论文标题
分数订单延迟微分方程是否可以混乱，整数阶的对应物是稳定的？
Can a Fractional Order Delay Differential Equation be Chaotic Whose Integer-Order Counterpart is Stable?
论文作者
论文摘要
对于分数订单系统\ [d^αx（t）= f（x），\ quad 0 <α\ leq 1，\ \]可以具有$α$ viz $α_*$的临界值，这样该系统的稳定性是$ 0 <α<α_*$ and stable的$ 0 <α<α_*$ and stable，并且对于$α_*_*<α\α\α\ leq。通常，如果某些$α_0\ in（0,1）$稳定，那么对于所有$α<α_0。$。在本文中，我们表明存在一些延迟微分方程\ [d^αx（t）= f（x（x（x（x（x），x（t-t），x（t-t-t）），x（t-t-t-t））\ y的分数顺序相反。这些系统对于较高的分数阶值不稳定，对于较低的值而言稳定。引人注目的观察是$α= 0.27 $的混乱的示例，但稳定为$α= 1 $。在分数微分方程（FDE）中无法观察到这一点。我们在标量FDE中提供完整的分叉场景。
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.
