Generalized Hopf Bifurcation for Planar Vector Fields via the Inverse Integrating Factor

Abstract

In this paper we study the maximum number of limit cycles that can bifurcate from a focus singular point p 0 of an analytic, autonomous differential system in the real plane under an analytic perturbation. We consider p 0 being a focus singular point of the following three types: non-degenerate, degenerate without characteristic directions and nilpotent. In a neighborhood of p 0 the differential system can always be brought, by means of a change to (generalized) polar coordinates (r, θ), to an equation over a cylinder in which the singular point p 0 corresponds to a limit cycle γ 0 . This equation over the cylinder always has an inverse integrating factor which is smooth and non-flat in r in a neighborhood of γ 0 . We define the notion of vanishing multiplicity of the inverse integrating factor over γ 0 . This vanishing multiplicity determines the maximum number of limit cycles that bifurcate from the singular point p 0 in the non-degenerate case and a lower bound for the cyclicity otherwise.Moreover, we prove the existence of an inverse integrating factor in a neighborhood of many types of singular points, namely for the three types of focus considered in the previous paragraph and for any isolated singular point with at least one non-zero eigenvalue.

A new approach to center conditions for simple analytic monodromic singularities

“…The existential part of (a) is proved in [7] while the uniqueness part is showed in [9], see also [10]. Our first main contribution is the part (b).…”

Mutiple Hopf Bifurcation on Center Manifold

“…where z; w; T are complex variables and for any positive integer k; j, we have a kj ¼ b kj , then systems (25) and (26) "# (27) such that (28) is called the mth singular point quantity at the degenerate singular point for system (26) …”

“…Take the degenerate singular point with a zero linear part in planar system, for example, the investigation of Hopf bifurcation from the equilibrium has to involve detecting the monodromy and distinguishing between a center and a focus [18,19]. For that matter, several available approaches and corresponding results can be seen in [18][19][20][21][22][23][24][25], and one can easily find that the results on the bifurcation of limit cycles are very less. Remarkably, the author of reference [26] in 2001 gave the formal series method of calculating the singular point quantities of the degenerate critical point, which made it possible to investigate multiple Hopf bifurcation…”

Centers: their integrability and relations with the divergence

“…There are systems with the degenerate form (4) for which such a parametrization is also possible, for instance the ones which do not have characteristic directions, see for instance [9]. The stability of the origin is clearly given by the sign of the first nonzero α i (that is, it is unstable if α i > 0, and stable if α i < 0), and if all the α i are zero then the origin is a center.…”

“…The following example is a particular case of Example 3 in [9], the stability of the origin can be studied using Theorem 8, for more details see [13].…”