Transcritical Bifurcation without Parameters in Memristive Circuits

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. Ministerio de Economía y Competitividad (MINECO)/Fondo Europeo de Desarrollo Regional (FEDER).1. Let P be a matrix that drives f ′ (0) to Jordan form:2. Under the change of coordinates x = P y, the system x ′ = f (x) is transformed into y ′ =f (y) = P −1 f (P y), which reads as1. The circuit displays neither VMC-loops nor ILC-cutsets.2. There is a unique VML-loop, which includes the memristor and at least one inductor.

Homogeneous Models of Nonlinear Circuits

“…In [7] the memristor is assumed to be flux-controlled, with a cubic characteristic which can be written in the form σ m = −ϕ m + ϕ 3 m . Two stability changes are reported in that paper to occur along a line of equilibria and for the flux values ϕ m = ± 1/3; more precisely, this circuit can be shown to undergo two transcritical bifurcations without parameters by checking that it satisfies the general requirements characterizing this bifurcation in [18]. By contrast, in [8] the memristor is assumed to have the dual charge-controlled form ϕ m = −σ m + σ 3 m , which is responsible for the presence of two impasse manifolds, defined by the charge values σ m = ± 1/3, where trajectories collapse in finite time with infinite speed.…”

“…whose eigenvalues are given by the roots of the polynomial λ(λq(u m ) + p(u m )); these are λ = 0 and λ = −p(u m )/q(u m ). Worth remarking is the fact that the null eigenvalue reflects that equilibrium points are not isolated but define a line, a phenomenon which is well-known to happen systematically in the presence of a memristor (see [18] and references therein). Now, the zeros of p and of q in each of the cases defined by the characteristics of [7,8] are located at u m = ± 1/3.…”

Associate Submersions and Qualitative Properties of Nonlinear Circuits with Implicit Characteristics

“…Equilibrium points are obtained after annihilating the right-hand side; it is known in the circuit-theoretic literature that such equilibria are never isolated in the presence of at least one memristor [9,22,31]. Now, using a Schur reduction [15], the characteristic polynomial at a given equilibrium may in this context be checked to be defined by…”

“…Indeed, should the resistor be voltage-controlled (and described by the conductance G) and/or the memristor charge-controlled (with memristance M), singularities due to the eventual vanishing of G or M would not be captured in (31). This means that there is no chance to describe such singularities (and the corresponding order reduction, possibly responsible for impasse phenomena: see [9] in this regard) by looking at (31). By contrast, the general form depicted in (30) captures these cases in a smooth manner simply by fixing respectively the parameter values P r = 0 or Q m = 0, both of which annihilate the leading term of (30).…”

Classification of Codimension-1 Singular Bifurcations in Low-dimensional DAEs

“…Regularization of such a singular DAE often leads to an ODE with higher dimensional manifolds of equilibria in phase space, which can manifest bifurcations without parameters (cf. [35]).…”