Bifurcations Leading to Nonlinear Oscillations in a 3D Piecewise Linear Memristor Oscillator

Abstract

In this paper, we make a bifurcation analysis of a mathematical model for an electric circuit formed by the four fundamental electronic elements: one memristor, one capacitor, one inductor and one resistor. The considered model is given by a discontinuous piecewise linear system of ordinary differential equations, defined on three zones in ℝ3, determined by |z| < 1 (called the central zone) and |z| > 1 (the external zones). We show that the z-axis is filled by equilibrium points of the system, and analyze the linear stability of the equilibria in each zone. Due to the existence of this line of equilibria, the phase space ℝ3 is foliated by invariant planes transversal to the z-axis and parallel to each other, in each zone. In this way, each solution is contained in a three-piece invariant set formed by part of a plane contained in the central zone, which is extended by two half planes in the external zones. We also show that the system may present nonlinear oscillations, given by the existence of infinitely many periodic orbits, each one belonging to one such invariant set and passing by two of the three zones or passing by the three zones. These orbits arise due to homoclinic and heteroclinic bifurcations, obtained varying one parameter in the studied model, and may also exist for some fixed sets of parameter values. This intricate phase space may bring some light to the understanding of these memristor properties. The analytical and numerical results obtained extend the analysis presented in [Itoh & Chua, 2009; Messias et al., 2010].

Memristor Circuits: Flux—Charge Analysis Method

“…Note that this relation does not involve the electric variables (v C , i C ). In the (ϕ, q)-domain the ideal capacitor C described by (20) admits of the equivalent circuit representation given in Fig. 3.…”

Variable cubic-polynomial memristor based canonical Chua’s chaotic circuit

“…Among various types of chaotic oscillators, Chua's canonical circuit is one of the most typical instances since it is simple and contains a wide range of dynamic behaviors. In many chaotic circuit studies, the constitutive relations of the memristors are non-smooth piecewise-linear functions, resulting in discontinuous nonlinear characteristics of the corresponding memristance (and memductance) [9,10,11]. Although most interesting chaotic phenomena and chaotic dynamics can be described by the piecewise-linear method; however, some subtle features of the real circuit may be missed by the piecewise-linear approximation [12].…”

“…Additionally, this makes the physical realization of such non-smooth memristors unrealistic [13]. Then, several chaotic circuit studies have been reported with smooth nonlinear memristors [13,14]; however, most of these published works are analytically developed with the numerical study without physical implantations [9,10,11,15]. As one of the most important perspective, controlling the behavior of the chaotic circuits is critical for chaos-based applications.…”

“…Another element to be considered in the chaotic circuits is the positive function chosen to represent the memductance of the memristor, which has a relevant effect on the circuit dynamics [13]. In fact, all of the published studies are limited within demonstrating chaotic dynamics of Chua's system with a memristor owning a fixed memductance profile [7,8,9,10,11,13,14,15]; thus, there is a lack of a view in which a change of the memductance profile of the memristor might vary the dynamics of Chua's circuit.…”

Organic memristive device as key element for neuromorphic networks

“…However, such system was reported [31], while other works [32,33] are dedicated to theoretical modeling of similar systems.…”