TY - JOUR

T1 - Superstability and rigorous asymptotics in singularly perturbed 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 study the singularly perturbed state-dependent delay-differential equation. εẋ(t)=-x(t)-kx(t-r),r=r(x(t))=1+x(t), which is a special case of the equation. εẋ(t)=g(x(t),x(t-r)),r=r(x(t)). One knows that for every sufficiently small ε>0, Eq. (*) possesses at least one so-called slowly oscillating periodic solution, and moreover, the graph of every such solution approaches a specific sawtooth-like shape as ε→0. In this paper we obtain higher-order asymptotics of the sawtooth, including the location of the minimum and maximum of the solution with the form of the solution near these turning points, and as well an asymptotic formula for the period. Using these and other asymptotic formulas, we further show that the solution enjoys the property of superstability, namely, the nontrivial characteristic multipliers are of size O(ε) for small ε. This stability property implies that this solution is unique among all slowly oscillating periodic solutions, again for small ε.

AB - We study the singularly perturbed state-dependent delay-differential equation. εẋ(t)=-x(t)-kx(t-r),r=r(x(t))=1+x(t), which is a special case of the equation. εẋ(t)=g(x(t),x(t-r)),r=r(x(t)). One knows that for every sufficiently small ε>0, Eq. (*) possesses at least one so-called slowly oscillating periodic solution, and moreover, the graph of every such solution approaches a specific sawtooth-like shape as ε→0. In this paper we obtain higher-order asymptotics of the sawtooth, including the location of the minimum and maximum of the solution with the form of the solution near these turning points, and as well an asymptotic formula for the period. Using these and other asymptotic formulas, we further show that the solution enjoys the property of superstability, namely, the nontrivial characteristic multipliers are of size O(ε) for small ε. This stability property implies that this solution is unique among all slowly oscillating periodic solutions, again for small ε.

KW - Delay-differential equations

KW - Periodic solution

KW - Singular perturbation

KW - Stability

KW - State-dependent delay

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U2 - 10.1016/j.jde.2010.10.024

DO - 10.1016/j.jde.2010.10.024

M3 - Article

AN - SCOPUS:79952193631

VL - 250

SP - 4037

EP - 4084

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 11

ER -