Hidden attractor and its dynamical characteristic in memristive self-oscillating system

Abstract

The classical attractors, defined as self-excited attractors, such as Lorenz attractor, Rssler attractor, Chua's attractor and many other well-known attractors, are all excited from unstable index-2 saddle-foci, namely, an attractor with an attraction basin corresponds to an unstable equilibrium. A new type of attractors, defined as hidden attractors, was first found and reported in 2011, whose attraction basin does not intersect with small neighborhoods of the equilibria of the system. Due to the existences of hidden attractors, some particular dynamical systems associated with line equilibrium, or no equilibrium, or stable equilibrium have attracted much attention recently. Additionally, by introducing memristors into existing oscillating circuits or substituting nonlinear resistors in classical chaotic circuits with memristors, a variety of memristor based chaotic and hyperchaotic circuits are simply established and has been broadly investigated in recent years. Motivated by these two considerations, in this paper, we present a novel memristive system with no equilibrium, from which an interesting and striking phenomenon of coexistence of the behaviors of hidden multiple attractors and the corresponding multistability is perfectly demonstrated by numerical simulations and experimental measurements. According to a newly proposed circuit realization scheme, a new type of four-dimensional memristive self-oscillated system is easily implemented by directly replacing a linear coupling resistor in an existing three-dimensional self-oscillated system circuit with a voltage-controlled memristor. The proposed system has no equilibrium, but can generate various hidden attractors including periodic limit cycle, quasi-periodic limit cycle, chaotic attractor, and coexisting attractors and so on. Based on bifurcation diagram, Lyapunov exponent spectra, and phase portraits, complex hidden dynamics with respect to a system parameter of the memristive self-oscillated system are studied. Specially, when different initial conditions are used, the system displays the coexistence phenomenon of chaotic attractors with different topological structures or quasi-periodic limit cycle and chaotic attractor, as well as the phenomenon of multiple attractors of quasi-periodic limit cycle and chaotic attractors with multiple topological structures. The results imply that some coexisting hidden multiple attractors reflecting the emergences of multistability can be observed in the proposed memristive self-oscillated system, which are well illustrated by several conventional dynamical analysis tools. Based on PSIM circuit simulation model, the memristive self-oscillated system is easily made in at a hardware level on a breadboard and two kinds of dynamical behaviors of coexisting hidden multiple attractors are captured in hardware experiments. Hardware experimental measurements are consistent with numerical simulations, which demonstrates that the proposed memristive self-oscillated system has very abundant and complex hidden dynamical characteristics.