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.

References in corpus (1)

• Averaging theory at any order for computing limit cycles of discontinuous piecewise differential systems with many zones

Cited by in corpus (5)

• Periodic solutions and invariant torus in the Rössler System
• Invariant tori via higher order averaging method: existence, regularity, convergence, stability, and dynamics
• A mechanism for detecting normally hyperbolic invariant tori in differential equations
• Averaging method for quasi-periodic response solutions
• Canonical models for torus canards in elliptic bursters