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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Bifurcation from a heteroclinic solution in differential delay equations

Author: Hans-Otto Walther
Journal: Trans. Amer. Math. Soc. 290 (1985), 213-233
MSC: Primary 34K15; Secondary 58F22
MathSciNet review: 787962
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Abstract: We study a class of functional differential equations $ \dot x(t) = af(x(t - 1))$ with periodic nonlinearity $ f:{\mathbf{R}} \to {\mathbf{R}},0 < f$ in $ (A,0)$ and $ f < 0$ in $ (0,B),f(A) = f(0) = f(B) = 0$ . Such equations describe a state variable on a circle with one attractive rest point (given by the argument $ \xi = 0$ of $ f$) and with reaction lag $ a$ to deviations. We prove that for a certain critical value $ a = {a_0}$ there exists a heteroclinic solution going from the equilibrium solution $ t \to A$ to the equilibrium $ t \to B$. For $ a - {a_0} > 0$, this heteroclinic connection is destroyed, and periodic solutions of the second kind bifurcate. These correspond to periodic rotations on the circle.

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Keywords: Characteristic equation, saddle point property, heteroclinic solution, Poincaré map, periodic solution of the second kind
Article copyright: © Copyright 1985 American Mathematical Society