Hidden transient chaotic attractors of Rabinovich–Fabrikant system

Abstract

In [1], it is shown that the Rabinovich-Fabrikant (RF) system admits self-excited and hidden chaotic attractors. In this paper, we further show that the RF system also admits a pair of symmetric transient hidden chaotic attractors. We reveal more extremely rich dynamics of this system, such as a new kind of "virtual saddles".Keywords Hidden transient chaotic attractor; Hidden attractor; Self-excited attractor; Rabinovich-Fabrikant system 1 Introduction Nowadays, the notion of self-excited and hidden attractor introduced by Leonov and Kuznetsov [2,3,4], has become a common subject (see e.g. [5,6,7,8,9,10,11,12,13,14,15,16]). The main characteristic of hidden attractors is that their basins of attraction do not intersect with arbitrary small neighborhood of any equilibrium point, while a basin of attraction of a self-excited attractor is associated with some unstable equilibrium. In this context, stationary points are less important for finding hidden attractors than for finding self-excited attractors. Self-excited attractors can be localized (excited) by standard computational procedures, by starting from a point in some neighborhood of an unstable equilibrium, while for localization of hidden attractors it is necessary to develop special numerical procedures. Some well-known classical chaotic and regular attractors (such as Lorenz 1 , Chen, Rösler, van der Pol, some Sprott systems, etc.) are self-excited attractors, 1 The possible existence in the Lorenz of a hidden chaotic attractor in the Lorenz system is an open problem [4].

Hidden Attractors on One Path: Glukhovsky–Dolzhansky, Lorenz, and Rabinovich Systems

“…The classification of attractors as being hidden or self-excited was introduced by G. Leonov and N. Kuznetsov in connection with the discovery of the first hidden Chua attractor [16,17,[27][28][29] and has captured much attention of scientists from around the world (see, e.g. [30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48]).…”

Creation and circuit implementation of two‐to‐eight–wing chaotic attractors using a 3D memristor‐based system

“…To identify whether the chaotic attractors shown in Figure 2 are self-excited or hidden attractors, [59][60][61][62][63][64][65][66] we need to check the stability of the origin with system parameters varying, and all attractors are hidden if all roots of the characteristic equation corresponding to the matrix in (5) have negative real parts, otherwise the attractors are nonhidden. 63 It is easy to see that the system (3) has the two-to-eight-wing self-excited chaotic attractors under the given system parameters.…”

Complex dynamics, hidden attractors and continuous approximation of a fractional-order hyperchaotic PWC system

“…In general, sustained chaos is numerically indistinguishable from transient chaos, which can persist for a long time (see [16,41]). For example, for q = 0.9725 and b = 1.77, because one of the LEs {0.0058, −0.0000, −0.0042, −0.0447} (measured with a precision of 1E − 5 and from initial conditions (1, 2, 0.0, 0.1)) is 0, the system evolves first for a relatively long time (t ∈ [0, t * ], with t * ≈ 220, with hidden transient chaos, after which the trajectory is attracted by a hidden torus which, as discussed in Subsection 4.2, is characterized by a numerical periodic trajectory (see phase projections (x 1 , x 2 , x 3 ) and time series in Figs.…”