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Hence condition (64) cannot be satisfied, and we have an obstruction to the asymptotic resonance each time there are ternary relations of the type (63) satisfying (65). Now, the list (63) depends of course on the choice of the indexes i = j. To prove the statement we have then to distinguish many different cases, each time deciding whether (65) is fulfilled or not. 44 Assume first that 1 1 ≤ i < (N +1), 3 and notice that we can fit one of the ternary relations (77) only if γ = i. If we moreover set j = 2(N +1)/3 + i, then (65) is satisfied, preventing the asymptotic resonance of µi with µj .
14] W. Narkiewicz. Elementary and analytic theory of algebraic numbers. , Springer–Verlag, Berlin; PWN–Polish Scientific Publishers, Warsaw, 1990.  B. Rink. Symmetry and resonance in periodic FPU chains. Comm. Math. Phys. 3, 665–685.  B. Rink. Geometry and dynamics in Hamiltonian latices with applications to tha Fermi–Pasta–Ulam problem. D. Thesis, Universiteit Utrecht, 2003. A. Stuart. An introduction to bifurcation theory based on differential calculus. Nonlinear Anal. : Heriot–Watt Symposium, vol.
Geometry and dynamics in Hamiltonian latices with applications to tha Fermi–Pasta–Ulam problem. D. Thesis, Universiteit Utrecht, 2003. A. Stuart. An introduction to bifurcation theory based on differential calculus. Nonlinear Anal. : Heriot–Watt Symposium, vol. IV pp. 76–135, Res. Notes in Math. 39, Pitman, Boston, 1979.  M. Toda. Theory of nonlinear lattices. Second ed. Springer Series in Solid– State Sciences, 20. Springer–Verlag, Berlin, 1989.