Periodic Orbits in the Muthuswamy-Chua Simplest Chaotic Circuit

In 2010, Muthuswamy and Chua presented an autonomous chaotic circuit using only three elements in series: an inductor, a capacitor and a memristor. This circuit is known as the simplest chaotic circuit and it is determined by a three-dimensional differential system, which depends on the real parameters C, L, α and β. Although the Muthuswamy-Chua system is simpler in formulation than other chaotic systems, its dynamics has proven to be complicated. Here we analytically prove the existence of periodic orbits in this system for suitable choice of the parameter values α and β leading to interesting phenomena as multistability and formation of chaotic attractors. In order to do that, we consider the existence of first integrals, invariant algebraic surfaces and a result from averaging theory. In addition, we relate the obtained results to the memristance and to the physical characteristics of the memristor.