09-77 Renato Calleja

Existence and persistence of invariant objects in dynamical systems and mathematical physics (2511K, PDF) May 18, 09

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Abstract. In this dissertation we present four papers as chapters. In Chapter 2, we extended the techniques used for the Klein-Gordon Chain by Iooss, Kirchg\"assner, James, and Sire, to chains with non-nearest neighbor interactions. We look for travelling waves by reducing the Klein-Gordon chain with second nearest neighbor interaction to an advance-delay equation. Then we reduce the equation to a finite dimensional center manifold for some parameter regimes. By using the normal form expansion on the center manifold we were able to prove the existence of three different types of travelling solutions for the Klein Gordon Chain: periodic, quasi-periodic and homoclinic to periodic orbits with exponentially small amplitude. In Chapter 3 we include numerical methods for computing quasi-periodic solutions. We developed very efficient algorithms to compute smooth quasi-periodic equilibrium states of models in 1-D statistical mechanics models allowing non-nearest neighbor interactions. If we discretize a hull function using $N$ Fourier coefficients, the algorithms require $O(N)$ storage and a Newton step for the equilibrium equation requires only $O(N \log(N))$ arithmetic operations. This numerical methods give rise to a criterion for the breakdown of quasi-periodic solutions. This criterion is presented in Chapter 4. In Chapter 5, we justify rigorously the criterion in Chapter 4. The justification of the criterion uses both Numerical KAM algorithms and rigorous results. The hypotheses of the theorem concern bounds on the Sobolev norms of a hull function and can be verified rigorously by the computer. The argument works with small modifications in all cases where there is an \emph{a posteriori} KAM theorem.

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