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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