A Shilnikov Phenomenon Due to State-Dependent Delay, by Means of the Fixed Point Index

• Published in: Journal of dynamics and differential equations Vol. 28; no. 3-4; pp. 627 - 688
• Main Authors: Lani-Wayda, Bernhard, Walther, Hans-Otto
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.09.2016; Springer Nature B.V
• Subjects: Applications of Mathematics; Differential equations; Mathematics; Mathematics and Statistics; Ordinary Differential Equations; Partial Differential Equations; Symbolic dynamics; Fixed point index; State-dependent delay; 34K23; 37B10
• ISSN: 1040-7294; 1572-9222
• DOI: 10.1007/s10884-014-9420-z

The first part of this paper is a general approach towards chaotic dynamics for a continuous map f : X ⊃ M → X which employs the fixed point index and continuation. The second part deals with the differential equation x ′ ( t ) = - α x ( t - d Δ ( x t ) ) . with state-dependent delay. For a suitable parameter α close to 5 π / 2 we construct a delay functional d Δ , constant near the origin, so that the previous equation has a homoclinic solution, h ( t ) → 0 as t → ± ∞ , with certain regularity properties of the linearization of the semiflow along the flowline t ↦ h t . The third part applies the method from the beginning to a return map which describes solution behaviour close to the homoclinic loop, and yields the existence of chaotic motion.