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

Подождите немного. Документ загружается.

P ERIODIC ORBITS AND L IMIT S ETS

If an orbit not shown limits on this set, it must wind around and around the

connecting arc, with arrows indicating the direction of the ﬂow going in opposite

directions on either side of the arc. This type of trajectory is only possible if every

point on the arc is an equilibrium, a situation ruled out by the hypothesis that

equilibria are isolated.

The structure and dynamics of

␻

-limit sets in higher dimensions include

many more possibilities than in ⺢

. What are the simplest examples of bounded

␻

limit sets that do not contain equilibria and are not periodic orbits? We describe a

differential equation deﬁned on a bounded subset of ⺢

, none of whose trajectories

limit on a periodic orbit or an equilibrium.

E XAMPLE 8.17

Begin with the unit square in the plane. Identify the left and right edges by

gluing them together, and likewise identify the top and bottom edges, as shown

in Figure 8.9(a). The result is a torus, a two-dimensional surface shaped like the

surface of a doughnut, as shown in Figure 8.9 (b).

Also shown in the ﬁgure is a picture of a vector ﬁeld on the square. Each

vector has slope q,whereq is an irrational number. When the corresponding edges

B B

(a) (b) (c)

Figure 8.9 A dense orbit on the torus.

(a) The vector ﬁeld has the same irrational slope at each point in the unit square.

(b) The square is made into a torus by gluing together the top and bottom (marked

A) to form a cylinder and then gluing together the ends (marked B). Each orbit

winds densely around the torus. For each point u of the torus, u belongs to

␻

(u)and

␣

(u). (c) There is no analogue of the Poincar

e-Bendixson Theorem on the torus

because there is no Jordan Curve Theorem on the torus. A simple closed curve that

does not divide the torus into two parts is shown.

348

8.3 PROOF OF THE P OINCAR

E -BENDIXSON T HEOREM

of the square are matched to form the torus, the vectors on either side of the glued

edges match up, so that the vector ﬁeld still is continuous, considered as a vector

ﬁeld on the torus. A solution of the differential equation described by this vector

ﬁeld wraps around the torus and cannot intersect itself. Such a solution is drawn

in Figure 8.9(b).

Given an initial location (x

) on the torus, we describe intersections of

the orbit F(t, (x

)) with the vertical line segment I ⫽ 兵(x

,y):0ⱕ y ⱕ 1其.

First notice that successive intersections are given by (x

, (y

⫹ nq)mod1),n ⫽

1, 2, 3,.... This set of points is dense in the interval I. Hence each point in I

is in the

␻

-limit set

␻

)). Since x

and y

are arbitrary numbers, the entire

torus is in the

␻

-limit set.

The torus is a type of

␻

-limit set that can occur in ⺢

when n ⱖ 3. We

see that the conclusions of the Poincar

e-Bendixson Theorem do not hold on the

torus, even though it is a two-dimensional manifold. Since there is no analogue

of the Jordan Curve Theorem on the torus (see Figure 8.9(c)), we can’t expect

the arguments we made in the planar case to apply here.

349

P ERIODIC ORBITS AND L IMIT S ETS

☞ C HALLENGE 8

Two Incommensurate Frequencies

Form a Torus

HE POINCAR

E-BENDIXSON THEOREM says that possible limit sets for tra-

jectories of an autonomous system in the plane are limited. They can be equilibria,

connecting arcs asymptotic to equilibria, or periodic orbits, which are topological

circles. Once we move out of the plane, however, new phenomena emerge. For

example, for autonomous equations in three dimensions, it is possible for a single

trajectory to wind indeﬁnitely around a two-dimensional torus, ﬁlling up the torus

while never returning to itself. Although this type of complicated behavior is pos-

sible on a two-dimensional torus, it is strictly prohibited in a two-dimensional

plane.

Step 1 Begin with the equation

x ⫹ 4x ⫽ 0 (8.4)

for scalar x. Show that all solutions can be written in the form x(t) ⫽ a sin(2t ⫹ b),

where a and b can be computed from initial conditions.

Step 2 Write (8.4) as a ﬁrst order autonomous system in ⺢

.Inthiscon-

text, the Poincar

e-Bendixson Theorem applies. Show that the geometric form

of the solution curve (x(t),

x(t)) in ⺢

is an ellipse, a periodic orbit. The ellipse

is traversed every

␲

radians, therefore the angular frequency of the solution is

␲

 period ⫽ 2 radians per unit time.

Step 3 Now introduce a second frequency by forcing the system with

natural frequency 2 at a frequency of 1. For example, consider the equation

x ⫹ 4x ⫽ 3sint (8.5)

x(0) ⫽ 0

x(0) ⫽ 3.

Find the solution of this initial value problem.

Step 4 Consider the equation as a ﬁrst-order system in ⺢

.Itisnonau-

tonomous, since the vector ﬁeld depends on t, so Poincar

e-Bendixson does not ap-

ply. Sketch the solution trajectory (x(t),

x(t)) in ⺢

. Is the fact that the trajectory

350

C HALLENGE 8

crosses itself consistent with what we have learned? Prove that x(t) ⫽ sin t ⫹ sin 2t

cannot be the solution of any equation of type

x ⫹ a

x ⫹ bx ⫽ f(x) (8.6)

x(0) ⫽ x

Step 5 More generally, for any real c, show that x(t) ⫽ sin t ⫹ sin ct is the

solution of

x ⫹ c

x ⫽ (c

⫺ 1) sin t (8.7)

x(0) ⫽ 0

x(0) ⫽ c ⫹ 1.

Step 6 In order to construct a phase space for this process, we need to

move to ⺢

, where the system of equations can be made autonomous. Consider

the following autonomous system in ⺢

x ⫽ y (8.8)

y ⫽⫺c

x ⫹ (c

⫺ 1) sin

␪

⫽ 1

with initial conditions x(0) ⫽ 0, y(0) ⫽ c ⫹ 1,

␪

(0) ⫽ 0. Check that the solution

of system (8.8) is

x(t) ⫽ sin t ⫹ sin ct (8.9)

y(t) ⫽ cos t ⫹ c cos ct

␪

(t) ⫽ t.

Step 7 In moving to ⺢

, we have ﬁctionalized the original problem in one

way. The angle

␪

should only have meaning modulo 2

␲

. In other words, the true

phase space for our original system is the inﬁnite slab ⺢

⫻ [0, 2

␲

], with

␪

⫽ 0

and

␪

⫽ 2

␲

identiﬁed together. We want to examine the shape of the solution

trajectory

x(t) ⫽ sin t ⫹ sin ct (8.10)

y(t) ⫽ cos t ⫹ c cos ct

␪

(t) ⫽ t modulo 2

␲

351

P ERIODIC ORBITS AND L IMIT S ETS

of equation (8.7), which we have reconceived as equation (8.8). There are two

cases. The ﬁrst case is when 1 and c are commensurate, meaning that their ratio

is a rational number. Show that in this case, the trajectory is a periodic orbit. If c

is a ratio of the two integers p q in reduced terms, what is the period?

Step 8 To see the torus, suppose that c is an irrational number. Fix an

angle

␪

⫽

␪

. Show that the trajectory at times t ⫽

␪

⫹ 2

␲

k, for each integer k,

lies on an ellipse in the

␪

⫽

␪

plane centered at (sin

␪

, cos

␪

). The ellipses ﬁt

together for 0 ⱕ

␪

ⱕ 2

␲

into a two-dimensional torus.

Postscript. A trajectory is dense in the torus in the irrational case. A trajectory with

two or more incommensurate natural frequencies is called quasiperiodic. The Poincar

emap

deﬁned on a cross-sectional circle is given by rotation through an angle incommensurate

with its circumference. This map was introduced in Chapter 3 as a quasiperiodic circle

map.

352

E XERCISES

8.1. Explain why x(t) ⫽ sin t cannot be the solution of a one-dimensional equation

x ⫽ f(x).

8.2. Find the

␻

-limit sets of the orbits of

r ⫽ r(r ⫺ 1)(3 ⫺ r)

␪

⫽ 1

with initial conditions (r,

␪

) ⫽ (0, 0), (1 2, 0), (1, 0), (2, 0).

8.3. Let A bea2⫻ 2 (real) matrix.

(a) Describe the

␻

-limit sets for orbits of ˙v ⫽ Av.

(b) What condition(s) on a matrix A guarantee(s) that the equilibria of ˙v ⫽ Av

are isolated?

8.4. Assume a differential equation in the plane has a single equilibrium v

, which is

attracting. Describe the possible limit sets of a trajectory v(t) that is bounded for

t ⱖ 0, but does not converge to the single point v

as t →

⬁

8.5. Find the

␣

-limit sets and

␻

-limit sets for the ﬂows depicted in Figure 8.4.

