Abstract
We consider the parametrized family of equations∂ tt ,u-∂ xx u-au+∥u∥ 2α2 u=O,x∃(0,πL), with Dirichlet boundary conditions. This equation has finite-dimensional invariant manifolds of solutions. Studying the reduced equation to a four-dimensional manifold, we prove the existence of transversal homoclinic orbits to periodic solutions and of invariant sets with “chaotic” dynamics, provided that α=2, 3, 4,.... For α=1 we prove the existence of infinitely many first integrals pairwise in involution.
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Grotta Ragazzo, C. Chaos and integrability in a nonlinear wave equation. J Dyn Diff Equat 6, 227–244 (1994). https://doi.org/10.1007/BF02219194
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DOI: https://doi.org/10.1007/BF02219194