Nonlinear Normal Modes of Symmetric Hamiltonian Systems
In the classical dynamics of anharmonic molecules and crystals an important role is played by nonlinear normal modes — families of periodic trajectories (parametrized by energy) with periods close to those of the normal modes of the linearization. These serve as organizing centres, a skeleton about which the rest of the dynamics can be arranged. By the KAM Theorem, elliptic periodic trajectories are surrounded by invariant tori and so carry with them regions of regular motion in which the only instability is caused by Arnold diffusion. Hyperbolic periodic orbits are typically connected by heteroclinic orbits, which in turn give rise to chaotic regions of phase space, CHURCHILL and ROD . An excellent example of this organizing behaviour can be seen in the Hénon-Heiles system, see HÉNON , ROD and CHURCHILL , and §§1–2.
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