3.3 Spectrum of Lyapunov Exponents
81
3.3 Spectrum of Lyapunov Exponents
Le’s spectrum gives a possibility for a qualitative estimation of the features of a local
attractor/replier stability.
We consider the phase trajectory x(t) of the dynamical system (3.1) starting from
the point x(0) as well as its neighbourhood trajectory
x 1 (t) = x(t) + ε(t).
(3.6)
Consider the function
λ [ε (0)] = lim
t→∞
ln
|ε(t)|
|ε(0)|
t
(3.7)
defined on the vectors of initial displacement ε (0) such that |ε (0)| = ε, where
ε → 0.
All possible rotations of the vector of initial displacement with regard to n directions in the N -dimensional phase space imply a jump like dynamics of the function
(3.7) tending to the finite number of the values λ 1 , λ 2 , λ 3 , . . . , λ n . The mentioned
values of λ are known as the Lyapunov exponents. The positive (negative) values of
LEs present a measure of the average exponential divergence (convergence) of the
neighbourhood trajectories.
A sum of the LEs describes an average divergence of the flow of the phase trajectories which in the case of a dissipative system should be negative, i.e. the system
has an attractor. However, owing to the numerical results, some dissipative systems
exhibit LEs being invariant with respect to all chosen initial conditions. This means
that the LEs can be used to quantify properties of attractors.
The LEs are presented in the decreased order. For instance, the symbols (+, 0, −)
mean that a certain attractor in the 3D state-space exhibits an exponential elongation
along one of the directions, in the second direction the phase flow has a neutral
stability, and finally, in the remaining third direction the trajectories are compressed
in an exponential way. It should be emphasized that attractors other than stable
stationary points always have one LE equal to zero. It means that in an average sense
the points on a trajectory cannot be either divergent or convergent.
In what follows, we consider a correspondence of the LEs with the features and
types of attractors:
(i) n = 1 (n stands for dimension of a phase space). Only stable non-movable
point (node or focus) can be an attractor. In this case, there is only one LE
λ 1 = (−) having a negative value.
(ii) n = 2. In 2D dynamical systems there are two types of attractors, i.e. either
stable fixed points or limiting cycles. The LEs follow
(λ 1 , λ 2 ) = (−, −) − stable fixed point;
81
3.3 Spectrum of Lyapunov Exponents
Le’s spectrum gives a possibility for a qualitative estimation of the features of a local
attractor/replier stability.
We consider the phase trajectory x(t) of the dynamical system (3.1) starting from
the point x(0) as well as its neighbourhood trajectory
x 1 (t) = x(t) + ε(t).
(3.6)
Consider the function
λ [ε (0)] = lim
t→∞
ln
|ε(t)|
|ε(0)|
t
(3.7)
defined on the vectors of initial displacement ε (0) such that |ε (0)| = ε, where
ε → 0.
All possible rotations of the vector of initial displacement with regard to n directions in the N -dimensional phase space imply a jump like dynamics of the function
(3.7) tending to the finite number of the values λ 1 , λ 2 , λ 3 , . . . , λ n . The mentioned
values of λ are known as the Lyapunov exponents. The positive (negative) values of
LEs present a measure of the average exponential divergence (convergence) of the
neighbourhood trajectories.
A sum of the LEs describes an average divergence of the flow of the phase trajectories which in the case of a dissipative system should be negative, i.e. the system
has an attractor. However, owing to the numerical results, some dissipative systems
exhibit LEs being invariant with respect to all chosen initial conditions. This means
that the LEs can be used to quantify properties of attractors.
The LEs are presented in the decreased order. For instance, the symbols (+, 0, −)
mean that a certain attractor in the 3D state-space exhibits an exponential elongation
along one of the directions, in the second direction the phase flow has a neutral
stability, and finally, in the remaining third direction the trajectories are compressed
in an exponential way. It should be emphasized that attractors other than stable
stationary points always have one LE equal to zero. It means that in an average sense
the points on a trajectory cannot be either divergent or convergent.
In what follows, we consider a correspondence of the LEs with the features and
types of attractors:
(i) n = 1 (n stands for dimension of a phase space). Only stable non-movable
point (node or focus) can be an attractor. In this case, there is only one LE
λ 1 = (−) having a negative value.
(ii) n = 2. In 2D dynamical systems there are two types of attractors, i.e. either
stable fixed points or limiting cycles. The LEs follow
(λ 1 , λ 2 ) = (−, −) − stable fixed point;