8.6. Assuming equilibria are isolated, which of the following letters can be

␻

-limit sets

of planar ﬂows? Explain. (Compare with Example 8.16.)

ABCDPQY

8.7. Repeat Exercise 8.6, without the assumption that equilibria are isolated.

8.8. Find the

␻

-limit sets for the phase planes of x



⫹ P



(x) ⫽ 0 where the potentials

P(x) are as given in Figure 7.26.

8.9. Let x



⫹ bx



⫹ g(x) ⫽ 0, where b is a positive constant.

(a) Show that this system has no non-equilibrium periodic solutions.

(b) Find the

␻

-limit sets for x



⫹ bx



⫹ P



(x) ⫽ 0, where the potentials P(x)

are as given in Figure 7.26.

8.10. The version of the Poincar

e-Bendixson Theorem stated here assumes that equilibria

are isolated. State a more general theorem in which this hypothesis is omitted.

Describe how the proof must be adapted for this more general case. In particular, if

a limit set of a point contains a periodic orbit, must the periodic orbit be the entire

limit set?

8.11. Use Continuous Dependence on Initial Conditions (Theorem 7.16) to give an

alternate proof of Lemma 8.15.

353

P ERIODIC ORBITS AND L IMIT S ETS

8.12. Consider a vector ﬁeld in the plane.

(a) Assume that p is a point in the plane, and that

␣

(p)and

␻

(p) are the same

periodic orbit. Prove that p lies on this periodic orbit.

(b) For an initial condition p,deﬁnethewidth of the forward (respectively,

backward) orbit of p to be the maximum distance between two points on the

forward (respectively, backward) orbit (this could be inﬁnite). For example, a

point is an equilibrium if and only if its forward orbit has zero width. Prove that

the existence of a periodic orbit of width w implies the existence of another

(forward or backward) orbit of width less than w.

have an equilibrium.

8.13. (a) Show that the

␣

-limit set of the equation v



⫽ f(v)isthe

␻

-limit set of



⫽⫺f(v).

(b) State and prove a theorem analogous to the Poincar

e-Bendixson Theorem

for

␣

⫺limit sets.

8.14. Let v(t) be a trajectory for which |v(t)|

→

⬁

as t

→

⬁

. Show that

␻

)isthe

empty set.

8.15. Sketch a vector ﬁeld in the plane for which the

␻

-limit set of a single trajectory is

two (unbounded) parallel lines.

8.16. Consider the two-dimensional equation ˙v ⫽ f(v), and let L be a Lipschitz constant

for f;thatis,|f(v) ⫺ f(y)| ⱕ L|x ⫺ y|.Letx(t) be a periodic orbit with period T.

Show that T ⱖ

␲

. This theorem is also true in higher dimensions, but the proof is

considerably more difﬁcult. See (Busenberg, Fisher and Martelli, 1989).

8.17. State an analogue of the Diamond Construction Lemma for ﬂows in

⺢

with n ⱖ 3.

8.18. Let L be a transversal to a nonequilibrium point of a planar vector ﬁeld. Show that

if an orbit crosses L at times t

and t

, then it crosses L at most a ﬁnite number of

times between t

and t

354

L AB V ISIT 8

☞ L AB V ISIT 8

Steady States and Periodicity

in a Squid Neuron

TEADY STATES and periodic orbits are the two simplest forms of behavior

for discrete and continuous dynamical systems. For two-dimensional systems of

differential equations, all bounded solutions must converge either to periodic

orbits or to sets containing equilibrium points and perhaps connecting arcs,

according to the Poincar

e-Bendixson Theorem. In this Lab Visit, we will see

evidence of coexisting attractors in a biological system, one a steady state and the

other a periodic orbit.

Periodic orbits are ubiquitous in biological systems, and interruption of pe-

riodicity is often a symptom of malfunction or disease. In physiology alone, a

signiﬁcant proportion of medical problems exhibit abnormal dynamical behav-

ior, including tremor and Parkinsonism, respiratory and cardiac disorders, sleep

apnea, epileptic seizures, migraine, manic-depressive illness, endocrinological ir-

regularities, and hiccups. (For the last, see (Whitelaw et al., 1995).)

In particular, the nervous systems of animals thread together a vast hierarchy

of oscillatory processes. Cyclic behavior goes on at many scales, and neurophysi-

ologists search for mechanisms that form the control point of these processes. The

eventual goal is to ﬁnd the critical parameters that affect the temporal pattern of

a given process, and to determine how to recover from an abnormal setting.

