Can a Fractional Order Delay Differential Equation be Chaotic Whose Integer-Order Counterpart is Stable?

TL;DR

This paper investigates fractional order delay differential equations, revealing cases where systems are chaotic at low fractional orders but stable at higher orders, contrary to typical fractional systems.

Contribution

It demonstrates the existence of fractional delay differential equations that exhibit opposite stability behavior compared to non-delay fractional systems, including chaos at low order and stability at high order.

Findings

Existence of fractional delay systems that are chaotic at low  and stable at high .

Complete bifurcation scenarios for scalar fractional delay differential equations.

Chaotic behavior observed at .27 and stability at .

Abstract

For the fractional order systems \[D^\alpha x(t)=f(x),\quad 0<\alpha\leq 1,\] one can have a critical value of $\alpha$ viz $\alpha_*$ such that the system is stable for $0<\alpha<\alpha_*$ and unstable for $\alpha_*<\alpha\leq 1$. In general, if such system is stable for some $\alpha_0\in(0,1)$ then it remains stable for all $\alpha<\alpha_0.$ In this paper, we show that there are some delay differential equations \[D^\alpha x(t)=f(x(t),x(t-\tau))\] of the fractional order which behave in an exactly opposite way. These systems are unstable for higher values of fractional order and stable for the lower values. The striking observation is the example which is chaotic for $\alpha=0.27$ but stable for $\alpha=1$. This cannot be observed in the fractional differential equations (FDEs) without delay. We provide the complete bifurcation scenarios in the scalar FDEs.