Metastable periodic patterns in singularly perturbed state-dependent delayed equations

•We show the existence of metastable oscillatory transients in state-dependent DDEs.•Transition layer equations are introduced for state dependent delay equations.•Transition layer solutions characterize the occurrences of metastability.•In the positive feedback case, metastability depends on a symmetry condition.•State dependent delays can render metastable otherwise short oscillatory transients.

We consider the scalar delayed differential equation ϵẋ(t)=−x(t)+f(x(t−r)), where ϵ>0ϵ>0, r=r(x,ϵ)=1+η(ϵ)R(x)r=r(x,ϵ)=1+η(ϵ)R(x), and ff represents either a monotone positive feedback df/dx>0df/dx>0 or a monotone negative feedback df/dx<0df/dx<0, plus some other hypotheses. When the delay is a constant, i.e. r(x,ϵ)=1r(x,ϵ)=1, this equation can support metastable rapidly oscillating solutions that are transients whose duration is of order exp(c/ϵ)exp(c/ϵ), for some c>0c>0. In this paper, combining analytic and numerical techniques, we investigate whether this metastable behavior persists when the delay r(x,ϵ)r(x,ϵ) depends non trivially on the state variable xx. More precisely, we proceed in three steps. First we explore theoretically and numerically the existence of branches of rapid periodic solutions for these equations. Then, to predict conditions for metastability, we introduce transition layer equations that depict the asymptotic shape of transient oscillations when ϵϵ tends to zero and η(ϵ)∼ϵη(ϵ)∼ϵ. Finally, these predictions are validated by numerical simulations. Numerical explorations are also performed for other scalings of η(ϵ)η(ϵ) with ϵϵ. Our conclusions are: (i) when η(0)≠0η(0)≠0, there are no metastable transients; (ii) when η(0)=0η(0)=0 and η′(0)η′(0) is finite, for monotone negative feedback, the metastable oscillations exist irrespective of choices of ff and RR; (iii) for monotone positive feedback, they require that the feedback ff and the delay RR satisfy extra conditions, such as ff being an odd function and the delay R(x)R(x) an even function. One novel result is that state-dependent delays may lead to metastable dynamics in equations that cannot support such regimes when the delay is constant.