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.
Highlights
Chaos has attracted appreciable attention due to its potential applications in weather forecasting, aircraft control, and secure communications [1]
Due to the technical drawbacks and high cost in fabricating nanoscale devices, most of the memristors employed in these memristive circuits are equivalently implemented by operational amplifiers and analog multipliers [16,17,18] as well as memristive diode bridges cascaded with RC [20, 21], LC [23], or RLC filters [36]
To fully demonstrate the dynamical behaviors, two-dimensional (2D) bifurcation plots depicted by the periodicities of the variable x and dynamical maps described by the largest Lyapunov exponent in the d–f parameter plane are drawn in Figures 3(a) and 3(b), respectively, where the initial conditions (0.1, 0.1, 10− 9, 10− 9) are utilized
Summary
Chaos has attracted appreciable attention due to its potential applications in weather forecasting, aircraft control, and secure communications [1]. Ese gave us the inspiration of that the memristor-based circuit can generate the initial conditions associated behaviors of riddled attraction basin and multistability simultaneously. An interesting work addresses the fact that two oscillators can share a common element to construct a chaotic circuit [1] Inspired by these two considerations, a simple three-element-based memristive circuit including two memristors and one capacitor connected in parallel is presented, which can be regarded as two memristor-capacitor oscillating units with a sharing capacitor. Us, only the corresponding stability distributions for the equilibrium point P+ are shown in Figure 2(b) in the considered model parameter ranges, where the yellow and green regions represent stable node-foci (marked as SNF), and the red region stands for the unstable saddle-foci (marked as USF), respectively. Since the stabilities of the nonzero equilibrium points are associated with the two model parameters, system (8) can display rich dynamical behaviors when varying these model parameters
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