Exploring three periodic point dynamics in $ 2 $D spatiotemporal discrete systems

Mohamed Lamine Sahari; Abdel-Kaddous Taha; Louis Randriamihamison; Mohamed Lamine Sahari; Abdel-Kaddous Taha; Louis Randriamihamison

doi:10.3934/math.2025230

AIMS Mathematics

2025, Volume 10, Issue 3: 5021-5051. doi: 10.3934/math.2025230

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Exploring three periodic point dynamics in $ 2 $D spatiotemporal discrete systems

1.
LANOS Laboratory, Department of Mathematics, Badji Mokhtar-Annaba University, P. O. Box 12, 23000 Annaba, Algeria
2.
INSA, Federal University of Toulouse Midi-Pyrénées, 135 Avenue de Rangueil, 31077 Toulouse Cedex 4, France
3.
IPST-Cnam, Institut National Polytechnique de Toulouse, University of Toulouse, 118, route de Narbonne, 31062 Toulouse Cedex 9, France

Received: 23 October 2024 Revised: 08 January 2025 Accepted: 13 February 2025 Published: 06 March 2025
MSC : 47A10, 37G10, 37G35, 37L15

This paper explores the dynamics of $ 2 $D spatiotemporal discrete systems, focusing on the stability and bifurcations of periodic solutions, particularly $ 3 $-cycles. After introducing the concept of a third-order cycle, we discuss both numerical and analytical techniques used to analyze these cycles, defining four types of $ 3 $-periodic points and their associated stability conditions. As a specific case, this study examines a spatiotemporal quadratic map, analyzing the existence of $ 3 $-cycles and various bifurcation scenarios, such as fold and flip bifurcations, as well as chaotic behavior. In $ 2 $D spatiotemporal systems, quadratic maps intrinsically offer better conditions that favor the emergence of chaos, which is characterized by high sensitivity to initial conditions. The findings emphasize the complexity of these systems and the crucial role of bifurcation curves in understanding stability regions. The paper concludes with key insights and suggestions for future research in this field.
- Chaos,
- bifurcation,
- bifurcation curves,
- $ 2 $D spatiotemporal discrete systems,
- spectrum,
- stability
Citation: Mohamed Lamine Sahari, Abdel-Kaddous Taha, Louis Randriamihamison. Exploring three periodic point dynamics in $ 2 $D spatiotemporal discrete systems[J]. AIMS Mathematics, 2025, 10(3): 5021-5051. doi: 10.3934/math.2025230

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Abstract

This paper explores the dynamics of $ 2 $D spatiotemporal discrete systems, focusing on the stability and bifurcations of periodic solutions, particularly $ 3 $-cycles. After introducing the concept of a third-order cycle, we discuss both numerical and analytical techniques used to analyze these cycles, defining four types of $ 3 $-periodic points and their associated stability conditions. As a specific case, this study examines a spatiotemporal quadratic map, analyzing the existence of $ 3 $-cycles and various bifurcation scenarios, such as fold and flip bifurcations, as well as chaotic behavior. In $ 2 $D spatiotemporal systems, quadratic maps intrinsically offer better conditions that favor the emergence of chaos, which is characterized by high sensitivity to initial conditions. The findings emphasize the complexity of these systems and the crucial role of bifurcation curves in understanding stability regions. The paper concludes with key insights and suggestions for future research in this field.

