Arnold tongues replace period-doubling cascades in a memristor circuit.

Abstract

The memristor, theorized by Leon Chua in 1971, functions as a fundamental electronic component, directly linking electric charge and magnetic flux. As a result of their nonlinear characteristics, memristive circuits generally exhibit chaotic attractors in addition to periodicity. In this work, we consider the Muthuswamy-Chua system, a chaotic circuit consisting of an inductor, a capacitor, and a memristor. In two-dimensional parameter spaces, this system displays periodic shrimp-shaped domains, which are periodic windows surrounded by chaotic regions. Applying a weak harmonic perturbation, we observe the replacement of periodicity by quasiperiodicity, followed by the formation of Arnold tongues (periodic structures) acting as boundaries between quasiperiodic and chaotic regions. Moreover, as an additional result of the perturbation, we identify another interesting feature: the metamorphosis of quasiperiodic shrimp-shaped domains into Arnold tongues. In both instances, Arnold tongues emerge in regions previously dominated by period-doubling cascades under unperturbed conditions.