Complex dynamical behavior in memristor–capacitor systems

The parallel and series circuits of a Hewlett–Packard memristor and a capacitor are foundational building blocks for realistic memristive circuits. Due to the nonlinearity of the memristor, traditional studies with a single sinusoidal stimulus are limited in their ability to reveal the complex characteristics of memristors. By converting the circuits to an autonomous dimensionless dynamical system, we show that both memristors and capacitors can generate complex dynamical behaviors such as high periodic limit cycles and chaos under a combined periodic stimulus of two sinusoidal signals. To verify the existence of chaos, we present a computer-assisted rigorous proof by a topological horseshoe as well as a circuit implementation. In this way, we uncover a new property of the memristor–capacitor systems: for a typical memristor, e.g., $R_{OFF}/R_{ON}=100$, no matter what values other parameters take, there often exists a periodic stimulus to make the circuits chaotic. Furthermore, under combined excitation of multiple periodic stimulus, the chaos in the systems still exists irrespective of a certain pattern that the frequencies of these stimulus have.