On the dynamics of a Riccati differential equation with perturbed delay

El-Sayed, Ahmed M. A.; Salman, Sanaa M.; AbdElfattah, AbdAllah A. F.

doi:10.21608/ejmaa.2023.192905.1007

On the dynamics of a Riccati differential equation with perturbed delay

Document Type : Regular research papers

Authors

¹ Department of Mathematics, Faculty of Science, Alexandria University, Egypt.

² Department of Mathematics, Faculty of Education, Alexandria University

³ Department of Mathematics and computer science, Faculty of Science, Alexandria University

10.21608/ejmaa.2023.192905.1007

Abstract

Within the scope of this study, we discuss the new concept of a perturbed delay. As a simple example, we will focus on a Riccati differential equation with a perturbed delay to illustrate this concept. We look at both the solution’s existence and its continuous dependence on the initial conditions. Analyses of Hopf bifurcations and the local stability of the fixed points are presented. In order to solve the delay differential equation with piecewise constant arguments, we adopt a discretization procedure. We do an analysis of the local stability of the discrete system. We use numerical simulations to draw out the results, like bifurcation diagrams, Lyapunov exponents, and phase diagrams. This helps us confirm our research and unearth more complex dynamics. We contrast the results of theoretical studies of the delayed Riccati differential equation and its perturbed equation. Our results show that, under certain conditions, the Riccati differential equation with perturbed delay is equivalent to the Riccati differential equation with the same dynamical properties.

Keywords