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Symbolic Description of Orbits: The Stable Manifolds of C1 and C2

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Part of the Applied Mathematical Sciences book series (AMS,volume 41)

Abstract

At various points in the last four chapters we have found it useful to be able to describe periodic orbits and trajectories with sequences of two symbols. When we studied homoclinic explosions in Chapter 2, we saw that we were rigorously justified in describing the periodic orbits and trajectories born (or destroyed) in these bifurcations with sequences of two symbols; each symbol corresponded to one of the two tubes surrounding (at the critical r-value) the two branches of the homoclinic orbit, and trajectories were assigned symbolic sequences according to the order in which they journeyed through these tubes. These descriptions were only local. In Chapters 3 through 5, we found we had need for a global method of describing orbits and trajectories. In Chapters 3 and 5, we defined symbol sequences according to the order in which trajectories intersected two halves of some suitable return surface. In Chapter 4, we assigned symbol sequences to periodic orbits according to whether their successive local maxima in the z-variable lay in x > 0 or x < 0. These two methods appeared to be equivalent.

Keywords

  • Periodic Orbit
  • Unstable Manifold
  • Homoclinic Orbit
  • Stable Manifold
  • Symbol Sequence

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 1982 Springer-Verlag New York Inc.

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Sparrow, C. (1982). Symbolic Description of Orbits: The Stable Manifolds of C1 and C2. In: The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors. Applied Mathematical Sciences, vol 41. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-5767-7_6

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  • DOI: https://doi.org/10.1007/978-1-4612-5767-7_6

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-387-90775-8

  • Online ISBN: 978-1-4612-5767-7

  • eBook Packages: Springer Book Archive