Absolute cyclicity, Lyapunov quantities and center conditions

Abstract

In this paper we consider analytic vector fields X 0 having a non-degenerate center point e. We estimate the maximum number of small amplitude limit cycles, i.e., limit cycles that arise after small perturbations of X 0 from e. When the perturbation (X λ ) is fixed, this number is referred to as the cyclicity of X λ at e for λ near 0. In this paper, we study the so-called absolute cyclicity; i.e., an upper bound for the cyclicity of any perturbation X λ for which the set defined by the center conditions is a fixed linear variety. It is known that the zero-set of the Lyapunov quantities correspond to the center conditions (Caubergh and Dumortier (2004) [6]). If the ideal generated by the Lyapunov quantities is regular, then the absolute cyclicity is the dimension of this so-called Lyapunov ideal minus 1. Here we study the absolute cyclicity in case that the Lyapunov ideal is not regular.

Essential perturbations of polynomial vector fields with a period annulus

“…The Poincaré-Liapunov constants are the basic tool to solve the center problem, see e.g. [2,5,20,27,28]. We remark that the Poincaré-Liapunov constants are polynomials in λ and that their number N + 1 only depends on the considered family (3).…”

“…In fact, finite cyclicity of any limit periodic set, in terms of the degree of the system, implies the solution to Hilbert's 16th problem. The cyclicity problem has been studied by several authors also in its relation with the center problem, see for instance [4,5,7,10,11,12,18]. Roughly speaking, the cyclicity of a limit periodic set of a polynomial system of degree at most d, is the maximum number of limit cycles that can bifurcate from the given limit periodic set inside the family of all polynomial systems of degree d; see Definition 12 in [25] for a precise definition.…”

“…Then, given a fixed value of ε = 0 there is a finite number of intervals in the set {(−ε n , ε n ) : n ∈ N} in which ε belongs to. Thus, the functions ϑ n (ε) only exist for this finite number of intervals and, as a consequence, there is only a finite number of limit cycles for system (5) for the considered fixed value of ε. If we take a value of ε closer to 0 we may have more limit cycles and since ε n > 0 for all n ∈ N and lim n→∞ ε n → 0, we have that given a number ℓ, there is always a value of ε close enough to 0 such that the corresponding system (5) has at least ℓ limit cycles bifurcating from the periodic orbits of P. Therefore, even though for a fixed system (5) the number of limit cycles is finite, one has that the cyclicity of the period annulus P is infinite.…”