Time-periodic perturbation of a Liénard equation with an unbounded homoclinic loop
The paper considers time-periodic perturbations of a special Liénard system \[ \left\{ \begin{aligned} \dot x&=y-a_2x^2+\varepsilon g(\omega t),\\ \dot y&=-x, \end{aligned} \right. \eqno(1) \] where \(a_2\) is a real constant, \(\varepsilon\) is a small real parameter and the perturbation function \(g\) is \(2\pi/\omega\)-periodic. It is known [\textit{A.~Lins, W.~de Melo} and \textit{C.C.~Pugh}, Lect. Notes Math. 597, 335--357 (1977; Zbl 0362.34022)] that system (1) with \(\varepsilon=0\) has an unbounded homoclinic loop, which is a solution tending in forward and backward time to a non-hyperbolic equilibrium point located at infinity. Under small time-periodic perturbation, this equilibrium becomes a normally hyperbolic line of singularities at infinity. In the paper, it is shown that the perturbed system may exhibit homoclinic bifurcations leading to the existence of transverse intersections between the stable and unstable manifolds of such a normally hyperbolic line of singularities. The global study concerning the infinity is performed using the Poincaré compactification in polar coordinates, from which a system defined on a solid torus in \(\mathbb R^3\) is obtained. The transversality of the manifolds is proved by using the Melnikov method and implies a complex dynamical behavior of the perturbed system solutions in the finite part of the phase space. Numerical simulations that illustrate this behaviour are presented. Although applied to a particular case of the Liénard system, the analysis presented provides a geometrical approach to study periodic perturbations of homoclinic (heteroclinic) loops to infinity of other planar polynomial vector fields.
scientific article; zbMATH DE number 5956454
scientific article