Topological Analysis of Chaotic Solution of a Three-Element Memristive Circuit

In classical electronics, there are three passive circuit elements: the resistor, the capacitor and the inductor. In 1971, Leon Chua introduced a fourth “missing” element which he named a memristor — for memory resistor — by using symmetry arguments [Chua, 1971]. Chua also derived the properties of this element. However, it is only in 2008 that Strukov and co-workers [Strukov et al., 2008] found a memristance arising in a nanoscale system in which solid-state electronic and ionic transport are coupled under an external bias voltage. In 1976, Chua and Kang generalized the memristor to a broader class of nonlinear dynamical systems they called memristive systems [Chua & Kang, 1976]. Few chaotic memristive electronic circuits were already proposed [Itoh & Chua, 2008] and [Muthuswamy & Kakate, 2010] but they were four-dimensional. It is only recently that a threedimensional system was proposed to describe a memristive circuit [Muthuswamy & Chua, 2010]. This 3D system is the simplest three-element electronic circuit producing chaotic behaviors since it is only made of two linear passive energy-storage elements, and an active memristive device [Chua & Kang, 1976]. Such a simple electronic circuit is not an algebraically minimal system for which only five terms in the three right members of the set of ordinary differential equations are allowed [Zhang & Heidel, 1997]. The system here studied has five linear terms and two nonlinear terms. The fact that