Alien limit cycles in Liénard equations
The second part of Hilbert's 16th problem deals about the existence of a finite upper bound, only depending on the degree of the vector field, of the number of limit cycles of a planar real polynomial differential system. An approach to this problem is by using perturbation methods, i.e., to ask for the maximum number of limit cycles when perturbing from polynomial Hamiltonian systems (the infinitesimal Hilbert's 16th problem). In perturbing from a polynomial Hamiltonian system, large part of the cycliclity can be reduced to studying the zeros of the so-called abelian integrals. However, this study does not finish the investigation perturbing Hamiltonian systems and some attention has to be paid to the singular cycles located at the boundary of the annuli of periodic orbits. The limit cycles that can be perturbed from the boundary cannot always been detected by the abelian integrals. Such limit cycles are called alien limit cycles. To prove the existence of alien limit cycles for concrete families of vector fields is far from being trivial and in some cases impossible. In a previous work [the second author and \textit{R. Roussarie}, J. Differ. Equations 227, No. 1, 116--165 (2006; Zbl 1111.34028)], it was shown that the simplest possibility to get alien limit cycles is the hyperbolic two-saddle cycle, and a first example was provided by \textit{S. Luca} et al. [Discrete Contin. Dyn. Syst. 25, No. 4, 1081--1108 (2009; Zbl 1194.34084)]. In the present paper, an example of a family of polynomial Liénard equations exhibiting an alien limit cycle is given. This limit cycle is perturbed from a two-saddle cycle in the boundary of an annulus of periodic orbits given by a Hamiltonian vector field. The Hamiltonian represents a truncated pendulum of degree four. In comparison to the former polynomial example, not only the equations are simpler, but a lot of tedious calculations can be avoided, making this example also interesting with respect to simplicity in treatment.
scientific article; zbMATH DE number 6126900