Oscillation Types and Bifurcations of a Nonlinear Second-Order Differential-Difference Equation

Abstract

This paper considers the second-order differential difference equation

$$X\left( t \right) = f\left( {X\left( {t - \tau } \right)} \right) - X\left( t \right)$$

with the constant delay τ > 0 and the piecewise constant function $f:\mathbb{R} \to \{ a,b\}$ with

$$\begin{gathered} f:\mathbb{R} \to \{ a,b\} \hfill \\ f\left( \xi \right): = \left\{ {_a^a } \right.{\text{ }}_{{\text{if }}}^{{\text{if }}} {\text{ }}_\xi ^\xi {\text{ }}_ \geqslant ^{\text{ < }} {\text{ }}_\Theta ^\Theta \hfill \\ \end{gathered} $$

Differential equations of this type occur in control systems, e.g., in heating systems and the pupil light reflex, if the controlling function is determined by a constant delay τ > 0 and the switch recognizes only the positions “on” [f(>) = a] and “off” [f(>) = b], depending on a constant threshold value Θ. By the nonsmooth nonlinearity the differential equation allows detailed analysis. It turns out that there is a rich solution structure. For a fixed set of parameters a, b, Θ, τ, infinitely many different periodic orbits of different minimal periods exist. There may be coexistence of three asymptotically stable periodic orbits (“multistability of limit cycles”). Stability or instability of orbits can be proven.