Analysis of 2-Term Fractional-Order Delay Differential Equations

Abstract

The value of a state variable at past time usually affects its rate of change at the present time. So, it is very natural to consider the delay while modeling the real-life systems. Further, the nonlocal fractional derivative operator is also useful in modeling memory in the system. Hence, the models involving delay as well as fractional derivative are very important. In this chapter, we review the basic results regarding the dynamical systems, fractional calculus, and delay differential equations. Further, we analyze 2-term nonlinear fractional-order delay differential equation & #x0024; & #x0024;D^\alpha x + c D^\beta x = f\left(x,x_\tau \right) & #x0024; & #x0024;, with constant delay & #x0024; & #x0024;\tau > 0 & #x0024; & #x0024; and fractional orders & #x0024; & #x0024;0 < \alpha < \beta < 1 & #x0024; & #x0024;. We present a numerical method for solving such equations and present an example exhibiting chaotic oscillations.