Finite-time Lyapunov dimension and hidden attractor of the Rabinovich system

The Rabinovich system, describing the process of interaction between waves in plasma, is considered. It is shown that the Rabinovich system can exhibit a {hidden attractor} in the case of multistability as well as a classical {self-excited attractor}. The hidden attractor in this system can be localized by analytical-numerical methods based on the {continuation} and {perpetual points}. For numerical study of the attractors' dimension the concept of {finite-time Lyapunov dimension} is developed. A conjecture on the Lyapunov dimension of self-excited attractors and the notion of {exact Lyapunov dimension} are discussed. A comparative survey on the computation of the finite-time Lyapunov exponents by different algorithms is presented and an approach for a reliable numerical estimation of the finite-time Lyapunov dimension is suggested. Various estimates of the finite-time Lyapunov dimension for the hidden attractor and hidden transient chaotic set in the case of multistability are given.

(1) This system describes [1,2] the interaction of three resonantly coupled waves two of them being parametrically excited. Here the parameter h is proportional to the pumping amplitude and the parameters ν 1 and ν 2 are normalized dumping decrements.
Note that since parameters ν 1 , ν 2 , h are positive then parameter a is negative. Also, due to (3) one can obtain the following relation between σ, a and r: σ = −ar.
For system (1) with fixed parameters ν 1 = 1, ν 2 = 4 and increasing h it is possible to observe [2] the classical scenario of transition to the chaos (via homoclinic and Andronov-Hopf subcritical bifurcations) similar to the scenario in the Lorenz system. Thereby for 4.84 h 13.4 system (1) has a self-excited chaotic attractor. The same scenario can be obtained for system (2) when parameters b > 0 and a < 0 are fixed and r is increasing. Let us note that in [12,13] system (2) with a > 0 was studied and it was shown numerically the existence of the hidden attractor. In this report we localize a hidden chaotic attractor in system (2) with a < 0.

A. Attractors from the computational point of view
An oscillation in dynamical system can be easily localized numerically if the initial conditions from its open neighborhood lead to the long-time behavior that approaches the oscillation. Thus, from a computational point of view, it is natural to suggest the following classification of attractors, based on the simplicity of finding the basin of attraction in the phase space: Definition [14][15][16][17] An attractor is called a self-excited attractor if its basin of attraction intersects with any open neighborhood of a stationary state (an equilibrium), otherwise it is called a hidden attractor.
For a self-excited attractor its basin of attraction is connected with an unstable equilibrium and, therefore, self-excited attractors can be localized numerically by the standard computational procedure in which after a transient process a trajectory, started in a neighborhood of an unstable equilibrium (from a point of its unstable manifold), is attracted to the state of oscillation and traces it. Thus self-excited attractors can be easily visualized.
For a hidden attractor its basin of attraction is not connected with equilibria. The hidden attractors, for example, are the attractors in the systems with no equilibria or with only one stable equilibrium (a special case of multistability -multistable systems and coexistence of attractors). One of the first well-known problems of analyzing hidden periodic oscillations arose in connection with the second part of Hilbert's 16th problem (1900) [18] on the number and mutual disposition of limit cycles in two-dimensional polynomial systems (see, e.g., [19,20]). Later the study of hidden attractors arose in connection with various fundamental problems and applied models (see, e.g., [21][22][23][24]). Recent examples of hidden periodic oscillations and hidden chaotic attractors can be found in . Note that while coexisting self-excited attractors can be found by the standard computational procedure, there is no regular way to predict the existence or coexistence of hidden attractors.

B. Hidden attractor visualization
One of the effective methods for numerical localization of hidden attractors in multidimensional dynamical systems is based on a homotopy and numerical continuation: it is necessary to construct a sequence of similar systems such that for the first (starting) system the initial point for numerical computation of oscillating solution (starting oscillation) can be obtained analytically, e.g, it is often possible to consider the starting system with self-excited starting oscillation. Then the transformation of this starting oscillation is tracked numerically in passing from one system to another. The last system corresponds to the system in which hidden attractor is searched.
Let us fix parameter b = 1 and construct on the plane (a, r) a line segment, intersecting a boundary of the domain of stability of the equilibria S 1,2 with the end point P 1 (r = 6.8, a = −0.5). The eigenvalues of equilibria S 0,1,2 of system (2) with parameters P 1 are the following: S 0 : 2.5576, −1, −7.5576, S 1,2 : − 0.0215 ± 3.6026i, −5.957, i.e. the equilibria S 1,2 become stable focus-nodes. Let us choose the point P 0 (r = 7.1, a = −0.5) as the initial point of the line segment. This point corresponds to the parameters for which in system (2) there exists a self-excited attractor, such that it can be computed by the standard procedure. Then for the considered line segment a sufficiently small partition step is chosen and a chaotic attractor in the phase space of system (2) at each iteration step of the procedure is computed. The last computed point at each step is used as the initial point for the computation of the next step.
In our experiment the length of the line segment is 0.02 and there are 15 iterations. At each iteration the largest Lyapunov exponent (LLE) and the Lyapunov dimension (LD) [11,46,47] are computed 1 .
Here for the selected path and selected partition it is possible to visualize a hidden attractor of system (2) (see Fig. 1). (see, e.g., its MatLab implementation in [49]) fails to compute the correct values (see discussion and examples in [47,50]). The existence of different definitions, computational methods, and related assumptions led to the appeal "Whatever you call your exponents, please state clearly how are they being computed" [51].
Here we use an algorithm based on SVD decomposition.