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Limit cycles in the Josephson equation. (English) Zbl 0604.58041

The paper deals with the Josephson equation \[ (1.1)\quad \beta d^ 2\Phi /d\tau^ 2+(1+\gamma \cos \Phi)d\Phi /d\tau +\sin \Phi =\alpha, \] where \(\Phi \in S^ 1\), \(\alpha\),\(\beta\),\(\gamma\in {\mathbb{R}}\). Supposing that \(\epsilon =\beta^{-1/2}\) is a small positive parameter and putting \(\alpha =\epsilon a\), \(\tau =\epsilon t\), the equation (1.1) can be written as \[ (1.2)\quad {\dot \Phi}=y,\quad \dot y=-\sin \Phi +\epsilon [a-(1+\gamma \cos \Phi)y]. \] Using the averaging method and some techniques from bifurcation theory, the author gives the a-\(\gamma\) plane bifurcation diagram and corresponding phase portraits on the cylinder \(TS^ 1\) for the system (1.2).
Reviewer: J.Kalas

MSC:

37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
34C29 Averaging method for ordinary differential equations
37G99 Local and nonlocal bifurcation theory for dynamical systems
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
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