Bifurcation Analysis and 0-1 Chaos Test of a Discrete T System

Yıl 2023, Cilt: 5 Sayı: 2, 90 - 104, 31.07.2023

Sarker Md Sohel Rana

https://doi.org/10.51537/chaos.1214284

Cited By: 1

Öz

This study examines discrete-time T system. We begin by listing the topological divisions of the system's fixed points. Then, we analytically demonstrate that a discrete T system sits at the foundation of a Neimark Sacker(NS) bifurcation under specific parametric circumstances. With the use of the explicit Flip-NS bifurcation criterion, we establish the flip-NS bifurcation's reality. Center manifold theory is then used to establish the direction of both bifurcations. We do numerical simulations to validate our theoretical findings. Additionally, we employ the $0-1$ test for chaos to demonstrate whether or not chaos exists in the system. In order to stop the system's chaotic trajectory, we ultimately employ a hybrid control method.

Anahtar Kelimeler

T system, Stability, Bifurcations, Chaos control

Teşekkür

There is no funding for this research work. Thank you.

Kaynakça

• Abdelaziz, M. A., A. I. Ismail, F. A. Abdullah, and M. H. Mohd, 2020 Codimension one and two bifurcations of a discrete-time fractional-order seir measles epidemic model with constant vaccination. Chaos, Solitons & Fractals 140: 110104.
• Babloyantz, A., J. Salazar, and C. Nicolis, 1985 Evidence of chaotic dynamics of brain activity during the sleep cycle. Physics letters A 111: 152–156.
• Camouzis, E. and G. Ladas, 2007 Dynamics of third-order rational difference equations with open problems and conjectures, volume 5. CRC Press.
• Chakraborty, P., S. Sarkar, and U. Ghosh, 2020 Stability and bifurcation analysis of a discrete prey–predator model with sigmoid functional response and allee effect. Rendiconti del Circolo Matematico di Palermo Series 2 pp. 1–21.
• Chen, G., 1999 Controlling chaos and bifurcations in engineering systems. CRC press.
• Chen, G. and X. Dong, 1998 From chaos to order: methodologies, perspectives and applications, volume 24. World Scientific.
• Din, Q. andW. Ishaque, 2019 Bifurcation analysis and chaos control in discrete-time eco–epidemiological models of pelicans at risk in the salton sea. International Journal of Dynamics and Control 8: 132–148.
• El Naschie, M., 2003 Non-linear dynamics and infinite dimensional topology in high energy particle physics. Chaos, Solitons & Fractals 17: 591–599.
• Fei, L., X. Chen, and B. Han, 2021 Bifurcation analysis and hybrid control of a discrete-time predator–prey model. Journal of Difference Equations and Applications 27: 102–117.
• Feng, G., D. Yin, and L. Jiacheng, 2021 Neimark–sacker bifurcation and controlling chaos in a three-species food chain model through the ogy method. Discrete Dynamics in Nature and Society 2021.
• Gottwald, G. A. and I. Melbourne, 2004 A new test for chaos in deterministic systems. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 460: 603–611.
• Hu, Z., Z. Teng, and L. Zhang, 2014 Stability and bifurcation analysis in a discrete sir epidemic model. Mathematics and computers in Simulation 97: 80–93.
• Ishaque, W., Q. Din, M. Taj, and M. A. Iqbal, 2019 Bifurcation and chaos control in a discrete-time predator–prey model with nonlinear saturated incidence rate and parasite interaction. Advances in Difference Equations 2019: 1–16.
• Jiang, B., X. Han, and Q. Bi, 2010 Hopf bifurcation analysis in the t system. Nonlinear Analysis: Real World Applications 11: 522–527.
• Kengne, J., Z. Njitacke, and H. Fotsin, 2016 Dynamical analysis of a simple autonomous jerk system with multiple attractors. Nonlinear Dynamics 83: 751–765.
• Khan, A. and M. Javaid, 2021 Discrete-time phytoplankton– zooplankton model with bifurcations and chaos. Advances in Difference Equations 2021: 1–30.
• Khan, M. S., M. Ozair, T. Hussain, J. Gómez-Aguilar, et al., 2021 Bifurcation analysis of a discrete-time compartmental model for hypertensive or diabetic patients exposed to covid-19. The European Physical Journal Plus 136: 1–26.
• Kuznetsov, Y. A., 2013 Elements of applied bifurcation theory, volume 112. Springer Science and Business Media, New York, USA.
• Li, B. and Q. He, 2019 Bifurcation analysis of a two-dimensional discrete hindmarsh–rose type model. Advances in difference equations 2019: 1–17.
• Li, X.-F., Y.-D. Chu, J.-G. Zhang, and Y.-X. Chang, 2009 Nonlinear dynamics and circuit implementation for a new lorenz-like attractor. Chaos, Solitons & Fractals 41: 2360–2370.
