Generalized exchange lemmas and orbits heteroclinic to invariant manifolds

Christopher K. R. T. Jones; Siu-Kei Tin

doi:10.3934/dcdss.2009.2.967

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2009, Volume 2, Issue 4: 967-1023. Doi: 10.3934/dcdss.2009.2.967

This issue Previous Article A survey of some results on overstability and bifurcation delay Next Article

Generalized exchange lemmas and orbits heteroclinic to invariant manifolds

Christopher K. R. T. Jones^1, and
Siu-Kei Tin^2,

1.
Department of Mathematics, University of North Carolina, Chapel Hill NC 27599-3250
2.
Division of Applied Mathematics, Brown University, Providence, RI 02912

Received: September 30, 2008

Revised: May 31, 2009

Published: November 2009

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Abstract

The construction of orbits with specific asymptotic properties, such as orbits that are heteroclinic or homoclinic to certain invariant sets, involves tracking stable and unstable manifolds around the system's phase space. This work addresses how, in some generality, the tracking can be achieved during the passage near a distinguished invariant manifold in the phase space. This leads to a very general form of the Exchange Lemma and it is further shown how the lemma can be used in the construction of distinguished homoclinic and heteroclinic orbits.

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Mathematics Subject Classification: Primary: 34C37; Secondary: 34C26.

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