December  2009, 2(4): 967-1023. doi: 10.3934/dcdss.2009.2.967

Generalized exchange lemmas and orbits heteroclinic to invariant manifolds

1. 

Department of Mathematics, University of North Carolina, Chapel Hill NC 27599-3250, United States

2. 

Division of Applied Mathematics, Brown University, Providence, RI 02912, United States

Received  October 2008 Revised  June 2009 Published  September 2009

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.
Citation: Christopher K. R. T. Jones, Siu-Kei Tin. Generalized exchange lemmas and orbits heteroclinic to invariant manifolds. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 967-1023. doi: 10.3934/dcdss.2009.2.967
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