Transcritical bifurcation without parameters in memristive circuits

TL;DR

This paper characterizes the transcritical bifurcation without parameters in memristive circuits, extending mathematical theory to differential equations and applying graph theory to analyze stability changes in circuits with a memristor.

Contribution

It develops new mathematical results extending TBWP theory to explicit ODEs and DAEs, and applies these to analyze memristive circuits with a systematic approach.

Findings

Extended TBWP theorem to arbitrary dimension ODEs

Characterized TBWP in semiexplicit DAEs for circuits

Analyzed passive and non-passive memristive circuits, including neural networks

Abstract

The transcritical bifurcation without parameters (TBWP) describes a stability change along a line of equilibria, resulting from the loss of normal hyperbolicity at a given point of such a line. Memristive circuits systematically yield manifolds of non-isolated equilibria, and in this paper we address a systematic characterization of the TBWP in circuits with a single memristor. To achieve this we develop two mathematical results of independent interest; the first one is an extension of the TBWP theorem to explicit ordinary differential equations (ODEs) in arbitrary dimension; the second result drives the characterization of this phenomenon to semiexplicit differential-algebraic equations (DAEs), which provide the appropriate framework for the analysis of circuit dynamics. In the circuit context the analysis is performed in graph-theoretic terms: in this setting, our first working scenario is restricted to passive problems (exception made of the bifurcating memristor), and in a second step some results are presented for the analysis of non-passive cases. The latter context is illustrated by means of a memristive neural network model.