Hopf Bifurcation and Chaos in Simplest Fractional-Order Memristor-based Electrical Circuit

In this article, we investigate the bifurcation and chaos in a simplest fractional-order memristor-based electrical circuit composed of only three circuit elements: a linear passive capacitor, a linear passive inductor and a non-linear active memristor with two-degree polynomial memristance and a second-order exponent internal state. It is shown that this fractional circuit can exhibit a drastically rich non-linear dynamics such as a Hopf bifurcation, coexistence of two, three and four limit cycles, double-scroll chaotic attractor, four-scroll chaotic attractor, coexistence of one (or two) chaotic attractor with one limit cycle and new chaotic attractor which is not observed in the integer case. Finally, the presence of chaos is confirmed by the application of the recently introduced 0–1 test.