Advances in Differential Equations

Complicated histories close to a homoclinic loop generated by variable delay

Hans-Otto Walther

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There exist a delay functional $d$ on an open set $U \subset C([-2,0],$ $\mathbb R)$ and a parameter $\alpha\gt 0$ so that the equation $$ x'(t)=-\alpha\,x(t-d(x_t)) $$ has a solution which is homoclinic to zero, the zero equilibrium is hyperbolic with 2-dimensional unstable manifold, and the stable manifold and unstable manifold intersect transversely along the homoclinic flowline. We prove that close to the homoclinic loop there exists a kind of complicated motion, which involves partially unstable behavior. This does not require any relation between the growth rate in the unstable space and the decay rate in the leading stable plane of the linearized semiflow.

Article information

Adv. Differential Equations, Volume 19, Number 9/10 (2014), 911-946.

First available in Project Euclid: 1 July 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34K23: Complex (chaotic) behavior of solutions 37D45: Strange attractors, chaotic dynamics 37L99: None of the above, but in this section


Walther, Hans-Otto. Complicated histories close to a homoclinic loop generated by variable delay. Adv. Differential Equations 19 (2014), no. 9/10, 911--946.

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