On the torus bifurcation in averaging theory

Abstract

In this paper, we take advantage of the averaging theory to investigate a torus bifurcation in two-parameter families of 2D nonautonomous differential equations. Our strategy consists in looking for generic conditions on the averaged functions that ensure the existence of a curve in the parameter space characterized by a Neimark-Sacker bifurcation in the corresponding Poincaré map. A Neimark-Sacker bifurcation for planar maps consists in the birth of an invariant closed curve from a fixed point, as the fixed point changes stability. In addition, we apply our results to study a torus bifurcation in a family of 3D vector fields.2010 Mathematics Subject Classification. Primary: 34C23, 34C29, 34C45.

Periodic solutions and invariant torus in the Rössler system

“…The stability properties of these periodic solutions are studied in section 3, using mainly the theory of k-determined hyperbolicity for perturbed matrices (see [15]). In section 4, we first introduce the recently developed theory for detecting invariant tori through the averaging theory (see [6]). Then, we apply this theory to study the existence of an invariant torus for case A of the Rössler system (1).…”

“…The previous result says that a simple zero of the first non-vanishing averaged function g m corresponds to a periodic solution of system (6). In the case that the zero is not simple but isolated, one can still use some topological version of theorem 1 to ensure the existence of periodic solutions (see, for instance, [4,13]).…”

“…. , k, such that their simple zeros lead to isolated T-periodic solutions of system (6). These functions are obtained as follows.…”

“…With these functions the classical averaging method for finding periodic solutions can be summarized by the following theorem, which relates zeros of the first non-vanishing averaged function to the existence of periodic solutions of the non-autonomous differential system (6).…”

“…If there exists z * ∈ Ω such that g m (z * ) = 0 and |Dg m (z * )| = 0, then for |ε| = 0 sufficiently small there exists an isolated T-periodic solution ϕ(t, ε) of system (6) such that ϕ(0, 0) = z * .…”