Bifurcations of rough heteroclinic loops with three saddle points
The perturbed \(C^r\)-system \[ \dot z=f(z)+g(z,\mu)\tag{1} \] is considered with \(r\geq 4\), \(z\in \mathbb{R}^{m}\), \(\mu \in \mathbb{R}^k\), \(k\geq 3\), \(0\leq |\mu |\ll 1\), \(g(z,0)=0\). The unperturbed system (1) with \(\mu =0\) has a heteroclinic loop \(\Gamma \) connecting three hyperbolic critical points. Under certain transversal and nontwisted conditions, it is shown, for system (1) with \(\mu \neq 0\) small, the existence, uniqueness and incoexistence of heteroclinic loops with three or two saddle points along with homoclinic and periodic orbits near \(\Gamma\). The case with a heteroclinic loop in (1) connecting several hyperbolic critical points is also studied.
scientific article; zbMATH DE number 1757459