0.

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E XERCISES

(b) Show that (11.22) implies there is a neighborhood of (a, x) in which there

are no other ﬁxed points (a

ⴱ

) satisfying the period-doubling condition

ⴱ

) ⫽⫺1.

11.10. Unlike a saddle node orbit, a period-doubling bifurcation can be either supercrit-

ical or subcritical. In other words, the bifurcation orbit for a family of scalar maps

can be either attracting or repelling. In this problem, two coefﬁcients in the Taylor

series expansion for the map at the bifurcation value are shown to determine the

stability of the bifurcation orbit.

(a) Let f(x) ⫽ x ⫹ cx

. Show that the ﬁxed point x ⫽ 0 is attracting if c ⬍ 0

and repelling if c ⬎ 0.

(b) Assume that x ⫽ 0 is a ﬁxed point of a scalar map g and that g



(0) ⫽⫺1.

(In this exercise there is no parameter: g is the map at the bifurcation parameter

value.) Let

g(x) ⫽⫺x ⫹ bx

⫹ dx

⫹ higher order terms.

Show that 0 is attracting if b

⫹ d ⬎ 0 and repelling if b

⫹ d ⬍ 0.

11.11. For a real number c, deﬁne the one-parameter family f

(x) ⫽ (x ⫺ a)(2x ⫺ 3a) ⫹

x ⫹ c. For what values of c is there a bifurcation in this family? Describe the

bifurcations and list the bifurcation points (

a, x). Note that c ⫽ 0 corresponds to

generic bifurcations while c ⫽ 0 corresponds to a special one.

11.12. Denote by ˙v ⫽ f(v) the system which gives the supercritical Hopf bifurcation

shown in Figure 11.23. Now draw the corresponding two diagrams for ˙v ⫽⫺f(v).

Which type of Hopf bifurcation is this?

11.13. Consider the Chua circuit equations as deﬁned in (9.6), with parameters set as in

Figure 9.10. Use Exercise T11.21 to ﬁnd the parameter c

for which the system

undergoes a Hopf bifurcation.

11.14. Let

x ⫽ r



⫺ d



⫹ ax ⫺ bxy

y ⫽ r



b ⫺



⫺ cy ⫹ dxy (11.23)

where a, b, c, and d are positive constants, and r is a ﬁxed parameter and x and y

are scalar variables, x ⫽ 0,y⫽ 0. Notice that when r ⫽ 0, the system models a

simple predator-prey interaction, as we described in Chapter 7.

(a) Show that (x, y) ⫽ (

) is an equilibrium for all r.

(b) Show there is a Hopf bifurcation at (

)whenr ⫽ 0.

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☞ L AB V ISIT 11

Iron + Sulfuric Acid −→ Hopf Bifurcation

HIS EXAMPLE of electric current oscillations in an electrochemical reac-

tion exhibits a subcritical Hopf bifurcation and hysteresis. These dynamical ideas

are fairly new compared to the long history of electrochemistry. The ﬁrst elec-

trochemical system showing periodic orbits dates from G. Fechner in 1828, who

found an oscillating current caused by the competing forces of depositing and

dissolving of silver on an iron electrode placed in a silver nitrate solution.

The experiment shown here was designed to explore oscillatory phenomena

in the dissolving of iron in a sulfuric acid solution. A 99.99% iron rod with a

diameter of 3 mm is lowered into 200 ml of the acid. When a potential difference

is applied, the current of the electrochemical system is a measure of the overall

reaction rate between the electrode surface and the electrolytic solution. The

behavior of the current as a function of time shows considerable complication at

certain parameter settings. The so-called electrodissolution problem is far from

completely understood, due to the large number of coupled chemical reactions

involved.

