Double Hamiltonian Hopf Bifurcation: normalization and normal form non-integrability

TL;DR

This paper analyzes the double Hamiltonian Hopf bifurcation in four-degree-of-freedom Hamiltonian systems, deriving its normal form, demonstrating non-integrability in most cases, and exploring bifurcations and homoclinic solutions numerically.

Contribution

It derives the normal form for the double Hamiltonian Hopf bifurcation and proves the non-integrability of the reduced system for generic parameters.

Findings

Normal form has two quadratic integrals generating a $T^2$ action.

Reduced system with two degrees of freedom is non-integrable for most parameters.

Bifurcation analysis and numerical investigation of homoclinic solutions.

Abstract

The double Hamiltonian Hopf bifurcation is studied, i.e. a generic two-parametric unfolding of a smooth Hamiltonian system with four degrees of freedom which has at the critical value of parameters the equilibrium with two pairs of double non semi-simple pure imaginary eigenvalues $\pm i\omega_1,$ $\pm i\omega_2,$ $\omega_1\ne \omega_2$ under an assumption of absence of strong resonances between $\omega_1,\omega_2$. We derive the normal form of the unfolding, when the ratio $\omega_1/\omega_2$ is irrational and study the truncated normal form of the fourth order. This truncated normal form is the same under the absence of strong resonances. The normal form has two quadratic integrals generating a symplectic periodic action of the abelian group $T^2.$ After reduction by means of these integrals we come to the reduced system with two degrees of freedom that is proven to be non-integrable for almost all values of its coefficients. Integrable such systems are also possible at some special values of coefficients, related examples are presented. Some investigations of this truncated system are presented along with its bifurcations when varying small detuning parameters. As an example of a system where this bifurcation is met, the system derived in \cite{KuLe} is investigated. Its homoclinic solutions are examined numerically when the system parameters correspond to a main equilibrium of the twofold saddle-focus type.