Melnikov-type vectors and principal normals
The problem of the persistence of a singular orbit asymptotic to two hyperbolic saddles of an autonomous system under autonomous perturbations is considered. It is assumed that the initial system \(\dot x= f(x)\), \((x\in \mathbb{R}^ n,\;f\in C^ 2)\) has two hyperbolic saddles \(\rho_ 1\), \(\rho_ 2\), and the heteroclinic (homoclinic when \(\rho_ 1= \rho_ 2\)) orbit \(\Gamma\) tending to \(\rho_ 1\) \((\rho_ 2)\) as \(t\to+ \infty\) \((t\to-\infty)\). Using Melnikov-type vectors, the author finds conditions under which, for the perturbed system \[ \dot x= f(x)+ h(x,\mu)\qquad (h\in C^ 2,\;h(x,0)= 0), \] where \(\mu\in \mathbb{R}^ m\) is a small parameter, in the parameter space, there is a surface \(H\subset \mathbb{R}^ m\) with the following property: for \(\mu\in H\) there exists a heteroclinic orbit \(\Gamma_ \mu\), \(\text{dist}(\Gamma_ \mu,\Gamma)= O(|\mu|)\) connecting two hyperbolic saddles \(\rho_ 1(\mu)\) \((\rho_ 2(\mu))\) near to \(\rho_ 1\) \((\rho_ 2)\). Examples of the uniquely persisting singular orbits lying on a 2-dimensional cylinder (torus) consisting of heteroclinic (homoclinic) orbits are given. The obtained results extend the corresponding ones of \textit{K. J. Palmer} [J. Differ. Equations 55, 225-256 (1984; Zbl 0539.58028)] and \textit{F. Battelli} and \textit{C. Lazzarri} [J. Differ. Equations 86, 342-366 (1990; Zbl 0713.34055)] to the case of autonomous perturbed systems and the case where singular orbits may have much higher degeneracy.
scientific article; zbMATH DE number 700532