Floquet Multipliers of a Periodic Solution Under State-Dependent Delay
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Floquet Multipliers of a Periodic Solution Under State-Dependent Delay Therese Mur Voigt1 · Hans-Otto Walther1
Received: 30 April 2020 / Revised: 21 August 2020 / Accepted: 2 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We consider a periodic function p : R → R of minimal period 4 which satisfies a family of delay differential equations x (t) = g(x(t − d (xt ))), ∈ R,
(0.1)
with a continuously differentiable function g : R → R and delay functionals d : C([−2, 0], R) → (0, 2). The solution segment xt in Eq. (0.1) is given by xt (s) = x(t + s). For every ∈ R the solutions of Eq. (0.1) defines a semiflow of continuously differentiable solution operators S,t : x0 → xt , t ≥ 0, on a continuously differentiable submanifold X of the space C 1 ([−2, 0], R), with codim X = 1. At = 0 the delay is constant, d0 (φ) = 1 everywhere, and the orbit O = { pt : 0 ≤ t < 4} ⊂ X 0 of the periodic solution is extremely stable in the sense that the spectrum of the monodromy operator M0 = DS0,4 ( p0 ) is σ0 = {0, 1}, with the eigenvalue 1 being simple. For || ∞ there is an increasing contribution of variable, state-dependent delay to the time lag d (xt ) = 1 + · · · in Eq. (0.1). We study how the spectrum σ of M = DS,4 ( p0 ) changes if || grows from 0 to ∞. A main result is that at = 0 an eigenvalue () < 0 of M bifurcates from 0 ∈ σ0 and decreases to −∞ as || ∞. Moreover we verify the spectral hypotheses for a period doubling bifurcation from the periodic orbit O at the critical parameter ∗ where (∗ ) = −1. Keywords Delay differential equation · State-dependent delay · Periodic solution · Floquet multipliers Mathematics Subject Classification 34K13 · 34K18 · 37L99
Dedicated to the memory of Pavol Brunovský.
B
Hans-Otto Walther [email protected] Therese Mur Voigt [email protected]
1
Mathematisches Institut, Universität Gießen, Arndtstr. 2, 35392 Gießen, Germany
123
Journal of Dynamics and Differential Equations
1 Introduction The present paper is a case study of the impact of variable delay on periodic motion. We consider a periodic solution of an autonomous differential equation with a constant time lag and ask how stability properties of the periodic solution change when the constant time lag is replaced by a variable, state-dependent delay—in such a way that the periodic solution is preserved. Let an odd continuously differentiable function g : R → R be given with g(ξ ) = 1 on (−∞, b] and g (ξ ) < 0 on (−b, b), 1 for some b ∈ 0, 3 . We begin with the equation x (t) = g(x(t − 1))
(1.1)
which models negative feedback with respect to a stationary state (here given by ξ = 0), for a scalar variable and with a constant time lag. Proceeding as in [2, Section XV.1] we find a periodic solution: Take any continuous function φ : [−1, 0] → R with φ(t) ≤ −b on [−1, −b] and φ(t) = t on [−b, 0]. Integrate Eq. (1.1) successively over the intervals [0, 1 − b], [1 − b, 1 + b], [1 + b, 2], with the initial condition x(t) = φ(t) on
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