Nonintegrability of 3 DoF Hamiltonian Resonant Systems
article
We review a number of methods to prove nonintegrability of Hamiltonian systems and focus on 3 Degrees-of-Freedom (DoF) systems listing the known results for the prominent resonances. Associated with the Hamiltonian systems are the averaged-normal forms that provide us with geometric insight, approximations of orbits and measures of chaos. Symmetries do change the qualitative and quantitative pictures; we illustrate this for the 1:2:1 resonance with discrete symmetry in the 1st and 3rd DoF. In this case, the averaged-normal form is still nonintegrable, but it becomes integrable when adding discrete symmetry in all DoF. Apart from the short periodic solutions obtained by averaging, we find many periodic solutions. There is numerical evidence of the presence of Silnikov bifurcation which clarifies the presence of nonintegrability ˇ phenomena qualitatively and quantitatively.
Topics
TNO Identifier
1001008
Source
International Journal of Bifurcation and Chaos, 34(13), pp. 1-15.
Pages
1-15
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