Non-Periodicity of Complex Caputo Like Fractional Differences
Abstract
1. Introduction
2. Properties of the Map
2.1. Stability of Fixed Points
2.2. Periodic Boundary Problem of (1)
2.3. Periodic Boundary Problem of FOM for Mandelbrot Case
- (a)
- The only fixed point of (12) is .
- (b)
- Nonzero solutions of
- (c)
- For any and , equation
2.4. Asymptotic 2-Periodic Solutions of FOM
3. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
References
- Tavazoei, M.S.; Haeri, M. A proof for non existence of periodic solutions in time invariant fractional order systems. Automatica 2009, 45, 1886–1890. [Google Scholar] [CrossRef]
- Tavazoei, M.S. A note on fractional-order derivatives of periodic functions. Automatica 2010, 46, 945–948. [Google Scholar] [CrossRef]
- Yazdani, M.; Salarieh, H. On the existence of periodic solutions in time-invariant fractional order systems. Automatica 2011, 47, 1834–1837. [Google Scholar] [CrossRef]
- Kang, Y.-M.; Xie, Y.; Lu, J.-C.; Jiang, J. On the nonexistence of non-constant exact periodic solutions in a class of the Caputo fractional-order dynamical systems. Nonlinear Dyn. 2015, 82, 1259–1267. [Google Scholar] [CrossRef]
- Hoti, M. A note on the local stability theory for Caputo fractional planar system. J. Fract. Calc. Appl. 2022, 13, 1–8. [Google Scholar]
- Kaslik, E.; Sivasundaram, S. Non-existence of periodic solutions in fractional-order dynamical systems and a remarkable difference between integer and fractional-order derivatives of periodic functions. Nonlinear Anal. Real World Appl. 2012, 13, 1489–1497. [Google Scholar] [CrossRef]
- Fečkan, M. Note on periodic and asymptotically periodic solutions of fractional differential equations. In Applied Mathematical Analysis: Theory, Methods, and Applications; Dutta, H., Peters, J., Eds.; Studies in Systems, Decision and Control; Springer: Cham, Switzerland, 2020; Volume 177, pp. 153–185. [Google Scholar]
- Shen, J.; Lam, J. Non-existence of finite-time stable equilibria in fractional-order nonlinear systems. Automatica 2014, 50, 547–551. [Google Scholar] [CrossRef]
- Wang, J.R.; Fečkan, M.; Zhou, Y. Nonexistence of periodic solutions and asymptotically periodic solutions for fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 2013, 18, 246–256. [Google Scholar] [CrossRef]
- Diblíka, J.; Fečkan, M.; Pospíšil, M. Nonexistence of periodic solutions and S-asymptotically periodic solutions in fractional difference equations. Appl. Math. Comput. 2015, 257, 230–240. [Google Scholar] [CrossRef]
- Pospíšil, M. A note on fractional difference equations with periodic and S-asymptotically periodic right-hand sides. J. Math. Sci. 2022, 265, 669–681. [Google Scholar] [CrossRef]
- Area, I.; Nieto, J.L.J.J. On quasi-periodic properties of fractional sums and fractional differences of periodic functions. Appl. Math. Comput. 2016, 273, 190–200. [Google Scholar] [CrossRef]
- Abdelaziz, M.A.M.; Ismail, A.I.; Abdullah, F.A.; Mohd, M.H. Bifurcations and chaos in a discrete SI epidemic model with fractional order. Adv. Differ. Equ. 2018, 2018, 1–19. [Google Scholar] [CrossRef]
- Dzieliński, A.; Sierociuk, D. Stability of discrete fractional order state-space systems. J. Vib. Control 2008, 14, 1543–1556. [Google Scholar] [CrossRef]
- Wang, Y.; Li, X.; Wang, D.; Liu, S. A brief note on fractal dynamics of fractional Mandelbrot sets. Appl. Math. Comput. 2022, 432, 127353. [Google Scholar] [CrossRef]
- Danca, M.-F.; Fečkan, M.; Kuznetsov, N. Chaos control in the fractional order logistic map via impulses. Nonlinear Dyn. 2019, 98, 1219–1230. [Google Scholar] [CrossRef]
- Danca, M.-F. Puu system of fractional order and its chaos suppression. Symmetry 2020, 12, 340. [Google Scholar] [CrossRef]
- Danca, M.-F.; Fečkan, M.; Kuznetsov, N.; Chen, G. Coupled discrete fractional-order logistic maps. Mathematics 2021, 9, 2204. [Google Scholar] [CrossRef]
- Anastassiou, G.A. Discrete fractional calculus and inequalities. arXiv 2009, arXiv:0911.3370. [Google Scholar]
- Čermák, J.; Gyori, I.; Nechvátal, L. On explicit stability conditions for a linear fractional difference system. Fract. Calc. Appl. Anal. 2015, 18, 651–672. [Google Scholar] [CrossRef]
- Fečkan, M.; Pospíšil, M. Note on fractional difference Gronwall inequalities. Electron. J. Qual. Theory Differ. Equ. 2014, 44, 1–18. [Google Scholar] [CrossRef]
- Peterson, A.C.; Goodrich, C. Discrete Fractional Calculus; Springer: Cham, Switzerland, 2015. [Google Scholar]
- Danca, M.-F.; Fečkan, M. Mandelbrot set and Julia sets of fractional order. arXiv 2022, arXiv:2210.02037. [Google Scholar]
- Stuart, A.; Humphries, A.R. Dynamical Systems and Numerical Analysis; Cambridge University Press: Cambridge, UK, 1998. [Google Scholar]
- Gautschi, W. Some elementary inequalities relating to the gamma and incompletegamma function. J. Math. Phys. 1959, 38, 77–81. [Google Scholar] [CrossRef]
- Kershaw, D. Some extensions of W. Gautschi’s inequalities for the gamma function. Math. Comput. 1983, 41, 607–611. [Google Scholar]
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Fečkan, M.; Danca, M.-F. Non-Periodicity of Complex Caputo Like Fractional Differences. Fractal Fract. 2023, 7, 68. https://doi.org/10.3390/fractalfract7010068
Fečkan M, Danca M-F. Non-Periodicity of Complex Caputo Like Fractional Differences. Fractal and Fractional. 2023; 7(1):68. https://doi.org/10.3390/fractalfract7010068
Chicago/Turabian StyleFečkan, Michal, and Marius-F. Danca. 2023. "Non-Periodicity of Complex Caputo Like Fractional Differences" Fractal and Fractional 7, no. 1: 68. https://doi.org/10.3390/fractalfract7010068
APA StyleFečkan, M., & Danca, M.-F. (2023). Non-Periodicity of Complex Caputo Like Fractional Differences. Fractal and Fractional, 7(1), 68. https://doi.org/10.3390/fractalfract7010068