80
3 Lyapunov Exponents and Methods of Their Analysis
(this method was proposed by the authors of the monograph) makes it possible to
calculate the spectrum of Lyapunov exponents and hence to identify phenomena such
as the transition of the system into chaos, hyper-chaos, etc. in a fast and reliable way
[1, 2].
3.2 Largest Lyapunov Exponent (LLE)
Let there be a given dynamical system
˙
x = f (x),
(3.1)
where x stands for N -dimensional state vector.
We choose two closely located phase points x 1 and x 2 , belonging to trajectories
(x 1 (t) and x 2 (t)), and we follow how the distance between the mentioned points
d(t) = |ε (t)| = |x 2 (t) − x 1 (t)|
(3.2)
is changed during the system evolution governed by (3.1).
If nonlinear dynamics of system (3.1) is chaotic, thn d(t) is expected to increase
in the following exponential way
d(t) ≈ d(0)e
kt
.
(3.3)
Equation (3.3) yields an average velocity of the exponential divergence of the
trajectory as follows
k ≈
ln [d(t)/d(0)]
t
,
(3.4)
or equivalently
k = lim
d(0)→0
t→∞
ln [d(t)/d(0)]
t
.
(3.5)
The quantity k is known as the Kolmogorov-Sinai entropy or KS-entropy [3].
Based on the definition of KS-entropy, one may quantify dynamic regimes which
can be either regular or chaotic. In particular, if a system dynamics is either periodic
or quasi-periodic, i.e. d(t) does not increase in time, then KS-entropy equals zero
(k = 0). If in the system a stable periodic non-movable point occurs then d(t) → 0
and k < 0. In the case of chaotic system dynamics the KS-entropy k > 0.
3 Lyapunov Exponents and Methods of Their Analysis
(this method was proposed by the authors of the monograph) makes it possible to
calculate the spectrum of Lyapunov exponents and hence to identify phenomena such
as the transition of the system into chaos, hyper-chaos, etc. in a fast and reliable way
[1, 2].
3.2 Largest Lyapunov Exponent (LLE)
Let there be a given dynamical system
˙
x = f (x),
(3.1)
where x stands for N -dimensional state vector.
We choose two closely located phase points x 1 and x 2 , belonging to trajectories
(x 1 (t) and x 2 (t)), and we follow how the distance between the mentioned points
d(t) = |ε (t)| = |x 2 (t) − x 1 (t)|
(3.2)
is changed during the system evolution governed by (3.1).
If nonlinear dynamics of system (3.1) is chaotic, thn d(t) is expected to increase
in the following exponential way
d(t) ≈ d(0)e
kt
.
(3.3)
Equation (3.3) yields an average velocity of the exponential divergence of the
trajectory as follows
k ≈
ln [d(t)/d(0)]
t
,
(3.4)
or equivalently
k = lim
d(0)→0
t→∞
ln [d(t)/d(0)]
t
.
(3.5)
The quantity k is known as the Kolmogorov-Sinai entropy or KS-entropy [3].
Based on the definition of KS-entropy, one may quantify dynamic regimes which
can be either regular or chaotic. In particular, if a system dynamics is either periodic
or quasi-periodic, i.e. d(t) does not increase in time, then KS-entropy equals zero
(k = 0). If in the system a stable periodic non-movable point occurs then d(t) → 0
and k < 0. In the case of chaotic system dynamics the KS-entropy k > 0.