Riddled Attraction Basin and Multistability in Three-Element-Based Memristive Circuit

Abstract

By coupling a diode bridge-based second-order memristor and an active voltage-controlled memristor with a capacitor, a three-element-based memristive circuit is synthesized and its system model is then built. The boundedness of the three-element-based memristive circuit is theoretically proved by employing the contraction mapping principle. Besides, the stability distributions of equilibrium points are theoretically and numerically expounded in a 2D parameter plane. The results imply the memristive circuit has a zero unstable saddle focus and a pair of nonzero stable node-foci or unstable saddle-foci depending on the considered parameters. The dynamical behaviors include point attractor, period, chaos, coexisting bifurcation mode, period-doubling bifurcation route, and crisis scenarios, which are explored using some common dynamical methods. Of particular concern, riddled attraction basins and multistability are uncovered under two sets of specified model parameters nearing the tiny neighborhood of crisis scenarios by local attraction basins and phase plane plots. The riddled attraction basins with island-like structure demonstrate that their dynamical behaviors are extremely sensitive to the initial conditions, resulting in the coexistence of limit cycles with period-2 and period-6, as well as the coexistence of period-1 limit cycles and single-scroll chaotic attractors. Moreover, a feasible on-breadboard hardware circuit is manually made and the experimental measurements are executed, upon which phase plane trajectories for some discrete model parameters are captured to further confirm the numerically simulated ones.

Multistability route in a PWL multi-scroll system through fractional-order derivatives

“…To ilustrate the described behavior the corresponding basins of attraction are calculated, four of which are shown in Figure 7, each one considering different integration values. This allows us to see the deformation of the basin of attraction of the three-state system into a riddled basin of attraction [50], while as the order of integration increases, the double-scroll states become more probable until their regions of attraction collide and the global basin of the monostable three-state system emerges. The basins of attraction shown in Figure 7 are based on the discretization of the attractors shown in Figure 4, in which all possible behaviors of the multi-scroll system are represented when modeled with fractional order derivatives.…”

Discrete Memristor and Discrete Memristive Systems

“…Owing to their properties, including memory and intrinsic nonlinearity, memristors have many potential applications in fields such as flash memory [ 4 ], reconfigurable computing [ 5 ], logic circuits [ 6 ], synapses [ 7 ], neural networks [ 8 , 9 ], solving mazes [ 10 ] and oscillation circuits [ 11 ]. Among those, memristor-based nonlinear systems or circuits have rich dynamics such as multistability and chaos [ 12 , 13 ].…”