This paper deals with the existence of a homoclinic solution in planar systems with discontinuous right-hand side. In fact, these type of systems are more used in practical problems than the classical planar differential systems and this article serves to a generalization of the Melnikov function to these discontinuous right-hand side systems. If one considers a system with a homoclinic loop and then takes a parametrical perturbation of it, inside the same class of systems, the Melnikov function determines the distance between the stable and unstable manifolds in terms of the parameter. The main result of this paper consists in deriving an expression of the Melnikov function in the case of discontinuous systems with a periodic perturbation and giving a method of how to find it in concrete problems.
The considered discontinuous systems are of the form: \[ \mathbf{x}' = \mathbf{f}(\mathbf{x}) + \varepsilon \mathbf{g}(t,\mathbf{x}), \tag{1} \] where \(\varepsilon >0\), \(t \in \mathbb{R}\), \(\mathbf{x} \in \mathbb{R}^2\) and the function \(\mathbf{f}\) is split in two \(\mathcal{C}^2\) functions, \(\mathbf{f}_{-}\) and \(\mathbf{f}_{+}\), defined in two corresponding disjoint subsets \(V_{-}\) and \(V_{+}\) which are separated by a hypersurface \(\Sigma\) and are such that \(V_{-} \cup \Sigma \cup V_{+}= \mathbb{R}^2\). Some smooth conditions are assumed on the function \(\mathbf{f}\) so as to ensure the existence of \(\mathcal{C}^2\)-extensions of \(\mathbf{f}_{-}\) and \(\mathbf{f}_{+}\) over \(\Sigma\). The function \(\mathbf{g}\) is defined analogously, with the same conditions and in the same subsets \(V_{-}\) and \(V_{+}\), for all \(t \in \mathbb{R}\):
\[ \mathbf{g}(t,\mathbf{x}) = \begin{cases} \mathbf{g}_{-}(t,\mathbf{x}), & \mathbf{x} \in V_{-}, \;t \in \mathbb{R}, \\ \mathbf{g}_{+}(t,\mathbf{x}), & \mathbf{x} \in V_{+}, \;t \in \mathbb{R}. \end{cases} \] The unperturbed system \(\mathbf{x}'=\mathbf{f}(\mathbf{x})\) is assumed to be divergence-free and with a saddle point in \(V_{-}\) whose homoclinic trajectory has transversal intersections with \(\Sigma\) and the number of intersections is non-zero. The perturbation function \(\mathbf{g}(t,\mathbf{x})\) is assumed to be \(T\) periodic in \(t\), with \(T>0\).
The expression of this “nonsmooth” Melnikov function is a sum of the classical expression of the Melnikov function and a finite sum of terms corresponding to the discontinuity of system (1) over \(\Sigma\). We recall that the classical expression of the Melnikov function, in the case of smooth systems, is an integral of the wedge product of \(\mathbf{f}\) and \(\mathbf{g}\) evaluated over the homoclinic loop. The additional terms corresponding to the discontinuity involve the intersections of the homoclinic trajectory of the unperturbed system with \(\Sigma\). In order to have these terms well-defined, the existence and continuity of the second derivative of system (1) out of the discontinuity surface \(\Sigma\) is needed.
Furthermore, the paper provides a generalization of the result that a simple zero of the Melnikov function is a sufficient condition for the existence of a homoclinic trajectory in the perturbed system. It is proved that, under additional assumptions, this perturbed system maintains a homoclinic trajectory.
Last but not least, a method to compute this Melnikov function is described. This method needs a parameterization of the homoclinic trajectory, to solve certain systems of differential equations and to calculate the Melnikov integral. Although these computations can be difficult to be carried out in an analytical way, they provide a numerical method to solve the considered problem.