Dynamics of a fractional order locally-active Memristor with applications in oscillatory systems*

Abstract

A non-volatile fractional-order Memristor, with two asymptotically stable equilibrium points and locally-active characteristic is presented. A fractional-order small-signal equivalent circuit is used to describe the memristor's characteristics at an operating point within a locally-active region. Via the equivalent circuit, the memristor is shown to possess an edge of chaos within a voltage range; when connected in series with an inductor, it generates periodic oscillation about the locally-active operating point in the edge of chaos. The oscillating frequency and the external inductance are determined by the small-signal circuit's admittance. Adding external capacitors and inductors in series/parallel with the memristor, three- and four-dimensional circuits are realized which generates chaotic oscillations. Analysis of the resulting three- and four-dimensional circuits are carried out at the memristor's equilibrium point, the effects of the memristor's parameters and the fractional order indexes of the added components on the system dynamics are also investigated using Lyapunov and bifurcation analysis. Numerical simulations show the versatility of the memristor for usages in oscillatory systems.

On two-parameter bifurcation and analog circuit implementation of a Chameleon chaotic system

“…Since the discovery of the butterfly effect in nonlinear system, the research on chaotic oscillations and chaos mechanism has become a hotspot [1][2][3]. For most autonomous ordinary differential equations (ODE) chaotic systems, the chaotic motions can be triggered by choosing an initial point in the small neighborhood of the systems' unstable equilibrium (if exists) since these motions are common self-excited oscillations [4][5][6]. The generated attractors in these systems usually have relatively large basins of attraction wrapping up at least one unstable equilibrium, such as the scroll-type Chua's attractor [7][8][9], the butterfly-type Lorenz attractor [10][11][12], and many other chaotic attractors in various nonlinear circuits and systems [13][14][15].…”