The Nature of Attractor Basins in Multistable Systems

Abstract

In systems that exhibit multistability, namely those that have more than one coexisting attractor, the basins of attraction evolve in specific ways with the creation of each new attractor. These multiple attractors can be created via different mechanisms. When an attractor is formed via a saddle-node bifurcation, the size of its basin increases as a power-law in the bifurcation parameter. In systems with weak dissipation, the basins of low-order periodic attractors increase linearly, while those of high-order periodic attractors decay exponentially as the dissipation is increased. These general features are illustrated for autonomous as well as driven mappings. In addition, the boundaries of the basins can also change from being smooth to fractal when a new attractor appears. Transitions in the basin boundary morphology are reflected in abrupt changes in the dependence of the uncertainty exponent on the bifurcation parameter.

Organization and identification of solutions in the time-delayed Mackey-Glass model

“…The maps of initial conditions that result in different periodic solutions were found to exhibit complex structures, which are not uncommon in delayed systems. 39 A full characterization of the complex organization of these solutions in the system's phase space is an open issue which deserves further research.…”

Multimedia Security Application of a Ten-Term Chaotic System without Equilibrium

“…It is noted that the concept of hidden attractors has been suggested in connection with the occurrence of unpredictable attractors in multistable systems [21]. Researchers have shown that multistability is connected with the occurrence of unpredictable attractors [21][22][23][24][25][26][27][28][29][30]. Recently, hidden attractor has been investigated in numerous systems such as Chua system [19], drilling system [31], Lorenz-like system [32], Goodwin oscillator [33], electromechanical systems [34], twodimensional maps [35], phase-locked loop circuits [36], and Rabinovich-Fabrikant system [37].…”

Delay-coupled discrete maps: Synchronization, bistability, and quasiperiodicity

“…Earlier studies of either synchronization or amplitude death which have largely focused on time-continuous dynamical systems, namely flows. An advantage in studying mappings is that some aspects of the analysis become simpler, particularly with reference to multistability [10]. Although delay increases the dimensionality of the problem, unlike the case of flows, the system remains finite-dimensional in discrete delay mappings.…”