New method to analyze the invariant manifolds of memristor circuits

Abstract

The paper considers a wide class of circuits containing memristors, coupled and nonlinear capacitors and inductors, linear resistive multi-ports and independent voltage and current sources. A new method is proposed to analyze the invariants of motion and invariant manifolds of memristor circuits in this class. The method permits to show the existence of invariant manifolds, and analytically find their expressions, under very weak conditions, namely, when there exists the differential algebraic equations (DAE) describing the circuit and there is at least one hybrid representation of a linear resistive multi-port where memristors are connected. These conditions are satisfied by all memristor circuits in the class, except for pathological cases only. One salient feature of the method is that the dynamic problem of finding the invariant manifolds is brought back to a static problem involving the analysis of a linear resistive multi-port. In the one-memristor case, this boils down to the familiar problem of finding the Thévenin or Norton equivalent of a linear resistive one-port. The results in the paper improve previous work along several directions. The technique here developed is applicable to a broader class of memristor circuits including coupled nonlinear reactive elements and linear multi-port resistors. More importantly, it is applicable even if the state equation (SE) description of the circuit does not exist, as in the case where there are singular points as impasse points. Moreover, when the SE representation exists, additional assumptions introduced in previous work are not needed. The effectiveness of the method is illustrated via the application to selected examples.