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

Nonlinear Dynamical Systems 83

nearby trajectories. Namely, if two nearby trajectories are separated

by distance d

0

at t ¼ 0, the separation evolves as

d(t) ¼ d

0

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, X

0

and

X

0

þ Dx

0

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

function Dx(X

0

, t) defines evolution of the distance between these

points, then

l ¼ lim

1

t

ln

jDx(X

0

,t)j

jDx

0

j

,t!1, Dx

0

! 0(7:6:2)

When l < 0, the system is asymptotically stable. If l ¼ 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(t

k

)¼x

k

,k¼1,

..., n. First, select a point x

i

and another point x

j

close to x

i

. Then

calculate the distances

d

0

¼jx

i

x

j

j,d

1

¼jx

iþ1

x

jþ1

j, ...,d

n

¼jx

iþn

x

jþn

j (7:6:3)

If the distance between x

iþn

and x

jþn

evolves with n accordingly with

(7.6.1), then

l(x

i

) ¼

1

n

ln

d

n

d

0

(7:6:4)

The value of the Lyapunov exponent l(x

i

) in (7.6.4) is expected to be

sensitive to the choice of the initial point x

i

. Therefore, the average

value over a large number of trials N of l(x

i

) is used in practice

l ¼

1

N

X

N

i ¼1

l(x

i

)(7:6:5)

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.

84 Nonlinear Dynamical Systems

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