Bifurcation of limit cycles for a class of quintic Hamiltonian systems with double heteroclinic loops. (Chinese. English summary) Zbl 1289.34108
Summary: We consider bifurcation of limit cycles in a class of quintic Hamiltonian systems
\[
\dot{x}=y,\;\dot{y}=-(ax+bx^3+cx^5)
\]
with double heteroclinic loops under small perturbation of the form \(\varepsilon (\alpha+\beta x^2+x^4)y\frac{\partial}{\partial y}\), where \(a<0,\;b>0\) and \(3b^2=16ac\). It is proved that at most 2 limit cycles can bifurcate for \(0<|\varepsilon|\ll 1\), and a complete bifurcation diagram is obtained.
MSC:
| 34C23 | Bifurcation theory for ordinary differential equations |
| 34C08 | Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.) |
| 34C07 | Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations |
| 37G15 | Bifurcations of limit cycles and periodic orbits in dynamical systems |
| 34C37 | Homoclinic and heteroclinic solutions to ordinary differential equations |
