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

For the fractional order systems $D^αx(t)=f(x),\quad 0<α\leq 1,$ one can have a critical value of $α$ viz $α_*$ such that the system is stable for $0<α<α_*$ and unstable for $α_*<α\leq 1$. In general, if such system is stable for some $α_0\in(0,1)$ then it remains stable for all $α<α_0.$ In this paper, we show that there are some delay differential equations $D^αx(t)=f(x(t),x(t-τ))$ 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 $α=0.27$ but stable for $α=1$. This cannot be observed in the fractional differential equations (FDEs) without delay. We provide the complete bifurcation scenarios in the scalar FDEs.

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