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3 Lyapunov Exponents and Methods of Their Analysis
(λ 1 , λ 2 ) = (0, −) − stable limiting cycle (one of LEs is equal to zero);
(iii) n = 3. In 3D phase space there are four types of attractors: stable fixed points,
limiting cycles 2D tori and strange attractors. A possible order of the LEs is as
follows:
(λ 1 , λ 2 , λ 3 ) = (−, −, −) − stable fixed point;
(λ 1 , λ 2 , λ 3 ) = (0, −, −) − stable limiting cycle;
(λ 1 , λ 2 , λ 3 ) = (0, 0, −) − stable 2D tori;
(λ 1 , λ 2 , λ 3 ) = (+, 0, −) strange chaotic attractor.
Analytical estimation of the LEs is not possible for majority of the dynamical
systems since it requires getting analytical solutions to the system of evolutionary differential equations. However, nowadays there are efficient numerical
algorithms for their estimation.
KS-entropy describes the maximum (largest) Lyapunov exponent (LLE) allowing for the definition of a velocity responsible for the lack of information on
the system initial state.
3.4 Benettin’s Method [4]
We begin with an example of the numerical estimation of the Kolmogorov entropy
for the case of the Hénon-Heiles model. The numerical computations are carried out
with an accuracy up to 10
−14 and with the help of the method of integration known
as the method of central points (independently of the studies in reference [4]).
We have also investigated numerically the ergodic properties of the dissipative
dynamical systems with a few degrees of freedom using examples of the Lorenz
systems. The Lorenz system possesses LEs spectrum of the type (+, 0, −) with
the same values of all orbits originating in an arbitrary point lying on the attractor.
The obtained results imply that the ergodic property of dynamical systems can be
characterized properly by Les’ spectra. Now we briefly described the used method
of the LEs estimation.
Let the point x 0 belongs to the attractor A of a dynamical system. We follow evolution of the point x 0 along its trajectory. We choose the positive quality ε essentially
less than the investigated dimension of the studied attractor. In addition, we choose
an arbitrary perturbated point ˜
x 0 in a way to satisfy the condition ˜
x 0 − x 0 = ε. We
consider evolutions of the chosen points x 0 and ˜
x 0 within a short time interval T and
we denote the new obtained points associated with T by x 1 and ˜
x 1 , respectively. The
vector x 1 = ˜
x 1 − x 1 stands as the perturbation vector. Hence, one may estimate
the value of λ in the following way:
3 Lyapunov Exponents and Methods of Their Analysis
(λ 1 , λ 2 ) = (0, −) − stable limiting cycle (one of LEs is equal to zero);
(iii) n = 3. In 3D phase space there are four types of attractors: stable fixed points,
limiting cycles 2D tori and strange attractors. A possible order of the LEs is as
follows:
(λ 1 , λ 2 , λ 3 ) = (−, −, −) − stable fixed point;
(λ 1 , λ 2 , λ 3 ) = (0, −, −) − stable limiting cycle;
(λ 1 , λ 2 , λ 3 ) = (0, 0, −) − stable 2D tori;
(λ 1 , λ 2 , λ 3 ) = (+, 0, −) strange chaotic attractor.
Analytical estimation of the LEs is not possible for majority of the dynamical
systems since it requires getting analytical solutions to the system of evolutionary differential equations. However, nowadays there are efficient numerical
algorithms for their estimation.
KS-entropy describes the maximum (largest) Lyapunov exponent (LLE) allowing for the definition of a velocity responsible for the lack of information on
the system initial state.
3.4 Benettin’s Method [4]
We begin with an example of the numerical estimation of the Kolmogorov entropy
for the case of the Hénon-Heiles model. The numerical computations are carried out
with an accuracy up to 10
−14 and with the help of the method of integration known
as the method of central points (independently of the studies in reference [4]).
We have also investigated numerically the ergodic properties of the dissipative
dynamical systems with a few degrees of freedom using examples of the Lorenz
systems. The Lorenz system possesses LEs spectrum of the type (+, 0, −) with
the same values of all orbits originating in an arbitrary point lying on the attractor.
The obtained results imply that the ergodic property of dynamical systems can be
characterized properly by Les’ spectra. Now we briefly described the used method
of the LEs estimation.
Let the point x 0 belongs to the attractor A of a dynamical system. We follow evolution of the point x 0 along its trajectory. We choose the positive quality ε essentially
less than the investigated dimension of the studied attractor. In addition, we choose
an arbitrary perturbated point ˜
x 0 in a way to satisfy the condition ˜
x 0 − x 0 = ε. We
consider evolutions of the chosen points x 0 and ˜
x 0 within a short time interval T and
we denote the new obtained points associated with T by x 1 and ˜
x 1 , respectively. The
vector x 1 = ˜
x 1 − x 1 stands as the perturbation vector. Hence, one may estimate
the value of λ in the following way: