Persistence Properties of Normally Hyperbolic Tori

Abstract

Near-resonances between frequencies notoriously lead to small denominators when trying to prove persistence of invariant tori carrying quasi-periodic motion. In dissipative systems external parameters detuning the frequencies are needed so that Diophantine conditions can be formulated, which allow to solve the homological equation that yields a conjugacy between perturbed and unperturbed quasi-periodic tori. The parameter values for which the Diophantine conditions are not fulfilled form so-called resonance gaps. Normal hyperbolicity can guarantee invariance of the perturbed tori, if not their quasi-periodicity, for larger parameter ranges. For a 1-dimensional parameter space this allows to close almost all resonance gaps.

Normal-normal resonances in a double Hopf bifurcation

“…The gaps left open when Cantorising the attracting tori to prove persistence using kam theory can be closed using normal hyperbolicity to obtain invariant tori on which the flow is no longer quasi-periodic, compare with [8] and results cited therein. There seem to be no claims concerning the heteroclinic bifurcation in the literature.…”

“…Theorem 3.1 (Normal Form) Let ∈ Z 2 be such that 0 < 1 < 2 . Consider the real analytic vector field (8) where for κ > n − 1 and Γ > 0 the frequency vector (ω, α) at µ = 0 satisfies the Diophantine conditions…”

“…-One of the improvements of normalisation theory in the analytic setting is that the remainder term can be made exponentially small [7,8] in the perturbation parameter.…”

Bifurcations in Volume-Preserving Systems

“…What is not considered in [24] are volume-preserving perturbations on T n × R 2 that do not lift to symplectic perturbations on T n × R n × R 2 . Then trace Dh need no longer vanish and might appear to be of the order of the perturbation, although the possibility of normalization shows that trace Dh is infinitely flat and even exponentially small for analytic (31), compare with [10,12,39]. The approach to quasi-periodic bifurcation theory in [11] yields quasi-periodic stability also if the symplectic structure is not preserved, i.e.…”

“…Using a single parameter β ∈ R is still possible, but requires explicit Diophantine conditions on (ω(0), α(0)) ∈ R n+1 to avoid that the complete bifurcation scenario disappears in a resonance gap. Indeed, the considerations in [8,10,17] concerning the (dissipative) quasi-periodic Hopf bifurcation apply here as well. In the hyperbolic case it is simpler to describe the bifurcation for β decreasing through 0: two normally hyperbolic n-tori meet and acquire an additional frequency, resulting in one normally hyperbolic (n + 1)-torus after bifurcation.…”