Stability of periodic solutions of state-dependent delay-differential equations

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:250

Article

Malet-Paret, John¹  Nussbaum, Roger D.² 

[1] Brown Univ, Div Appl Math, Providence, RI 02912 USA

[2] Rutgers State Univ, Dept Math, Piscataway, NJ 08854 USA

关键词: Delay-differential equations;    State-dependent delay;    Periodic solution;    Orbital stability;    Asymptotic phase;   

DOI  :  10.1016/j.jde.2010.10.023

来源: Elsevier

【 摘 要 】

We consider a class of autonomous delay-differential equations (z) over dot(t) = f(z(t)) which includes equations of the form (z) over dot(t) = g(z(t), z(t - r(1)),..., z(t - r(n))), r(i) = r(i)(z(t)) for 1 <= i <= n, with state-dependent delays r(i)(z(t)) >= 0. The functions g and r(i) satisfy appropriate smoothness conditions. We assume there exists a periodic solution z = x(t) which is linearly asymptotically stable, namely with all nontrivial characteristic multipliers mu satisfying vertical bar mu vertical bar < 1. We prove that the appropriate nonlinear stability properties hold for x(t), namely, that this solution is asymptotically orbitally stable with asymptotic phase, and enjoys an exponential rate of attraction given in terms of the leading nontrivial characteristic multiplier. A principal difficulty which distinguishes the analysis of equations such as (*) from ones with constant delays, is that even with g and r(i) smooth, the associated function f is not smooth in function space. Techniques of Hartung, Krisztin, Walther, and Wu are employed to resolve these issues. (C) 2010 Published by Elsevier Inc.

【 授权许可】

Free