Stability and dynamics of complex order fractional difference equations

• Published in: Chaos, solitons and fractals solitons and fractals, 2022-05, Vol.158, p.112063, Article 112063
• Main Authors: Bhalekar, Sachin, Gade, Prashant M., Joshi, Divya
• Format: Article
• Language: English
• Subjects: Complex order; Fractional difference equation; Stability
• ISSN: 0960-0779; 1873-2887
• DOI: 10.1016/j.chaos.2022.112063

We extend the definition of n-dimensional difference equations to complex order. We investigate the stability of linear systems defined by an n-dimensional matrix and derive the conditions for the stability of zero solution of linear systems. For the one-dimensional case, we find that the stability region, if any is enclosed by a boundary curve and we obtain a parametric equation for the same. Furthermore, we find that there is no stable region if this parametric curve is self-intersecting. Even for the real eigenvalues, the solutions can be complex and dynamics in one-dimension is richer than the case for real order. These results can be extended to n-dimensions. For nonlinear systems, we observe that the stability of the linearized system determines the stability of the equilibrium point. •Extended the definition of n-dimensional difference equations to complex order•Provided the parametric equations for the boundary of stable region•If the boundary curve has multiple points, then the system is unstable.•The dynamics is richer than the real-order case.•Stability of the linearized system determines the stability of the equilibrium point of nonlinear system.