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5. INVARIANT MANIFOLDS
5.6. General Overview For discrete systems, i.e., the flow is given by a map, it is more convenient to use Hadamard’s method to prove invariant manifold and fiber theorems [84]. Even for continuous systems, Hadamard’s method was often utilized [64]. On the other hand, Perron’s method provides shorter proofs. It involves manipulation of integral equations. This method should be a favorite of analysts. Hadamard’s method deals with graph transform. The proof is often much longer, with clear geometric intuitions. It should be a favorite of geometers. For finite-dimensional continuous systems, N. Fenichel proved persistence of invariant manifolds under C 1 perturbations of flow in a very general setting [64]. He then went on to prove the fiber theorems in [65] [66] also in this general setting. Finally, he applied this general machinery to a general system of ordinary differential equations [67]. As a result, Theorems 5.1 and 5.2 hold for the following perturbed discrete cubic nonlinear Schr¨odinger equations [138], 1 qn+1 − 2qn + qn−1 + |qn |2 (qn+1 + qn−1 ) − 2ω 2 qn iq˙n = h2 1 (5.55) +iǫ − αqn + 2 (qn+1 − 2qn + qn−1 ) + β , h √ where i = −1, qn ’s are complex variables, qn+N = qn , (periodic condition);
h=
1 N,
and q−n = qn , (even condition);
and π 2π < ω < N tan , for N > 3, N N π 3 tan < ω < ∞, for N = 3. 3 ǫ ∈ [0, ǫ1 ), α (> 0), β (> 0) are constants. N tan
This is a 2(M + 1) dimensional system, where M = N/2, (N even);
and M = (N − 1)/2, (N odd).
This system is a finite-difference discretization of the perturbed NLS (5.2). For a general system of ordinary differential equations, Kelley [103] used the Perron’s method to give a very short proof of the classical unstable, stable, and center manifold theorem. This paper is a good starting point of reading upon Perron’s method. In the book [84], Hadamard’s method is mainly employed. This book is an excellent starting point for a comprehensive reading on invariant manifolds. There have been more and more invariant manifold results for infinite dimensional systems [140]. For the employment of Perron’s method, we refer the readers to [42]. For the employment of Hadamard’s method, we refer the readers to [18] [19] [20] which are terribly long papers.