Melnikov functions and limit cycles in piecewise smooth perturbations of a linear center using regularization method
Consider a perturbed planar system of the form \[ dX/dt=AX+\varepsilon G(X), \] where \(X=(x,y)\in\mathbb{R}^2\), \(\varepsilon\geq0\), \(A\) is a matrix such that the equilibrium \((0,0)\) is a center of the unperturbed system and \(G(X)\) is a piecewise smooth vector field, \[ G(X)=G^-(X)\text{ for }y<0, \;G(X)=G^+(X)\text{ for }y>0 \] with \(G^{\pm}\) being smooth. The main results are concerned with the existence and the stability of a limit cycle of the system. The results are obtained using the Melnikov function method. Two approaches are used: (i) the Taylor expansion for the Poincaré first return map of the discontinuous system, (ii) the classical Melnikov method for a regularized approximation of the system. The link between these two approaches is investigated.
scientific article; zbMATH DE number 6698075