Preservation of homoclinic orbits under discretization of delay differential equations

Pages: 275 - 299, Volume 31, Issue 1, September 2011

doi:10.3934/dcds.2011.31.275       Abstract        References        Full Text (518.5K)       Related Articles

Yingxiang Xu - School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China (email)
Yongkui Zou - School of Mathematics, Jilin University, Changchun 130012, China (email)

Abstract: In this paper, we propose a nondegenerate condition for a homoclinic orbit with respect to a parameter in delay differential equations. Based on this nondegeneracy we describe and investigate the regularity of the homoclinic orbit together with parameter. Then we show that a forward Euler method, when applied to a one-parameteric system of delay differential equations with a homoclinic orbit, also exhibits a closed loop of discrete homoclinic orbits. These discrete homoclinic orbits tend to the continuous one by the rate of $O(\varepsilon)$ as the step-size $\varepsilon$ goes to $0$. And the corresponding parameter varies periodically with respect to a phase parameter with period $\varepsilon$ while the orbit shifts its index after one revolution. We also show that at least two homoclinic tangencies occur on this loop. By numerical simulations, the theoretical results are illustrated, and the possibility of extending theoretical results to the implicit and higher order numerical schemes is discussed.

Keywords:  Delay differential equations, forward Euler method, homoclinic orbits, homoclinic tangencies.
Mathematics Subject Classification:  Primary: 37C29, 37N30; Secondary: 65P30.

Received: February 2010;      Revised: November 2010;      Published: June 2011.

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