Abstract
We introduce a framework to systematically investigate the resonant double Hopf bifurcation. We use the basic invariants of the ensuing T1-action to analyse the approximating normal form truncations in a unified manner. In this way we obtain a global description of the parameter space and thus find the organising resonance droplet, which is the present analogue of the resonant gap. The dynamics of the normal form yields a skeleton for the dynamics of the original system. In the ensuing perturbation theory both normal hyperbolicity (centre manifold theory) and kam theory are being used.
Highlights
A central question of the theory of dynamical systems is stability loss of equilibria
We find a number of subordinate bifurcations of the double Hopf bifurcations not met before in the literature
The resonant double Hopf bifurcation has already been studied by many authors
Summary
A central question of the theory of dynamical systems is stability loss of equilibria. The double Hopf bifurcation is obtained if at bifurcation there are two pairs of imaginary eigenvalues, or two pairs of imaginary Floquet exponents For this situation to occur stably in a family of vector fields, that is, occurring in all families that differ from the given one by a small perturbation, we have to consider a family that depends on at least two parameters. Normalisation and truncation yields two-dimensional reduced dynamics, as there are two radial variables, and there are twelve different possible scenarios how the orbit structure around a double Hopf singular equilibrium can change as parameters change [27,34]. We find a number of subordinate bifurcations of the double Hopf bifurcations not met before in the literature
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