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Stability and dynamics of complex order fractional difference equations

Author

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  • Bhalekar, Sachin
  • Gade, Prashant M.
  • Joshi, Divya

Abstract

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.

Suggested Citation

  • Bhalekar, Sachin & Gade, Prashant M. & Joshi, Divya, 2022. "Stability and dynamics of complex order fractional difference equations," Chaos, Solitons & Fractals, Elsevier, vol. 158(C).
    • Handle: RePEc:eee:chsofr:v:158:y:2022:i:c:s0960077922002739
    DOI: 10.1016/j.chaos.2022.112063
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    References listed on IDEAS

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    1. Pakhare, Sumit S. & Daftardar-Gejji, Varsha & Badwaik, Dilip S. & Deshpande, Amey & Gade, Prashant M., 2020. "Emergence of order in dynamical phases in coupled fractional gauss map," Chaos, Solitons & Fractals, Elsevier, vol. 135(C).
    2. Plastino, A.R & Giordano, C & Plastino, A & Casas, M, 2004. "Liouville equation and the q-statistical formalism," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 336(3), pages 376-390.
    3. Prashant M. Gade & Sachin Bhalekar, 2021. "On Fractional Order Maps And Their Synchronization," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 29(06), pages 1-9, September.
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