The experiment reported here demonstrated the possibility of intervening

in a biological system to move a trajectory from one of the coexisting attractors

to the other. The system is a neuronal circuit that is well-studied in the scientiﬁc

literature, the giant axon from a squid. The axon is the part of the neuron that is

responsible for transmitting information in the form of pulses to other neurons.

The squid giant axon is often chosen because of its size, and the fact that it can

be made to continue ﬁring in a fairly natural manner after it is dissected from the

squid.

The axon was bathed in an artiﬁcial seawater solution with abnormally

low calcium concentration, held at approximately 22

◦

C. Periodic ﬁring was

Guttman, R., Lewis, S., Rinzel, J., 1980. Control of repetitive ﬁring in squid axon

membrane as a model for a neuroneoscillator. J. Physiology 305:377–395.

355

P ERIODIC ORBITS AND L IMIT S ETS

initiated in the axon by applying a ﬁxed current over a 30 msec interval (1 msec,

or millisecond, is 0.001 seconds). Within this interval, a very brief (0.15 msec)

pulse of current was added on top of the ﬁxed bias current. This small shock was

often able to move the system from periodic to steady behavior.

Figure 8.10 shows four different experimental runs. In the upper half of part

(a), the upper trace shows the step of applied bias current of 30 msec duration.

The lower trace shows the electrical output of the axon, which has a repetitive

spiking behavior. This periodic motion is assumed to be the result of the states of

the axon moving along a periodic trajectory in its phase space.

The lower half of Figure 8.10(a) shows the 30 msec step current with a

dot above it, representing the 0.15 msec added pulse. The pulse, applied after

two oscillations of the system, annihilates the repetitive spiking. The experimen-

tal interpretation is that the perturbation provided by the pulse is sufﬁcient to

move the system trajectory from a periodic orbit to the basin of a nearby stable

equilibrium, where it subsequently stays.

Figure 8.10(b) shows two more experimental runs, and the effect of the

pulse strength on the spike annihilation. The upper pair of traces shows the step

current with added pulse, which stops the periodic spiking, but leaves behind a

low-amplitude transient that eventually damps out completely. The interpretation

here is that the perturbation moved the trajectory less directly to the equilibrium

Figure 8.10 Oscilloscope traces of the ﬁring of a squid axon.

Four experimental runs are shown, each represented by a pair of curves. (a) The upper

pair of traces represents the ﬁxed applied current and the spiking axon output. In

the lower half, the repetitive spiking is interrupted by a large applied pulse (shown

as a single dot) on top of the ﬁxed current. (b) Annihilation of spiking for two

different pulse magnitudes. The degree of damping of the oscillations depends on

the size of the pulse.

356

L AB V ISIT 8

in this case, and that we are seeing the spiraling in to the stable equilibrium. The

lower pair of traces represents the effect of a low pulse strength, which moves the

trajectory into the steady state basin but even farther from the steady state, so

that more time is needed to spiral in to ﬁxed behavior.

The authors of this study compared their results with a computer simu-

lation of the Hodgkin-Huxley equations, which are widely considered to be a

reasonably accurate model of neuronal circuit dynamics. Although they did not

attempt to directly match the experimental conditions of the squid axon to the

parameters of the computer simulation, they exhibit computer trajectories in or-

der to schematically represent possible interpretations using ideas from nonlinear

dynamics.

Figure 8.11 is an illustration of a computer simulation that may explain

the experimental results, at least in a qualitative sense. The curves shown are

trajectories of the voltage level in the simulated neuron plotted against the time

derivative of the voltage on the vertical axis. Part (a) shows two periodic orbits to-

gether with an equilibrium denoted with a plus sign. (None of the three intersect

but they are very close in the picture.) The outside orbit is a stable, attracting pe-

riodic orbit, and the inner orbit repels nearby trajectories. The ⫹ equilibrium lies

Figure 8.11 Diagram of solutions of the Hodgkin-Huxley nerve conduction

equations.

This diagram serves as a suggested qualitative explanation of the experimental

results of Figure 8.10. (a) Two periodic orbits and an equilibrium denoted by a plus

sign. The numbers 1-4 along the outside orbit correspond to parts of the axon spike

as shown in the schematic inset in the upper left corner. (b) Magniﬁcation of part

(a), showing the vicinity of the equilibrium. The brief pulse moves the trajectory

from A to B, which lies in the basin of the stable equilibrium.

357