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Limit cycles of a class of planar system with \(1:2\) resonance for linear cusp. (Chinese. English summary) Zbl 1212.34101

Summary: The authors discuss nonlocal bifurcations of the class of planar systems \[ \dot{x}=y,\;\dot{y}=x (1-x^2) (3-x^2)+\mu (\xi_0+\xi_1 x^2-x^4)y, \] where \(x,y\in \mathbb{R}\), parameters \(\xi_0,\xi_1\in \mathbb{R}\) and \(|\mu|\ll 1\). The authors give conditions for limit cycles arising by Poincaré bifurcation and heteroclinic bifurcation.

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
37G10 Bifurcations of singular points in dynamical systems
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
34C23 Bifurcation theory for ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
34E10 Perturbations, asymptotics of solutions to ordinary differential equations
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