On Discontinuous Piecewise Linear Models for Memristor Oscillators

In this paper, we provide for the first time rigorous mathematical results regarding the rich dynamics of piecewise linear memristor oscillators. In particular, for each nonlinear oscillator given in [Itoh & Chua, 2008], we show the existence of an infinite family of invariant manifolds and that the dynamics on such manifolds can be modeled without resorting to discontinuous models. Our approach provides topologically equivalent continuous models with one dimension less but with one extra parameter associated to the initial conditions. It is possible to justify the periodic behavior exhibited by three-dimensional memristor oscillators, by taking advantage of known results for planar continuous piecewise linear systems. The analysis developed not only confirms the numerical results contained in previous works [Messias et al., 2010; Scarabello & Messias, 2014] but also goes much further by showing the existence of closed surfaces in the state space which are foliated by periodic orbits. The important role of initial conditions that justify the infinite number of periodic orbits exhibited by these models, is stressed. The possibility of unsuspected bistable regimes under specific configurations of parameters is also emphasized.

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Invariant manifolds of periodic orbits for piecewise linear three-dimensional systems

IMA Journal of Applied Mathematics ◽

10.1093/imamat/69.1.71 ◽

2004 ◽

Vol 69 (1) ◽

pp. 71-91 ◽

Cited By ~ 13

Author(s):

V. Carmona

Keyword(s):

Periodic Orbits ◽

Invariant Manifolds ◽

Piecewise Linear ◽

Three Dimensional

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HOPF BIFURCATION FROM LINES OF EQUILIBRIA WITHOUT PARAMETERS IN MEMRISTOR OSCILLATORS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127410025521 ◽

2010 ◽

Vol 20 (02) ◽

pp. 437-450 ◽

Cited By ~ 54

Author(s):

MARCELO MESSIAS ◽

CRISTIANE NESPOLI ◽

VANESSA A. BOTTA

Keyword(s):

Hopf Bifurcation ◽

Periodic Orbits ◽

Initial Conditions ◽

Piecewise Linear ◽

Equilibrium Points ◽

Three Dimensional ◽

Fixed Set ◽

Electronic Element ◽

Stable Periodic ◽

Parameter Values

The memristor is supposed to be the fourth fundamental electronic element in addition to the well-known resistor, inductor and capacitor. Named as a contraction for memory resistor, its theoretical existence was postulated in 1971 by L. O. Chua, based on symmetrical and logical properties observed in some electronic circuits. On the other hand its physical realization was announced only recently in a paper published on May 2008 issue of Nature by a research team from Hewlett–Packard Company. In this work, we present the bifurcation analysis of two memristor oscillators mathematical models, given by three-dimensional five-parameter piecewise-linear and cubic systems of ordinary differential equations. We show that depending on the parameter values, the systems may present the coexistence of both infinitely many stable periodic orbits and stable equilibrium points. The periodic orbits arise from the change in local stability of equilibrium points on a line of equilibria, for a fixed set of parameter values. This phenomenon is a kind of Hopf bifurcation without parameters. We have numerical evidences that such stable periodic orbits form an invariant surface, which is an attractor of the systems solutions. The results obtained imply that even for a fixed set of parameters the two systems studied may or may not present oscillations, depending on the initial condition considered in the phase space. Moreover, when they exist, the amplitude of the oscillations also depends on the initial conditions.

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