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The evolution of periodic orbits close to homoclinic points

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Abstract

For conservative dynamical systems having two degrees of freedom Birkhoff has established the existence of two classes of periodic orbits. The first consists of stable-unstable pairs close to periodic orbits of the stable type, and the second of orbits having fixed points (in a suitable surface of section) close to homoclinic points. In this paper orbits of the latter type are listed, and their evolution followed as a function of the energy. For the energy at which they were first computed, all were unstable; but they evolved, with diminishing energy, into one orbit of the stable type which appears to be a member of the first class of orbits mentioned above.

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References

  • Birkhoff, G. D.: 1933,Dynamical Systems, published by the American Math. Soc., see Chapter 8, Section 2. Subsequent proofs, such as that by Birkhoff, G. D. and Lewis, D. C.: 1933,Annali di Matematica Pura ed Applicata, series 4,12, 117 are concerned with systems of higher degrees of freedom.

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Authors and Affiliations

  1. North Carolina State University, 27607, Raleigh, N.C., USA

    J. M. A. Danby

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  1. J. M. A. Danby
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Danby, J.M.A. The evolution of periodic orbits close to homoclinic points. Celestial Mechanics 8, 273–280 (1973). https://doi.org/10.1007/BF01231428

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  • DOI: https://doi.org/10.1007/BF01231428

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