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Homoclinic Explosions: The First Homoclinic Explosion

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

Abstract

When r > 1 there is a two-dimensional sheet of initial values in R3 from which trajectories tend towards the origin. This two-dimensional sheet is called the stable manifold of the origin. Near the origin we know that this sheet looks like a plane (the plane associated with the two negative eigenvalues of the flow linearized near the origin) and if we wished we could approximate its position elsewhere by integrating the equations in backwards time with initial conditions lying on this plane and near to the origin. It appears that when r is only moderately larger than one, the stable manifold of the origin divides R3 into two halves in a fairly simple way. Trajectories started in one half-space tend towards C1 and trajectories started in the other half-space tend towards C2. Trajectories started on the stable manifold of the origin tend, of course, towards the origin. (See Fig. 2.1.)

Keywords

  • Periodic Orbit
  • Unstable Manifold
  • Homoclinic Orbit
  • Stable Manifold
  • Simple Periodic Orbit

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). Homoclinic Explosions: The First Homoclinic Explosion. 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_2

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  • DOI: https://doi.org/10.1007/978-1-4612-5767-7_2

  • 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