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Existence of generalized homoclinic orbits for one parameter families of flows


Author: Konstantin Mischaikow
Journal: Proc. Amer. Math. Soc. 103 (1988), 59-68
MSC: Primary 58F14
DOI: https://doi.org/10.1090/S0002-9939-1988-0938645-7
MathSciNet review: 938645
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Abstract: We consider a one parameter family of flows generated by $ \dot x = f\left( {x,\lambda } \right)$, where $ \lambda \in \left[ {0,1} \right]$. We, also, assume that there exists an isolated invariant set, $ S$, which continues across the interval $ \left[ {0,1} \right]$, and that we know the connection matrices for $ \dot x = f\left( {x,1} \right)$ and $ \dot x = f\left( {x,0} \right)$. We then give conditions under which a Morse decomposition of $ S$ cannot continue across $ \left[ {0,1} \right]$. Furthermore, using the language of chain recurrence we define a generalized homoclinic orbit and give conditions under which such objects exist.


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DOI: https://doi.org/10.1090/S0002-9939-1988-0938645-7
Article copyright: © Copyright 1988 American Mathematical Society

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