Multi-cycle periodic solutions of a differential equation with delay that switches periodically (Q6163932)

Scalar delay differential equation of the form \[ x^{\prime}(t)=-d x(t)+f(x(t-\tau(t))) \tag{DDE} \] is considered with the piecewise constant nonlinearity \(f(x)=-\mathrm{sign}(x)\) and a \(T\)-periodic piecewise constant delay \(\tau\) defined as \(\tau(t)=\tau_2\) if \(\bmod(t,T)\ge\gamma\) and \(\tau(t)=\tau_1=\beta\tau_2\) if \(\bmod(t,T)<\gamma\) (where \(d, \tau_2, \beta, T>\gamma\) are all positive constants). Sufficient conditions for the existence of the so-called \(m\)-cycle periodic solutions with the period \(nT\) for some \(n\in\mathbf{N}\) are established. Roughly speaking, an \(m\)-cycle periodic solution is such that it has all distinct shapes on consecutive positive and negative oscillatory semi-cycles (distances between consecutive zeros). Conditions are also indicated when the equation (DDE) does not have such periodic solutions. The analysis is based on the explicit calculation of forward solutions, which is possible due to the piecewise constant values of \(f\) and \(\tau\). Some related numerical simulations are also provided. The motivation for consideration of this delay differential equation comes from a biological population model.