Limiting phase trajectories and dynamic transitions in nonlinear periodic systems

The intensity of energy exchange between parts of periodic nonlinear Frenkel-Kontorova and Klein-Gordon lattices is analyzed based on a concept of limiting phase trajectories introduced earlier. It is demonstrated that, with increasing nonlinearity parameter in these lattices, two dynamic transitions take place successively. The first transition is due to the bifurcation of the lower (with respect to frequency) normal mode because of its instability. It is accompanied by the occurrence of two additional normal modes and the separatrix between them. In this case, after this transition and before it, complete energy exchange between parts of the system is possible. The second transition takes place as a result of merging of the limiting phase trajectory with the separatrix, after which complete energy exchange between parts of the system is impossible. Analytical results are proven by numerical data.