References

[1]	K. R. Crounse, L. O. Chua, Methods for image processing and pattern formation in cellular neural networks: A tutorial, IEEE Trans. Circuits Syst. I, 42 (1995), 583–601. https://doi.org/10.1109/81.473566 doi: 10.1109/81.473566
[2]	Z. Lv, F. Sun, C. Cai, A new spatiotemporal chaotic system based on two-dimensional discrete system, Nonlinear Dyn., 109 (2022), 3133–3144. https://doi.org/10.1007/s11071-022-07585-2 doi: 10.1007/s11071-022-07585-2
[3]	V. I. Nekorkin, M. G. Velarde, Synergetic phenomena in active lattices: patterns, waves, solitons, chaos, Springer, 2002. https://doi.org/10.1007/978-3-642-56053-8
[4]	Y. Song, X. Zou, Spatiotemporal dynamics in a diffusive ratio-dependent predator–prey model near a Hopf–Turing bifurcation point, Comput. Math. Appl., 67 (2014), 1978–1997, 2014. https://doi.org/10.1016/j.camwa.2014.04.015 doi: 10.1016/j.camwa.2014.04.015
[5]	Y. Tao, W. Cui, Z. Zhang, Spatiotemporal chaos in multiple dynamically coupled map lattices and its application in a novel image encryption algorithm, J. Inf. Secur. Appl., 55 (2020), 102650. https://doi.org/10.1016/j.jisa.2020.102650 doi: 10.1016/j.jisa.2020.102650
[6]	C. Tian, On the pseudo-randomicity of discrete spatiotemporal systems, Int. J. Bifurcat. Chaos, 24 (2014), 1450013. https://doi.org/10.1142/s0218127414500138 doi: 10.1142/s0218127414500138
[7]	C. Tian, Chaos in the sense of Devaney for two-dimensional time-varying generalized symbolic dynamical systems, Int. J. Bifurcat. Chaos, 27 (2017), 1750060. https://doi.org/10.1142/s0218127417500602 doi: 10.1142/s0218127417500602
[8]	C. Tian, Hypercyclicity for a class of discrete spatiotemporal systems, Int. J. Bifurcat. Chaos, 27 (2017), 1750045. https://doi.org/10.1142/s0218127417500456 doi: 10.1142/s0218127417500456
[9]	C. Tian, G. Chen, Stability and chaos in a class of 2-dimensional spatiotemporal discrete systems, J. Math. Anal. Appl., 356 (2009), 800–815. https://doi.org/10.1016/j.jmaa.2009.03.046 doi: 10.1016/j.jmaa.2009.03.046
[10]	R. Chen, F. Zhang, L. Teng, X. Wang, Cross-image encryption algorithm based on block recombination and spatiotemporal chaos system, J. Opt., 52 (2023), 2109–2129. https://doi.org/10.1007/s12596-023-01104-1 doi: 10.1007/s12596-023-01104-1
[11]	E. Fornasini, A 2-D systems approach to river pollution modelling, Multidimens. Syst. Signal Process., 2 (1991), 233–265. https://doi.org/10.1007/bf01952235 doi: 10.1007/bf01952235
[12]	E. Fornasini, G. Marchesini, Doubly-indexed dynamical systems: State-space models and structural properties. Math. Syst. Theory, 12 (1978), 59–72. https://doi.org/10.1007/bf01776566 doi: 10.1007/bf01776566
[13]	J. Fritz, Partial Differential Equations, volume 1. Springer Science & Business Media, 1991. https://doi.org/10.1007/978-1-4419-1096-7_5
[14]	S. Liu, F. Sun, Spatial chaos-based image encryption design, Sci. China Ser. G, 52 (2009), 177–183. https://doi.org/10.1007/s11433-009-0032-2 doi: 10.1007/s11433-009-0032-2
[15]	Y. Ma, Y. Tian, L. Zhang, P. Zuo, Two-dimensional hyperchaotic effect coupled mapping lattice and its application in dynamic s-box generation, Nonlinear Dyn., 112 (2024), 17445–17476. https://doi.org/10.1007/s11071-024-09907-y doi: 10.1007/s11071-024-09907-y
[16]	R. Roesser, A discrete state-space model for linear image processing, IEEE Trans. Autom. Control, 20 (1975), 1–10. https://doi.org/10.1109/tac.1975.1100844 doi: 10.1109/tac.1975.1100844
[17]	F. Sun, S. Liu, Z. Li, Z. Lü, A novel image encryption scheme based on spatial chaos map, Chaos Soliton. Fract., 38 (2008), 631–640. https://doi.org/10.1016/j.chaos.2008.01.028 doi: 10.1016/j.chaos.2008.01.028
