×
Cánovas, Jose S.

Classical versus fractional difference equations: the logistic case. (English) Zbl 1537.39005

Fract. Calc. Appl. Anal. 26, No. 5, 2281-2297 (2023).

MSC:

39A13 Difference equations, scaling (\(q\)-differences)
39A12 Discrete version of topics in analysis
26A33 Fractional derivatives and integrals

Keywords:

fractional difference equations; periodicity; dynamical complexity
Cite Review PDF
Full Text: DOI

References:

[1] Adler, RL; Konheim, AG; McAndrew, MH, Topological entropy, Trans. Amer. Math. Soc., 114, 309-319 (1965) · Zbl 0127.13102 · doi:10.1090/S0002-9947-1965-0175106-9
[2] Alsedá, L.; Llibre, J.; Misiurewicz, M., Combinatorial Dynamics and Entropy in Dimension One (1993), Singapore: World Scientific Publishing, Singapore · Zbl 0843.58034 · doi:10.1142/1980
[3] Bandt, C.; Keller, G.; Pompe, B., Entropy of interval maps via permutations, Nonlinearity, 15, 1595-1602 (2002) · Zbl 1026.37027 · doi:10.1088/0951-7715/15/5/312
[4] Bandt, C.; Pompe, B., Permutation entropy-a natural complexity measure for time series, Phys. Rev. Lett., 88 (2002) · doi:10.1103/PhysRevLett.88.174102
[5] Block, LS; Coppel, WA, Dynamics in One Dimension (1992), Berlin-Heidelberg: Springer, Berlin-Heidelberg · Zbl 0746.58007 · doi:10.1007/BFb0084762
[6] Block, LS; Keesling, J.; Li, SH; Peterson, K., An improved algorithm for computing topological entropy, J. Stat. Phy., 55, 929-939 (1989) · Zbl 0714.54018 · doi:10.1007/BF01041072
[7] Cánovas, JS, Estimating topological entropy from individual orbits, Int. J. Comput. Math., 86, 1901-1906 (2009) · Zbl 1185.37097 · doi:10.1080/00207160902825184
[8] Cánovas, JS; Guillamón, A., Permutations and time series analysis, Chaos, 19 (2009) · Zbl 1311.37063 · doi:10.1063/1.3238256
[9] Cánovas, JS; Muñoz-Guillermo, M., On the complexity of economic dynamics: An approach through topological entropy, Chaos Solitons Fractals, 103, 163-176 (2017) · Zbl 1376.91113 · doi:10.1016/j.chaos.2017.05.030
[10] de Melo, W.; van Strien, S., One Dimensional Dynamics (1993), Berlin: Springer, Berlin · Zbl 0791.58003 · doi:10.1007/978-3-642-78043-1
[11] Devaney, RL, An introduction to Chaotic Dynamical Systems (2003), Boca Raton: CRC Press, Boca Raton · Zbl 1025.37001
[12] Edelman, M., Caputo standard \(\alpha \)-family of maps: fractional difference vs. fractional, Chaos, 24 (2014) · Zbl 1345.39012 · doi:10.1063/1.4885536
[13] Edelman, M.; Edelman, M.; Macau, E.; Sanjuan, M., Universality in Systems with power-law memory and fractional dynamics, Chaotic, Fractional, and Complex Dynamics: New Insights and Perspectives understanding complex systems (2018), Cham: Springer, Cham · Zbl 1390.34018
[14] Edelman, M., On stability of fixed points and chaos in fractional systems, Chaos, 28 (2018) · Zbl 1390.39024 · doi:10.1063/1.5016437
[15] Edelman, M.; Tarasov, VE, Dynamics of nonlinear systems with power-law memory, Volume 4 Applications in Physics Part A, Handbook of Fractional Calculus with Applications, 103-132 (2019), Berlin: De Gruyter, Berlin · Zbl 1410.26006 · doi:10.1515/9783110571707-005
[16] Edelman, M., Cycles in asymptotically stable and chaotic fractional maps, Nonlinear Dyn., 104, 2829-2841 (2021) · Zbl 1537.39012 · doi:10.1007/s11071-021-06379-2
[17] Edelman, M., Stability of fixed points in generalized fractional maps of the orders \(0<\alpha <1\), Nonlinear Dyn., 111, 10247-10254 (2023) · doi:10.1007/s11071-023-08359-0
[18] Edelman, M.; Helman, AB, Asymptotic cycles in fractional maps of arbitrary positive orders, Fract. Calc. Appl. Anal., 25, 181-206 (2022) · Zbl 1503.39003 · doi:10.1007/s13540-021-00008-w
[19] Edelman, M., Helman, A. B.: Asymptotically periodic points, bifurcations, and transition to chaos in fractional difference maps. http://arxiv.org/abs/2209.15462v1
[20] Elaydi, SN, Discrete Chaos. With Applications in Science and Engineering (2008), Boca Raton: Chapman and Hall CRC, Boca Raton · Zbl 1153.39002
[21] Čermák, J.; Györi, I.; Nechvátal, L., On explicit stability conditions for a linear fractional difference system, Fract. Calc. Appl. Anal., 18, 651-672 (2015) · Zbl 1320.39004 · doi:10.1515/fca-2015-0040
[22] Kawan, C., Metric entropy of nonautonomous dynamical systems, Nonautonomous Dyn. Syst., 1, 26-52 (2014) · Zbl 1296.37018 · doi:10.2478/msds-2013-0003
[23] Kolyada, S.; Snoha, L., Topological entropy of nonautononous dynamical systems, Random Comp. Dyn., 4, 205-233 (1996) · Zbl 0909.54012
[24] Li, TY; Yorke, JA, Period three implies chaos, Amer. Math. Monthly, 82, 985-992 (1975) · Zbl 0351.92021 · doi:10.2307/2318254
[25] May, RM, Simple mathematical models with very complicated dynamics, Nature, 261, 459-467 (1976) · Zbl 1369.37088 · doi:10.1038/261459a0
[26] Milnor, J., On the concept of attractor, Comm. Math. Phys., 99, 177-195 (1985) · Zbl 0595.58028 · doi:10.1007/BF01212280
[27] Peng, Y.; Sun, K.; He, S.; Wang, L., Comments on Discrete fractional logistic map and its chaos [Nonlinear Dyn. 75, 283-287 (2014)], Nonlinear Dyn., 75, 283-287 (2014) · doi:10.1007/s11071-019-05012-7
[28] Thunberg, H., Periodicity versus chaos in one-dimensional dynamics, SIAM Rev., 43, 3-30 (2001) · Zbl 1049.37027 · doi:10.1137/S0036144500376649
[29] Walters, P., An Introduction to Ergodic Theory (1982), New York: Springer, New York · Zbl 0475.28009 · doi:10.1007/978-1-4612-5775-2
[30] Wu, GC; Baleanu, D., Discrete fractional logistic map and its chaos, Nonlinear Dyn., 75, 283-287 (2014) · Zbl 1281.34121 · doi:10.1007/s11071-013-1065-7
[31] Xu, B.; Ye, X.; Wang, G.; Huang, Z.; Zhang, C., A fractional-order improved quantum logistic aap: chaos, 0-1 testing, complexity and control, Axioms, 12, 94 (2023) · doi:10.3390/axioms12010094
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.