Bifurcations of limit cycles from quintic Hamiltonian systems with an eye-figure loop. II. (English) Zbl 1161.34329
Summary: We consider Liénard equations of the form
\[ \begin{cases} \dot x=y,\\ \dot y=-(x-2x^3+x^5)-\varepsilon(\alpha+\beta x^2+\gamma x^4)y\end{cases} \]
where \(0<|\varepsilon|\ll 1\), \((\alpha,\beta,\gamma)\in\Lambda\subset \mathbb R^3\) and \(\Lambda\) is bounded. We prove that the least upper bound for the number of zeros of the related abelian integrals
\[ I(h)=\oint_{\Gamma_h}(\alpha+\beta x^2+\gamma x^4)y\,dx \]
for \(h\in (1/6,\infty)\) is three and for \(h\in (0,\infty)\) is four (counted with multiplicity) for all parameters \(\alpha,\beta\) and \(\gamma\). This implies that the number of limit cycles that bifurcated from periodic orbits of the unperturbed system for \(\varepsilon=0\) outside an eye-figure loop is less than or equal to three.
For part I, cf. ibid. 68, No. 10 (A), 2957–2976 (2008; Zbl 1158.34024).
\[ \begin{cases} \dot x=y,\\ \dot y=-(x-2x^3+x^5)-\varepsilon(\alpha+\beta x^2+\gamma x^4)y\end{cases} \]
where \(0<|\varepsilon|\ll 1\), \((\alpha,\beta,\gamma)\in\Lambda\subset \mathbb R^3\) and \(\Lambda\) is bounded. We prove that the least upper bound for the number of zeros of the related abelian integrals
\[ I(h)=\oint_{\Gamma_h}(\alpha+\beta x^2+\gamma x^4)y\,dx \]
for \(h\in (1/6,\infty)\) is three and for \(h\in (0,\infty)\) is four (counted with multiplicity) for all parameters \(\alpha,\beta\) and \(\gamma\). This implies that the number of limit cycles that bifurcated from periodic orbits of the unperturbed system for \(\varepsilon=0\) outside an eye-figure loop is less than or equal to three.
For part I, cf. ibid. 68, No. 10 (A), 2957–2976 (2008; Zbl 1158.34024).
MSC:
| 34C23 | Bifurcation theory for 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 |
| 34C08 | Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.) |
| 37G15 | Bifurcations of limit cycles and periodic orbits in dynamical systems |
| 34C37 | Homoclinic and heteroclinic solutions to ordinary differential equations |
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
Citations:
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