Snap-back repellers and chaos in a class of discrete-time memristor circuits

Abstract

In the last decade the flux-charge analysis method (FCAM) has been successfully used to show that continuous-time (CT) memristor circuits possess for structural reasons first integrals (invariants of motion) and their state space can be foliated in invariant manifolds. Consequently, they display an initial condition dependent dynamics, extreme multistability (coexistence of infinitely many attractors) and bifurcations without parameters. Recently, a new discretization scheme has been introduced for CT memristor circuits, guaranteeing that the first integrals are preserved exactly in the discretization. On this basis, FCAM has been extended to discrete-time (DT) memristor circuits showing that they also are characterized by invariant manifolds and they display extreme multistability and bifurcations without parameters. This manuscript considers the maps obtained via DT-FCAM for a circuit with a flux-controlled memristor and a capacitor and it provides a thorough and rigorous investigation of the presence of chaotic dynamics. In particular, parameter ranges are obtained where the maps have snap-back repellers at some fixed points, thus implying that they display chaos in the Marotto and also in the Li–Yorke sense. Bifurcation diagrams are provided where it is possible to analytically identify relevant points in correspondence with the appearance of snap-back repellers and the onset of chaos. The dependence of the bifurcation diagrams and snap-back repellers upon the circuit initial conditions and the related manifold is also studied.