Metastable periodic patterns in singularly perturbed state-dependent delayed equations

We consider the scalar delayed differential equation ϵ x ̇ ( t ) = − x ( t ) + f ( x ( t − r ) ) , where ϵ > 0 , r = r ( x , ϵ ) = 1 + η ( ϵ ) R ( x ) , and f represents either a monotone positive feedback d f / d x > 0 or a monotone negative feedback d f / d x < 0 , plus some other hypotheses. When the delay is a constant, i.e. r ( x , ϵ ) = 1 , this equation can support metastable rapidly oscillating solutions that are transients whose duration is of order exp ( c / ϵ ) , for some c > 0 . In this paper, combining analytic and numerical techniques, we investigate whether this metastable behavior persists when the delay r ( x , ϵ ) depends non trivially on the state variable x . 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 ϵ tends to zero and η ( ϵ ) ∼ ϵ . Finally, these predictions are validated by numerical simulations. Numerical explorations are also performed for other scalings of η ( ϵ ) with ϵ . Our conclusions are: (i) when η ( 0 ) ≠ 0 , there are no metastable transients; (ii) when η ( 0 ) = 0 and η ′ ( 0 ) is finite, for monotone negative feedback, the metastable oscillations exist irrespective of choices of f and R ; (iii) for monotone positive feedback, they require that the feedback f and the delay R satisfy extra conditions, such as f being an odd function and the delay 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.