AN INVARIANT MANIFOLD APPROACH FOR STUDYING WAVES IN A ONE-DIMENSIONAL ARRAY OF NONLINEAR OSCILLATORS

We consider a one-dimensional linear spring-mass array coupled to a on e-dimensional array of uncoupled pendula. The principal aim of this st udy is to investigate the non-linear dynamics of this large-scale syst em in the limit of weak non-linearities, i.e. when the (fast) non-line ar pendulum effects are small compared to the underlying (slow) linear dynamics of the linear spring-mass chain. We approach the dynamics in the context of invariant manifolds of motion. In particular, we prove the existence of an invariant manifold containing the (predominantly) slow dynamics of the system, with the fast pendulum dynamics providin g small perturbations to the motions on the invariant manifold. By res tricting the motion on the slow invariant manifold and performing asym ptotic analysis we prove that the non-linear large-scale system posses ses propagation and attenuation zones (PZs and AZs) in the frequency d omain, similarly to the corresponding zones of the linearized system. Inside PZs non-linear travelling wave solutions exist, whereas in AZs only attenuating waves are permissible. Copyright (C) 1996 Published b y Elsevier Science Ltd.