A Note on the Lyapunov and Period Constants

Abstract

It is well known that the number of small amplitude limit cycles that can bifurcate from the origin of a weak focus or a non degenerated center for a family of planar polynomial vector fields is governed by the structure of the so called Lyapunov constants, that are polynomials in the parameters of the system. These constants are essentially the coefficients of the odd terms of the Taylor development at zero of the displacement map. Although many authors use that the coefficients of the even terms of this map belong to the ideal generated by the previous odd terms, we have not found a proof in the literature. In this paper we present a simple proof of this fact based on a general property of the composition of one-dimensional analytic reversing orientation diffeomorphisms with themselves. We also prove similar results for the period constants. These facts, together with some classical tools like the Weirstrass preparation Theorem, or the theory of extended Chebyshev systems, are used to revisit some classical results on cyclicity and criticality for polynomial families of planar differential equations.

Lower bounds for the local cyclicity of centers using high order developments and parallelization

“…This proves that there exists a curve, in the parameters space, for v 11 small enough, of weak-foci of order 11 that is born at the origin. The cubic perturbation mechanism described in Section 2 following the scheme of Roussarie, see [28], proves that only 11 limit cycles can bifurcate from the origin of system (14). This is because, from the Weierstrass Preparation Theorem, see [31], the Poincaré return map is a polynomial with coefficients the Lyapunov constants, that write, after all the changes of variables, as L k = u k , for k = 1, .…”

“…Proof of Proposition 4.1. Clearly system (14) has a center at the origin because ( 15) is a first integral well defined at the origin and the corresponding level curves in a neighborhood of the origin are ovals. In fact…”

“…The first step is the computation of the Taylor approximations of the Lyapunov constants, L k , corresponding to a cubic perturbation of system (14) having only quadratic and cubic terms. We will study which are the principal parts, near the origin in the parameters space, of each Lyapunov constant when the previous vanish.…”

The local cyclicity problem: Melnikov method using Lyapunov constants

“…So, It is well known that in the above expression the odd exponents in corresponds to the property that the first non vanishing Lyapunov constant has always an odd index, see [3]. For a proof that the Lyapunov constants of even indices are in the ideal of the odd ones, we refer the reader to [16, 31]. Applying the ideas in [13, Sect.…”

T-S Fuzzy Control of Uncertain Fractional-Order Systems with Time Delay

“…e traditional integer-order theory is not suitable for FOSs because of its unique definition, so the researchers adopt two new methods to stabilize the FOS. e first solution is Lyapunov function, and the other is Laplace transformation to stabilize FOS [11][12][13][14][15]. In control analysis, it is necessary to make the state trajectory of the system follow the desired command.…”