2016/05/19 by Ricardo Riaza, Riaza, Ricardo
Mathematics · #34A09 #34C45 #34D35 #37G10 #94C05 #94C15 #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS #msc:34A09 #msc:34C45 #msc:34D35 #msc:37G10 #msc:94C05 #msc:94C15
paper · pdf · doi:10.48550/arxiv.1605.05925
arxiv created 2016/05/19 · arxiv updated 2016/05/20
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.