We consider state-dependent delay equations of the form x′(t)=f(x(t−d(x(t)))) where d is smooth and f is smooth, bounded, nonincreasing, and satisfies the negative feedback condition xf(x)x≠0. We identify a special family of such equations each of which has a ``rapidly oscillating" periodic solution p. The initial segment p0 of p is the fixed point of a return map R that is differentiable in an appropriate setting.
We show that, although all the periodic solutions p we consider are unstable, the stability can be made arbitrarily mild in the sense that, given ϵ>0, we can choose f and d such that the spectral radius of the derivative of R at p0 is less than 1+ϵ. The spectral radii are computed via a semiconjugacy of R with a finite-dimensional map.
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Kennedy, Benjamin B. “A State-Dependent Delay Equation with Negative Feedback and ‘Mildly Unstable’ Rapidly Oscillating Periodic Solutions.” Discrete and Continuous Dynamical System Series B 18.6 (2013): 1633-1650.
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