Memristor Circuits as Linear-Gradient Systems and Discrete Analogues Preserving a First Integral

The paper considers a wide class of nonlinear circuits with an ideal memristor, capacitors, inductors and current or voltage sources. A fundamental dynamical property is that each memristor circuit in this class admits a first integral (invariant of motion or preserved quantity), i.e., a function which is constant along the solutions. The first main result is that we can put the state equations of each circuit in a universal form, known as linear-gradient form, given by a state-dependent skew-symmetric matrix times the gradient of the first integral. This is a simple and general form which is both of theoretic and practical interest. First of all, it makes manifest the existence of a first integral. Moreover, it admits an elegant discrete-time (DT) analogue. Indeed, the linear-gradient form, combined with geometric discretization methods and the concept of discrete gradients, yields a DT version of each memristor circuit that exactly preserves the first integral for any discretization time step. This is relevant, since the existence of a first integral is a fragile property that is in general destroyed by typical discretization schemes used in the literature no matter how small the step size is. On one hand, the proposed discretization scheme can be used for constructing numerical algorithms that better approximate the solutions of memristor circuits for small step sizes. On the other hand, for larger step sizes the obtained DT memristor circuits can be of interest by themselves since they are able to easily generate complex dynamics potentially useful for engineering applications (computational chaos). Furthermore, the paper shows that, thanks to the decomposition of the state space in invariant manifolds, the derived DT circuits exhibit extreme multistability, i.e., the coexistence of infinitely many different attractors for fixed set of circuit parameters, memristor nonlinearity and step size.