Constructing the Second Order Poincaré Map Based on the Hopf-Zero Unfolding Method

Abstract EN

We investigate the Shilnikov sense homoclinicity in a 3D system and consider the dynamical behaviors in vicinity of the principal homoclinic orbit emerging from a third order simplified system.

It depends on the application of the simplest normal form theory and further evolution of the Hopf-zero singularity unfolding.

For the Shilnikov sense homoclinic orbit, the complex form analytic expression is accomplished by using the power series of the manifolds surrounding the saddle-focus equilibrium.

Then, the second order Poincaré map in a generally analytical style helps to portrait the double pulse dynamics existing in the tubular neighborhood of the principal homoclinic orbit.