Multistability: Uncovering hidden attractors

Abstract

Abstract. This topical issue collects contributions exemplifying the recent scientific progress in understanding the dynamics of multistable systems. The individual papers focus on different questions of present day interest in theory and applications of systems with multiple attractors. The particular attention is paid to uncovering and characterizing hidden attractors. Both theoretical and experimental studies are presented.Multistability is a system property which refers to systems that are neither stable nor totally instable, but that alternate between two or more mutually exclusive states (attractors) over time [1][2][3][4][5][6][7][8][9][10][11]. Multistable systems are very sensitive towards noise [10,11], initial conditions [2, 4,6] and system parameter [5] so to keep the system on the desired attractor one needs to apply anappropriate controlling scheme [4].Most of the common examples of both chaotic and regular attractors, like that of van der Pol, Beluosov-Zhabotinsky, Lorenz, Rossler, Chua and many others are located in the neighbouhoods of unstable fixed points (its basins of attraction touch unstable fixed points). Such attractors are called the self-exited attractor and can can be easily localized numerically by the standard computational procedure (one can start with the initial conditions in a small neighborhood of the unstable fixed point on unstable manifold and observe how it is attracted) [12,13].Recently, it has been shown that multistability is connected with the occurrence of unpredictable attractors [1][2][3][4][5][6][7][8][9][10][11] which have been called the hidden attractors [14][15][16][17]. An attractor is called hidden attractor if its basin of attraction does not intersect with small neighborhoods of the unstable fixed point, i.e., the basins of attraction of the hidden atttractors do not contain unstable fixed points and are located far away from such points. For example, the hidden attractor is the periodic or chaotic attractor in the system without equilibria or with the only stable equilibrium (a special case of multistability and coexistence of attractors). Various examples of hidden attractors are presented in [18][19][20][21][22][23][24][25].Contrary to the self-exited attractors for numerical localization of hidden attractors it is necessary to develop special analytical-numerical procedures, since there are no similar transient processes leading to such attractors from the neighborhoods of the unstable fixed points [26,27].

A symmetric pair of hyperchaotic attractors

“…When a = 0.25, b = 3, c = 0.05, and d = 0.5, and initial conditions are (0, 1, 3, 18), System (2) is hyperchaotic with Lyapunov exponents (LEs) of (0.1121, 0.0213, 0, −24.9268) and a Kaplan-Yorke dimension of D KY = 3-(λ 1 + λ 2 )/λ 4 ≈ 3.0054. 14,34,[37][38][39][40][41] Here, our computation of LEs is based on the algorithm of Wolf rather from Kuznetsov, 42 which is also a finite-time LE, and the time is 4e7. Figure 1 shows various projections of the coexisting hyperchaotic attractors, which resemble the attractor for System (1).…”

A new three-dimensional chaotic flow with one stable equilibrium: dynamical properties and complexity analysis

“…Another important aspect about our new proposed system is that it is multi-stable. Multistability is an important topic in nonlinear dynamics and chaos [28][29][30][31][32][33]. In some occasions multistability is unwanted, while in some cases it is desired.…”

Megastability: Coexistence of a countable infinity of nested attractors in a periodically-forced oscillator with spatially-periodic damping

“…This system is actually a linear oscillator but with a spatially-dependent nonlinear damping. Multistability is one of the most important phenomena in dynamical systems [67][68][69][70][71][72][73][74][75][76] since it occurs in many areas of science including physics, chemistry, biology, economics, and nature. The attracting state of a multistable system depends on the initial conditions in addition to the usual sensitive dependence on initial conditions that characterizes a chaotic system and precludes long-term predictability.…”