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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
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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  • [1] C. C. Conley, Isolated invariant sets and the Morse index, CBMS Regional Conf. Ser. Math., no. 38, Amer. Math. Soc., Providence, R.I., 1978. MR 511133 (80c:58009)
  • [2] -, The gradient structure of a flow. I, IBM Report RC 3932 (#17806), 1972.
  • [3] R. Franzosa, Index filtrations and connection matrices for partially ordered Morse decompositions, Thesis, Univ. of Wisconsin, Madison, 1984.
  • [4] -, The connection matrix theory for Morse decompositions, preprint.
  • [5] K. Mischaikow, Homoclinic orbits in Hamiltonian systems and heteroclinic orbits in gradient and gradient-like systems, LCDS Report #86-33. 1986.
  • [6] K. Mischaikow and G. Wolkowicz, Predator prey with group defense; A connection matrix approach, preprint.
  • [7] J. Reineck, Connecting orbits in one parameter families of flows, preprint. MR 967644 (89i:58128)
  • [8] D. Salamon, Connected simple systems and the Conley index of isolated invariant sets, Trans. Amer. Math. Soc. 29 (1985), 1-41. MR 797044 (87e:58182)
  • [9] J. Smoller, Shock waves and reaction diffusion equations, Springer-Verlag, New York, 1983. MR 688146 (84d:35002)

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