Nonlinear Sciences – Exactly Solvable and Integrable Systems
Scientific paper
2007-08-29
Regul. Chaotic Dyn., 2007, 12 (6), pp. 642-663
Nonlinear Sciences
Exactly Solvable and Integrable Systems
24 pages, 19 figures, corrected lemma 1 and 3
Scientific paper
10.1134/S156035470706007X
On the linear level elliptic equilibria of Hamiltonian systems are mere superpositions of harmonic oscillators. Non-linear terms can produce instability, if the ratio of frequencies is rational and the Hamiltonian is indefinite. In this paper we study the frequency ratio 1/2 and its unfolding. In particular we show that for the indefinite case (1:-2) the frequency ratio map in a neighbourhood of the origin has a critical point, i.e. the twist condition is violated for one torus on every energy surface near the energy of the equilibrium. In contrast, we show that the frequency map itself is non-degenerate (i.e. the Kolmogorov non-degeneracy condition holds) for every torus in a neighbourhood of the equilibrium point. As a byproduct we are able to obtain another proof of fractional monodromy in the 1:-2 resonance.
Cushman R. H.
Dullin Holger R.
Hanßmann Heinz
Schmidt Sven
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