A Lyapunov Function-Based Condition for Nonoscillatory Behaviors in a Class of Memristor Circuits

In recent years, many contributions have been reported in the literature showing that memristor circuits can exhibit very rich oscillatory and chaotic behaviors. In this paper, we look for conditions ensuring that memristor circuits are nonoscillatory, i.e. they do not display oscillations and more complex attractors and enable convergence toward some of the infinite nonisolated equilibrium points. Specifically, we consider the class of memristor circuits composed by the interconnection of a linear time-invariant two-terminal (one port) element and an ideal memristor, which can be either flux-controlled or charge-controlled and whose characteristic satisfies a slope-bounded condition. First, exploiting the well-known fact that any circuit with an ideal memristor admits first integrals, and hence its state space is decomposed in a continuum of invariant manifolds, a state space representation of the circuit dynamics on each invariant manifold is derived in an explicit way. Then, conditions to ensure the absence of oscillatory behaviors on each manifold are obtained by employing the 2-additive compound matrix of the Jacobian of these representations. It is shown that the memristor circuit is nonoscillatory if there exists a common Lyapunov function for two suitable 2-additive compound matrices, which is obtained by solving two linear matrix inequalities. Two memristor circuits are analyzed in some detail to illustrate the application of the results and their degree of conservatism.