[18]	F. H. Willeboordse, The spatial logistic map as a simple prototype for spatiotemporal chaos, Chaos, 13 (2003), 533–540. https://doi.org/10.1063/1.1568692 doi: 10.1063/1.1568692
[19]	C. Yuan, S. Liu, Exact regions of oscillation for a mixed 2D discrete convection system, J. Differ. Equ. Appl., 21 (2015), 37–52. https://doi.org/10.1080/10236198.2014.979167 doi: 10.1080/10236198.2014.979167
[20]	K. Sarkar, S. Khajanchi, Spatiotemporal dynamics of a predator-prey system with fear effect, J. Franklin Inst., 360 (2023), 7380–7414. https://doi.org/10.1016/j.jfranklin.2023.05.034 doi: 10.1016/j.jfranklin.2023.05.034
[21]	E. M. Izhikevich, Dynamical systems in neuroscience, MIT Press, 2007. https://doi.org/10.7551/mitpress/2526.001.0001
[22]	L. Randriamihamison, A. K. Taha, About the singularities and bifurcations of double indices recursion sequences, Nonlinear Dyn., 66 (2011), 795–808. https://doi.org/10.1007/s11071-011-9952-2 doi: 10.1007/s11071-011-9952-2
[23]	M. L. Sahari, A. K. Taha, L. Randriamihamison, Stability and bifurcations in 2D spatiotemporal discrete systems, Int. J. Bifurcat. Chaos, 28 (2018), 1830026. https://doi.org/10.1142/s0218127418300264 doi: 10.1142/s0218127418300264
[24]	M. L. Sahari, A. K. Taha, L. Randriamihamison, Bifurcations in 2D spatiotemporal maps, Int. J. Bifurcat. Chaos, 31 (2021), 2150091. https://doi.org/10.1142/s0218127421500917 doi: 10.1142/s0218127421500917
[25]	M. L. Sahari, A. K. Taha, L. Randriamihamison, A note on the spectrum of diagonal perturbation of weighted shift operator, Le Matematiche, 74 (2019), 35–47, 2019. http://dx.doi.org/10.48550/arXiv.1609.05203 doi: 10.48550/arXiv.1609.05203
[26]	P. R. Halmos, Introduction to Hilbert space and the theory of spectral multiplicity, Courier Dover Publications, 2017. https://doi.org/10.2307/3608269
[27]	Y. A. Kuznetsov, I. A. Kuznetsov, Y. Kuznetsov, Elements of applied bifurcation theory, volume 112. Springer, 1998. https://doi.org/10.1007/978-1-4757-2421-9
[28]	I. Gumowski, C. Mira, Dynamique chaotique: Transformations ponctuelles, transition, ordre-désordre. Cepadues, 1980. https://doi.org/10.1002/zamm.19810611023
[29]	C. Mira, Chaos and fractal properties induced by noninvertibility of models in the form of maps, Chaos Soliton. Fract., 11 (2000), 251–262. https://doi.org/10.1016/s0960-0779(98)00291-4 doi: 10.1016/s0960-0779(98)00291-4
[30]	A. L. Shields, Weighted shift operators and analytic function theory, Math. Surv., 13 (1974), 49–128. https://doi.org/10.1090/surv/013/02 doi: 10.1090/surv/013/02
[31]	P. R. Halmos, A Hilbert space problem book, volume 19, Springer Science & Business Media, 2012. https://doi.org/10.1007/978-1-4684-9330-6
[32]	W. B. Gordon, Period three trajectories of the logistic map, Math. Mag., 69 (1996), 118–120. https://doi.org/10.1080/0025570x.1996.11996403 doi: 10.1080/0025570x.1996.11996403
[33]	M. H. Lee, Three-cycle problem in the logistic map and Sharkovskii's theorem, Acta Phys. Pol. B, 42 (2011). https://doi.org/10.5506/APhysPolB.42.1071 doi: 10.5506/APhysPolB.42.1071
[34]	M. H. Lee, Solving for the fixed points of 3-cycle in the logistic map and toward realizing chaos by the theorems of Sharkovskii and Li-Yorke, Commun. Theor. Phys., 62 (2014), 485. https://doi.org/10.1088/0253-6102/62/4/06 doi: 10.1088/0253-6102/62/4/06
[35]	P. Saha, S. H. Strogatz, The birth of period three, Math. Mag., 68 (1995), 42–47. https://doi.org/10.1080/0025570x.1995.11996273 doi: 10.1080/0025570x.1995.11996273
[36]	J. C. Cathala, H. Kawakami, C. Mira, Singular points with two multipliers, s1 = - s2 = 1, in the bifurcation curves of maps, Int. J. Bifurcat. Chaos, 2 (1992), 1001–1004. https://doi.org/10.1142/s0218127492000616 doi: 10.1142/s0218127492000616

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