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Global dynamics of memristor oscillator. (English) Zbl 1352.34068

Summary: In this paper, we investigate the global dynamics of a memristor oscillator \[ \begin{cases} \dot{x}_1=ax_2-\alpha\bigg[bx_1+\frac{(a-b)(|x_{1}+1|-|x_{1}-1|)}{2}\bigg], \\ \dot{x}_2=-\xi x_1+\beta x_2, \end{cases} \] where \(a,b\in \mathbb R\), and \(\alpha,\beta,\xi\in\mathbb R_+\). Clearly, the case \(a=b\) is trivial. So far, all results of this oscillator were given only for the case \(a>b\), where the set of equilibria may change among a singleton, three points and a singular continuum and at most one limit cycle can arise and no limit cycles arise from the continuum. Compared with the case \(a>b\), this oscillator displays more complicated dynamics for the case when \(a<b\). More clearly, one limit cycle may arise from the continuum and at most three limit cycles appear in the case of three equilibria, where generalized pitchfork bifurcation, saddle-node bifurcation, generalized Hopf bifurcation, double limit cycle bifurcation and homoclinic bifurcation may occur. Finally all global phase portraits are given for 50 cases on the Poincaré disc, where a generalized normal sector method is applied. Moreover, our partial analytical results are demonstrated by numerical examples.

MSC:

34C60 Qualitative investigation and simulation of ordinary differential equation models
94C05 Analytic circuit theory
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
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