Anomalous scaling behavior in Takens-Bogdanov bifurcations

Interactions of Andronov-Hopf and Bogdanov-Takens Bifurcations

A codimension-three bifurcation, characterized by a pair of purely imaginary eigenvalues and a nonsemisimple double zero eigenvalue, arises in the study of a pair of weakly coupled nonlinear oscillators with Z2 Z2 symmetry. The methodology is based on Arnold's ideas of versal deformations of matrices for the linear analysis, and Poincar e normal forms for the nonlinear analysis of the system. B...

Takens-Bogdanov bifurcations without parameters, and oscillatory shock profiles

Takens-Bogdanov bifurcations of periodic orbits and Arnold's Tongues in a Three-Dimensional Electronic Model

In this paper we study Arnold’s tongues in a Z2-symmetric electronic circuit. They appear in a rich bifurcation scenario organized by a degenerate codimension-three Hopf–pitchfork bifurcation. On the one hand, we describe the transition open-to-closed of the resonance zones, finding two different types of Takens–Bogdanov bifurcations (quadratic and cubic homoclinic-type) of periodic orbits. The...

Improved Homoclinic Predictor for Bogdanov-Takens Bifurcation

An improved homoclinic predictor at a generic codim 2 Bogdanov-Takens (BT) bifucation is derived. We use the classical “blow-up” technique to reduce the canonical smooth normal form near a generic BT bifurcation to a perturbed Hamiltonian system. With a simple perturbation method, we derive explicit firstand second-order corrections of the unperturbed homoclinic orbit and parameter value. To ob...

Normal Forms for Coupled Takens-Bogdanov Systems

The set of systems of differential equations that are in normal form with respect to a particular linear part has the structure of a module of equivariants, and is best described by giving a Stanley decomposition of that module. In this paper Groebner basis methods are used to determine a Groebner basis for the ideal of relations and a Stanley decomposition for the ring of invariants that arise...