Abstract
Higher order resonances in two degrees of freedom Hamiltonian systems are studied by using Birkhoff normalization. The normal forms can be used as a starting point to develop a theory of asymptotic approximations on the natural time-scale of the resonances. The asymptotic expressions are used to obtain a geometric picture of the flow in 4-space. An application of the theory is found in the model problem of Contopoulos for the Hamiltonian H=1/2(x2+y2)+1/2(ω 21 x2+ω 22 y2)−εxy2. A comparison with numerical results obtained earlier yields excellent agreement and we put Contopoulos' formal «third» integral in a new perspective.
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Sanders, J.A., Verhulst, F. (1979). Approximations of higher order resonances with an application to Contopoulos' model problem. In: Verhulst, F. (eds) Asymptotic Analysis. Lecture Notes in Mathematics, vol 711. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0062955
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DOI: https://doi.org/10.1007/BFb0062955
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