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Bifurcation of limit cycles from a double homoclinic loop with a rough saddle. (English) Zbl 1059.34027

The paper deals with the bifurcation of limit cycles from a double homoclinic loop under multiple parameter perturbations for the general planar system

$\stackrel{˙}{x}=f\left(x,y\right)+\epsilon {f}_{0}\left(x,y,\epsilon ,\delta \right),\phantom{\rule{1.em}{0ex}}\stackrel{˙}{y}=g\left(x,y\right)+\epsilon {g}_{0}\left(x,y,\epsilon ,\delta \right),$

where $f$, ${f}_{0}$, $g$, and ${g}_{0}$ are ${C}^{r}$, $r\ge 3$, functions, $\epsilon >0$ is small and $\delta \in D\subset {ℝ}^{n}$, $n\ge 1$, with $D$ being compact. Conditions for the existence of 4 homoclinic bifurcation curves are given. The conditions obtained are illustrated by examples.

##### MSC:
 34C23 Bifurcation (ODE) 34C05 Location of integral curves, singular points, limit cycles (ODE) 37G25 Bifurcations connected with nontransversal intersection 34C37 Homoclinic and heteroclinic solutions of ODE
##### Keywords:
double homoclinic loop; bifurcation; limit cycle; planar system