Characteristic multipliers and stability of symmetric periodic solutions of $\dot x(t)=g(x(t-1))$

Abstract:

We study the scalar delay differential equation $\dot x(t) = g(x(t - 1))$ with negative feedback. We assume that the nonlinear function $g$ is odd and monotone. We prove that periodic solutions $x(t)$ of slowly oscillating type satisfying the symmetry condition $x(t) = - x(t - 2)$, $t \in {\mathbf {R}}$, are nondegenerate and have all nontrivial Floquet multipliers strictly inside the unit circle. This says that the periodic orbit $\{ {x_t}:t \in {\mathbf {R}}\}$ in the phase space $C[ - 1, 0]$ is orbitally exponentially asymptotically stable.