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The quasi-periodic centre-saddle bifurcation. (English) Zbl 0936.37028

Perturbations of a one-parameter family of Hamiltonian systems, \(H_{\lambda}\), are considered in the paper. It is assumed that for some value of ”bifurcation parameter” \(\lambda_{0}\) the Hamiltonian system \(H_{\lambda_{0}}\) has lower dimensional normally parabolic invariant tori and quasi-periodic center-saddle bifurcation takes place. In the integrable case such invariant tori bifurcate into normally hyperbolic and elliptic tori. With a KAM-theoretic approach the authors prove that the whole bifurcation scenario with all lower dimensional normally parabolic tori parameterized by pertinent Cantor sets of diophantine frequency vectors persist under small non-integrable perturbations. These result are applied to perturbations of a dynamically symmetric free rigid body, namely, the Euler top.

MSC:

37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37N05 Dynamical systems in classical and celestial mechanics
37J20 Bifurcation problems for finite-dimensional Hamiltonian and Lagrangian systems
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