• Liu, Y. and X. Li, 2021 Dynamics of a discrete predator-prey model with holling-ii functional response. International Journal of Biomathematics p. 2150068.
• Lorenz, E. N., 1963 Deterministic nonperiodic flow. Journal of atmospheric sciences 20: 130–141.
• Lü, J., G. Chen, and S. Zhang, 2002 Dynamical analysis of a new chaotic attractor. International Journal of Bifurcation and chaos 12: 1001–1015.
• Luo, W., Q. Ou, F. Yu, L. Cui, and J. Jin, 2020 Analysis of a new hidden attractor coupled chaotic system and application of its weak signal detection. Mathematical Problems in Engineering 2020.
• Pecora, L. M. and T. L. Carroll, 1991 Driving systems with chaotic signals. Physical review A 44: 2374.
• Qin, S., J. Zhang,W. Du, and J. Yu, 2016 Neimark–sacker bifurcation in a new three–dimensional discrete chaotic system. ICIC-EL 10: 1–7.
• Rabinovich, M. and H. Abarbanel, 1998 The role of chaos in neural systems. Neuroscience 87: 5–14.
• Rana, S. M. S., 2019a Bifurcations and chaos control in a discretetime predator-prey system of leslie type. Journal of Applied Analysis & Computation 9: 31–44.
• Rana, S. M. S., 2019b Dynamics and chaos control in a discrete-time ratio-dependent holling-tanner model. Journal of the Egyptian Mathematical Society 27: 1–16.
• Rössler, O. E., 1976 An equation for continuous chaos. Physics Letters A 57: 397–398.
• Sachdev, P. and R. Sarathy, 1994 Periodic and chaotic solutions for a nonlinear system arising from a nuclear spin generator. Chaos, Solitons & Fractals 4: 2015–2041.
• Singh, A. and P. Deolia, 2021 Bifurcation and chaos in a discrete predator–prey model with holling type-iii functional response and harvesting effect. Journal of Biological Systems 29: 451–478.
• Sparrow, C., 2012 The Lorenz equations: bifurcations, chaos, and strange attractors, volume 41. Springer Science and Business Media.
• Sundarapandian, V., 2011 Global chaos anti-synchronization of tigan and lorenz systems by nonlinear control. Int J Math Sci Appl 1: 679–690.
• Tang, S. and L. Chen, 2003 Quasiperiodic solutions and chaos in a periodically forced predator–prey model with age structure for predator. International Journal of Bifurcation and Chaos 13: 973–980.
• Tigan, G., 2005 Analysis of a dynamical system derived from the lorenz system. Sci. Bull. Politehnica Univ. Timisoara Tomu 50: 61–72.
• Ueta, T. and G. Chen, 2000 Bifurcation analysis of chen’s equation. International Journal of Bifurcation and Chaos 10: 1917–1931.
• Vaidyanathan, S. and K. Rajagopal, 2011 Anti-synchronization of li and t chaotic systems by active nonlinear control. In Advances in Computing and Information Technology: First International Conference, ACITY 2011, Chennai, India, July 15-17, 2011. Proceedings, pp. 175–184, Springer.
• Wen, G., 2005 Criterion to identify hopf bifurcations in maps of arbitrary dimension. Physical Review E 72: 026201.
• Xin, B., T. Chen, and J. Ma, 2010 Neimark-sacker bifurcation in a discrete-time financial system. Discrete Dynamics in Nature and Society 2010.
• Xin, B. and Y. Li, 2013 0-1 test for chaos in a fractional order financial system with investment incentive. In Abstract and Applied Analysis, volume 2013, Hindawi.
• Xin, B. and Z.Wu, 2015 Neimark–sacker bifurcation analysis and 0– 1 chaos test of an interactions model between industrial production and environmental quality in a closed area. sustainability 7: 10191–10209.
• Yang, T. and L. O. Chua, 1997 Impulsive stabilization for control and synchronization of chaotic systems: theory and application to secure communication. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 44: 976–988.
• Yao, S., 2012 New bifurcation critical criterion of flip-neimarksacker bifurcations for two-parameterized family of-dimensional discrete systems. Discrete Dynamics in Nature and Society 2012.
• Yong, C. and Y. Zhen-Ya, 2008 Chaos control in a new threedimensional chaotic t system. Communications in Theoretical Physics 49: 951.
• Yuan, L.-G. and Q.-G. Yang, 2015 Bifurcation, invariant curve and hybrid control in a discrete-time predator–prey system. Applied Mathematical Modelling 39: 2345–2362.
• Zhang, Y., Q. Cheng, and S. Deng, 2022 Qualitative structure of a discrete predator-prey model with nonmonotonic functional response. Discrete & Continuous Dynamical Systems-S .
• Zhao, M., 2021 Bifurcation and chaotic behavior in the discrete bvp oscillator. International Journal of Non-Linear Mechanics 131: 103687.