Nondegeneracy of hope points emanating from a Z2-symmetry-breaking Takens-Bogdanov point

Bogdanov–Takens Bifurcation Points and S̆il’nikov Homoclinicity in a Simple Power-System Model of Voltage Collapse

The bifurcation structure of a simple power-system model is investigated, with respect to changes to both the real and reative loads. Numerical methods for this bifurcation analysis are presented and discussed. The model is shown to have a Bogdanov–Takens bifurcation point and hence homoclinic orbits; these orbits can be of S̆il’nikov type with many coexisting periodic solutions. We may use 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...

Initialization of Homoclinic Solutions near Bogdanov-Takens Points: Lindstedt-Poincaré Compared with Regular Perturbation Method

To continue a branch of homoclinic solutions starting from a Bogdanov–Takens (BT) point in parameter and state space, one needs a predictor based on asymptotics for the bifurcation parameter values and the corresponding small homoclinic orbits in the phase space. We derive two explicit asymptotics for the homoclinic orbits near a generic BT point. A recent generalization of the Lindstedt–Poinca...

A Subshift of Finite Type in the Takens-Bogdanov Bifurcation with D 3 Symmetry

Bogdanov-takens Singularity of a Neural Network Model with Delay

In this article, we study Bogdanov-Takens (BT) singularity of a tree-neuron model with time delay. By using the frameworks of CampbellYuan [2] and Faria-Magalhães [4, 5], the normal form on the center manifold is derived for this singularity and hence the corresponding bifurcation diagrams such as Hopf, double limit cycle, and triple limit cycle bifurcations are obtained. Examples are given to ...