The electrical potential applied to the electrode is used as a bifurcation

parameter in this experiment. Figure 11.27 shows a small sample of the interest-

ing dynamical phenomena in this reaction. Parts (a)–(d) of the ﬁgure show the

measured current (in milliamperes) as a function of time (in seconds), for four

different settings of the potential. The potential increases from (a) to (d). Figure

11.27(a) shows a short transient followed by a constant current, which continues

for the slightly higher potential in (b). In Figure 11.27(c), small irregular oscilla-

tions have developed, which seem aperiodic. Another small change in potential

leads to clear periodic oscillations of larger amplitude in (d). Apparently, a Hopf

bifurcation has occurred.

The time series in Figure 11.28 are from the same electrochemical experi-

ment, except that in this case, the voltage parameter is decreased from part (a)

to (d). Viewed together, the behaviors shown in Figures 11.27 and 11.28 can be

Sazou, D., Karantonis, A., and Pagitsas, M. 1993. Generalized Hopf, saddle-node inﬁnite

period bifurcations and excitability during the electrodissolution and passivation of iron

in a sulfuric acid solution. Int. J. Bifurcations and Chaos 3: 981–997.

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L AB V ISIT 11

Figure 11.27 Current time series for increasing parameter.

Moving to the right along the solid a-axis in the subcritical Hopf bifurcation of

Figure 11.25. (a) Potential is 245 mV. Transient yields to steady state. (b) 246–260

mV show steady state. (c) 261.6 mV, oscillations of small amplitude and period.

(d) 262.6 mV, nine-second oscillations.

interpreted as being the result of a subcritical Hopf bifurcation, as seen earlier in

this chapter.

Referring to Figure 11.25, for large values of the parameter, the periodic

orbit is the only attractor, and so it is globally attracting. As the parameter

moves to the left, the trajectory continues its periodic behavior even as the Hopf

bifurcation point is passed. Only after the saddle-node bifurcation point is passed,

on the far left of Figure 11.25, does the system relax to the steady state. Thus

hysteresis is seen. As the parameter increases along the negative x-axis, we see

one system behavior, while when the parameter is decreased through the same

parameter range, we see a different behavior.

Figure 11.28 demonstrates this type of hysteresis in the electrochemical

experiment. The voltage parameter is decreased from part (a) to part (d), crossing

the same parameter range that was crossed in an increasing manner in Figure

11.27. This time, as expected for a subcritical Hopf bifurcation with hysteresis,

periodic behavior is recorded until the voltage is decreased sufﬁciently, to pass the

saddle-node bifurcation point. After this value, orbits are attracted to the steady

state as shown in Figure 11.28(d).

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Figure 11.28 Current time series for decreasing parameter.

Moving to the left along the upper (periodic) branch of Figure 11.25. (a) Potential

difference is 260 mV, periodic orbit. (b) 253 mV, periodic. (c) 248 mV, periodic.

(d) 245 mV, periodic orbit disappears, convergence to steady state after transient.

The small aperiodic oscillations that are apparent in Figure 11.27(c) are

unexplained by this theory and may be due to the presence of “noise” near the

Hopf bifurcation point. The theory of the Hopf bifurcation is developed for ﬁnite-

dimensional differential equations, and applies to the extent that this experiment

is well-modeled by them. Model inaccuracies may matter at some times more than

others. In particular, they may be magniﬁed at parameter values where stability is

in ﬂux, such as in Figure 11.27(c), thereby limiting the explanatory power of the

model.

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C HAPTER T WELVE

Cascades

ASCADES OF period-doubling bifurcations have been seen in the great majority

of low-dimensional systems that exhibit chaotic behavior. A “cascade” appears

as an inﬁnite sequence of period-doubling bifurcations. A stable periodic orbit is

seen to become unstable as a parameter is increased or decreased and is replaced

by a stable periodic orbit of twice its period. This orbit in turn becomes unstable

and is replaced by a new stable orbit with its period again doubled, and the process

continues through an inﬁnity of such period-doubling bifurcations.

The goal of this chapter is to study properties of cascades, and to understand

what general conditions imply their existence. We continue the study of bifur-

cations from Chapter 11 and develop a picture of families of orbits as connected

sets, where the primary bifurcations are period doublings and saddle nodes. Our

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