Discontinuous dynamical system represents the Logistic retarded functional equation with two different delays

Abstract

In this work we are concerned with the discontinuous dynamical system representing the problem of the logistic retarded functional equation with two different delays,
$$\begin{aligned} & x(t)=\rho x\left(t-r_1\right)\left[1-x\left(t-r_2\right)\right], \quad t \in(0, T], \\ & x(t)=x_0, \quad t \leq 0 . \end{aligned}$$
The existence of a unique solution $x \in L^1[0, T]$ which is continuously dependence on the initial data, will be proved. The local stability at the equilibrium points will be studied. The bifurcation analysis and chaos will be discussed.