Stability and Bifurcation Analysis of Two-term Fractional Difference Equation

We consider the linear equation including two fractional order difference operators, viz. ${\mathrm{\Delta }}^{\alpha }$ and ${\mathrm{\Delta }}^{\beta }$, $0<\beta <\alpha \le 1$. The sequence representation will be provided to find the solution in an easier way. The Z-transform will be used to find the boundary of the stable region in the complex plane. If the coefficient of the operator ${\mathrm{\Delta }}^{\beta }$ is negative (near 0), then we observe that the boundary curve has multiple points generating multiple stability regions. We provide all possible bifurcations in terms of parameters. An ample number of examples will be provided to support the results.