Parabolic resonances in 3 degree of freedom near-integrable Hamiltonian systems

Abstract

Perturbing an integrable 3 degree of freedom (d.o.f.) Hamiltonian system containing a normally parabolic 2-torus which is m-resonant (m = 1 or 2) creates a parabolic m-resonance (m-PR). PRs of different types are either persistent or of low co-dimension, hence they appear robustly in many applications. Energy-momenta bifurcation diagram is constructed as a tool for studying the global structure of 3 d.o.f. near-integrable systems. A link between the diagram shape, PR and the resonance structure is found. The differences between the dynamics appearing in 2 and 3 d.o.f. systems exhibiting PRs are studied analytically and numerically. The numerical study demonstrates that PRs are an unavoidable source of large and fast instabilities in typical 3 d.o.f. systems. © 2002 Elsevier Science B.V. All rights reserved.