Periodic solutions in a class of periodic switching delay differential equations

In this paper, we explore the dynamical properties of a class of nonlinear systems governed by delay differential equations with multitime periodic switching. The systems incorporate piecewise-smooth birth and death functions to capture complex population dynamics under seasonal variations. Assuming monotonicity for both birth and death functions, we obtain a novel equivalence result: when the delay is a positive integer multiple of the switching period, the existence and stability of periodic solutions for the systems are equivalent to those in the nondelay case. To illustrate and validate the theoretical findings, a logistic model with seasonal switching is presented. Numerical simulations further confirm that the system exhibits consistent dynamical behaviors across varying delay values.