Limit cycles near homoclinic and heteroclinic loops
The paper deals with a near-Hamiltonian system in the form of \[ \dot{x}=H_y + \varepsilon p(x,y,\varepsilon,\delta), \quad \dot{y}=-H_x + \varepsilon q(x,y,\varepsilon,\delta), \] where \(H(x,y)\), \(p\) and \(q\) are analytic functions in \((x,y)\in \mathbb{R}^2\), and \(p\) and \(q\) being \(C^1\) in a small real parameter \(\varepsilon \geq 0\), \(\delta \in D \subset \mathbb{R}^m\) is a vector parameter with \(D\) compact. The main purpose is to give an explicit formula to compute the first four coefficients appeared in the expansion of the first order Melnikov function for both homoclinic and heteroclinic cases and to present a way to find limit cycles near the loops by using these coefficients. An application of the obtained results to a class of polynomial systems with three limit cycles is presented.
scientific article; zbMATH DE number 5375873