Periodic solutions for nonlinear differential systems: the second order bifurcation function

Abstract

Abstract. We are concerned here with the classical problem of Poincaré of persistence of periodic solutions under small perturbations. The main contribution of this work is to give the expression of the second order bifurcation function in more general hypotheses than the ones already existing in the literature. We illustrate our main result constructing a second order bifurcation function for the perturbed symmetric Euler top.

Persistence of periodic solutions for higher order perturbed differential systems via Lyapunov–Schmidt reduction

“…Using now the functions g i as stated in (11) we define the functions f i , F k , and γ i given by (5), (6), and (7), respectively. Recently in [15] the Bell polynomials were used to provide an alternative formula for the recurrence (12).…”

“…Using now the functions g i as stated in (11) we define the functions f i , F k , and γ i given by ( 5), (6), and (7), respectively.…”

“…When dim(Z) = n this problem is studied at an arbitrary order of ε, see [9,11], even for nonsmooth systems. When dim(Z) < n, this approach has already been used in [4], up to order 1, and in [5,6], up to order 2. In [14] this approach was used up to order 3 relaxing some hypotheses assumed in those previous 3 works.…”

On the limit cycles of planar polynomial system with non-rational first integral via averaging method at any order

“…Roughly speaking, averaging method gives the qualitative relation between the number of limit cycles for differential system and the number of zeros for the averaged function. There are several papers [1,3,6] where the averaging method is extended to study the number of limit cycles which bifurcate from the unperturbed system with an invariant manifold of periodic solutions. In the paper [1], the authors obtain the first order averaged function.…”

“…If this function vanishes, then the number of limit cycles of perturbed systems depends on the second order averaged function. The authors of the paper [3] consider the second order averaged function. In a recent paper [6], the authors deduce the expression of the averaged function at any order.…”