Bifurcation Analysis of a Cubic Memristor Model

The memristor, named as a contraction for "memory resistor", is supposed to be the fourth fundamental electronic element, in addition to the well-known resistor, inductor, and capacitor. Its theoretical existence was postulated in 1971 by L. O. Chua [1], but its physical realization was announced only recently in a paper published in the May 2008 issue of Nature by a research team from Hewlett–Packard [2]. It has attracted worldwide attention due to its potential applications in the construction of electronic circuits, especially for newer-generation computers.
In [3] we present a bifurcation analysis of a memristor oscillator mathematical model, given by a cubic system of ordinary differential equations, depending on four parameters.
We show that, depending on the parameter values, the system may present the coexistence of both infinitely many stable periodic orbits and stable equilibrium points.
The periodic orbits arise from the change in the local stability of equilibrium points on a line of equilibria for a fixed set of parameter values. This is an interesting type of Hopf bifurcation, which occurs without varying parameters.
In these graphics we show the birth of periodic orbits by varying the initial conditions and parameters in the model.


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[1] L. O. Chua, "Memristor—The Missing Circuit Element," IEEE Trans. Circuit Theory, 18(5), 1971 pp. 507–519.
[2] D. B. Strukov, G. S. Snider, G. R. Stewart, and R. S. Williams, "The Missing Memristor Found," Nature, 453, 2008 pp. 80–83.
[3] M. Messias, C. Néspoli, and V. Botta, "Hopf Bifurcation from Lines of Equilibria without Parameters in Memristor Oscillators," Internationl Journal of Bifurcation and Chaos (to appear).


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