A STATE-DEPENDENT DELAY EQUATION WITH NEGATIVE FEEDBACK AND "MILDLY UNSTABLE" RAPIDLY OSCILLATING PERIODIC SOLUTIONS

作者：Kennedy Benjamin B^*

来源：Discrete and Continuous Dynamical Systems-Series B, 2013, 18(6): 1633-1650.

DOI：10.3934/dcdsb.2013.18.1633

• State-dependent delay
• periodic solution
• stability

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 x f (x) < 0 for x not equal 0. We identify a special family of such equations each of which has a "rapidly oscillating" periodic solution p. The initial segment p 0 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 epsilon > 0, we can choose f and d such that the spectral radius of the derivative of R at p 0 is less than 1 + epsilon. The spectral radii are computed via a semiconjugacy of R with a finite-dimensional map.

• 出版日期2013-8