TY - JOUR

T1 - Stability of periodic solutions of state-dependent delay-differential equations

AU - Mallet-Paret, John

AU - Nussbaum, Roger D.

N1 - Funding Information:
E-mail addresses: jmp@dam.brown.edu (J. Mallet-Paret), nussbaum@math.rutgers.edu (R.D. Nussbaum). 1 Partially supported by NSF DMS-0500674, and by the Center for Nonlinear Analysis, Rutgers University. 2 Partially supported by NSF DMS-0701171, and by the Lefschetz Center for Dynamical Systems, Brown University.

PY - 2011/6/1

Y1 - 2011/6/1

N2 - We consider a class of autonomous delay-differential equations. ż(t)=f(zt) which includes equations of the form. ż(t)=g(z(t),z(t-r1),...,z(t-rn)),ri=ri(z(t))for1≤i≤n, with state-dependent delays ri(z(t))≥0. The functions g and ri 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 μ satisfying |μ|<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 ri smooth, the associated function f is not smooth in function space. Techniques of Hartung, Krisztin, Walther, and Wu are employed to resolve these issues.

AB - We consider a class of autonomous delay-differential equations. ż(t)=f(zt) which includes equations of the form. ż(t)=g(z(t),z(t-r1),...,z(t-rn)),ri=ri(z(t))for1≤i≤n, with state-dependent delays ri(z(t))≥0. The functions g and ri 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 μ satisfying |μ|<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 ri smooth, the associated function f is not smooth in function space. Techniques of Hartung, Krisztin, Walther, and Wu are employed to resolve these issues.

KW - Asymptotic phase

KW - Delay-differential equations

KW - Orbital stability

KW - Periodic solution

KW - State-dependent delay

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U2 - 10.1016/j.jde.2010.10.023

DO - 10.1016/j.jde.2010.10.023

M3 - Article

AN - SCOPUS:79952191925

SN - 0022-0396

VL - 250

SP - 4085

EP - 4103

JO - Journal of Differential Equations

JF - Journal of Differential Equations

IS - 11

ER -