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when the system control parameter reaches the critical value. Three

types of local bifurcations are discerned: period-doubling, quasi-peri-

odicity, and intermittency. Period-doubling starts with a limit cycle at

some value of the system control parameter. With further change of

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Nonlinear Dynamical Systems

this parameter, the trajectory period doubles and doubles until it

becomes infinite. This process was proposed by Landau as the main

turbulence mechanism. Namely, laminar flow develops oscillations at

some sufficiently high velocity. As velocity increases, another (incom-

mensurate) frequency appears in the flow, and so on. Finally, the

frequency spectrum has the form of a practically continuous band. An

alternative mechanism of turbulence (quasi-periodicity) was proposed

by Ruelle and Takens. They have shown that the quasi-periodic

trajectories confined on the torus surface can become chaotic due to

high sensitivity to the input parameters. Intermittency is a broad

category itself. Its pathway to chaos consists of a sequence of periodic

and chaotic regions. With changing the control parameter, chaotic

regions become larger and larger and eventually fill the entire

space.

In the global bifurcations, the trajectories approach simple attract-

ors within some control parameter range. With further change of the

control parameter, these trajectories become increasingly complicated

and in the end, exhibit chaotic motion. Global bifurcations are parti-

tioned into crises and chaotic transients. Crises include sudden

changes in the size of chaotic attractors, sudden appearances of the

chaotic attractors, and sudden destructions of chaotic attractors and

their basins. In chaotic transients, typical trajectories initially behave

in an apparently chaotic manner for some time, but then move to

some other region of the phase space. This movement may asymptot-

ically approach a non-chaotic attractor.

Unfortunately, there is no simple rule for determining the condi-

tions at which chaos appears in a given flow. Moreover, the same

system may become chaotic in different ways depending on its par-

ameters. Hence, attentive analysis is needed for every particular

system.

7.6 MEASURING CHAOS

As it was noticed in in Section 7.1, it is important to understand

whether randomness of an empirical time series is caused by noise or

by the chaotic nature of the underlying deterministic process. To

address this problem, let us introduce the Lyapunov exponent. The

major property of a chaotic attractor is exponential divergence of its

84 Nonlinear Dynamical Systems

nearby trajectories. Namely, if two nearby trajectories are separated

by distance d0 at t Â¼ 0, the separation evolves as

d(t) Â¼ d0 exp (lt) (7:6:1)

The parameter l in (7.6.1) is called the Lyapunov exponent. For the

rigorous definition, consider two points in the phase space, X0 and

X0 Ã¾ Dx0 , that generate two trajectories with some flow (7.3.2). If the

function Dx(X0 , t) defines evolution of the distance between these

points, then

1 jDx(X0 , t)j

l Â¼ lim ln , t ! 1, Dx0 ! 0 (7:6:2)

jDx0 j

t

When l < 0, the system is asymptotically stable. If l Â¼ 0, the system

is conservative. Finally, the case with l > 0 indicates chaos since the

system trajectories diverge exponentially.

The practical receipt for calculating the Lyapunov exponent is as

follows. Consider n observations of a time series x(t): x(tk ) Â¼ xk , k Â¼ 1,

. . . , n. First, select a point xi and another point xj close to xi . Then

calculate the distances

d0 Â¼ jxi Ã€ xj j, d1 Â¼ jxiÃ¾1 Ã€ xjÃ¾1 j, . . . , dn Â¼ jxiÃ¾n Ã€ xjÃ¾n j (7:6:3)

If the distance between xiÃ¾n and xjÃ¾n evolves with n accordingly with

(7.6.1), then

1 dn

l(xi ) Â¼ ln (7:6:4)

n d0

The value of the Lyapunov exponent l(xi ) in (7.6.4) is expected to be

sensitive to the choice of the initial point xi . Therefore, the average

value over a large number of trials N of l(xi ) is used in practice

1X N

lÂ¼ l(xi ) (7:6:5)

N iÂ¼1

Due to the finite size of empirical data samples, there are limitations

on the values of n and N, which affects the accuracy of calculating the

Lyapunov exponent. More details about this problem, as well as other

chaos quantifiers, such as the Kolmogorov-Sinai entropy, can be

found in [5] and references therein.

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Nonlinear Dynamical Systems

The generic characteristic of the strange attractor is its fractal

dimension. In fact, the non-integer (i.e., fractal) dimension of an

attractor can be used as the definition of a strange attractor. In

Chapter 6, the box-counting fractal dimension was introduced.

A computationally simpler alternative, so-called correlation dimen-

sion, is often used in nonlinear dynamics [3, 5].

Consider a sample with N trajectory points within an attractor. To

define the correlation dimension, first the relative number of points

located within the distance R from the point i must be calculated

X N

1

pi (R) Â¼ u(R Ã€ jxj Ã€ xi j) (7:6:6)

N Ã€ 1 j Â¼ 1, j 6Â¼ i

In (7.6.6), the Heaviside step function u equals

0, x < 0

uÂ¼ (7:6:7)

1, x ! 0

Then the correlation sum that characterizes the probability of finding

two trajectory points within the distance R is computed

1X N

C(R) Â¼ pi (R) (7:6:8)

N iÂ¼1

It is assumed that C(R) $ RDc . Hence, the correlation dimension Dc

equals

Dc Â¼ lim [ ln C(R)= ln R] (7:6:9)

R!0

There is an obvious problem of finding the limit (7.6.9) for data

samples on a finite grid. Yet, plotting ln[C(R)] versus ln(R) (which

is expected to yield a linear graph) provides an estimate of the

correlation dimension.

An interesting question is whether a strange attractor is always

chaotic, in other words, if it always has a positive Lyapunov expo-

nent. It turns out there are rare situations when an attractor may be

strange but not chaotic. One such example is the logistic map at the

period-doubling points: Its Lyapunov exponent equals zero while the

fractal dimension is about 0.5. Current opinion, however, holds that

the strange deterministic attractors may appear in discrete maps

rather than in continuous systems [5].

86 Nonlinear Dynamical Systems

7.7 REFERENCES FOR FURTHER READING

Two popular books, the journalistic report by Gleick [8] and the

â€˜â€˜first-handâ€™â€™ account by Ruelle [9], offer insight into the science of

chaos and the people behind it. The textbook by Hilborn [5] provides

a comprehensive description of the subject. The interrelations be-

tween the chaos theory and fractals are discussed in detail in [10].

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