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Stability and bifurcation of spatially coherent solutions of the damped- driven NLS equation. (English) Zbl 0702.58067

Numerical experiments have identified low dimensional chaotic attractors with rich structure for the small amplitude responses of the damped driven pendulum chain. In order to give a theoretical explanation of this fact such solutions are approximated by the solutions of a damped driven NLS equation.
Locked states are found for which the spatial structure consists of coherent excitations localized about \(x=0\) or \(x=L/2\). A bifurcation analysis reveals the relationship of these spatially localized solutions to the spatially independent ones and provides a cutoff wave number above which there are no spatially dependent solutions. A linear stability analysis shows that solutions spatially localized undergo some Hopf bifurcations.
The analytical results compare favorably with numerical solutions and provide the ingredients for constructing chaotic attractors for this dynamical system.
Reviewer: L.Nicolaescu

MSC:

35B35 Stability in context of PDEs
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37C75 Stability theory for smooth dynamical systems
37G99 Local and nonlocal bifurcation theory for dynamical systems
35B32 Bifurcations in context of PDEs

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