Complete Synchronization, Anti-Synchronization and Hybrid Synchronization of Non-Identical Chaotic Financial Systems
DOI:
https://doi.org/10.62292/njtep.v2i2.2024.45Keywords:
Chaos theory, Financial system, Complete synchronization, Anti-synchronization, Hybrid synchronizationAbstract
Chaos theory, being a branch of mathematics that deals with disordered or random-seeming mathematical systems, is receiving more attention in market-related circumstances as financial markets become more unstable and the level of unpredictability becomes more prevalent. Understanding the potential of chaos theory, its limitations, and its relationship to traditional economic theories is essential for anyone working in the finance sector. Chaos theory is ideally suited for comprehending the financial market, which is subject to both internal and external influences due to its high degree of instability and growing randomness. This work examines the complete synchronization, anti-synchronization and hybrid synchronization of two non-identical financial systems. The nonlinear active controllers are designed, and the error dynamics stability for each phenomenon is accomplished by two theoretical approaches - linear system theory and Lyapunov second method. Controllers are designed by using the relevant variables of drive and response systems in such a way that the error variables are stable. The controllers, when activated, enable the drive and response state variables to achieve identical dynamics despite starting from different initial conditions. Numerical simulations are performed using the ODE45 algorithm embedded in MATLAB software package to show the feasibility and effectiveness of the designed controllers.
References
Aguilar-LOpez, R., Martnez-Guerra, R., Perez-Pinacho, C. (2014). Nonlinear observer for synchronization of chaotic systems with application to secure data transmission. European Physical Journal, Special Topics, 223, 1541. https://doi.10.1140/epjst/e2014-02116-0.
Arashi, M., Rounaghi, M. (2022). Analysis of market efficiency and fractal feature of NASDAQ stock exchange: Time series modeling and forecasting of stock index using ARMA-GARCH model. Future Business Journal, 8, 1. https://doi.10.1186/s43093-022-00125-9.
Baleanu, D., Jajarmi, A., Sajjadi, S. S., Asad, J. H. (2020). The fractional features of a harmonic oscillator with position-dependent mass. Commun. Theor., 72, 1. https://doi.10.1088/1572-9494/ab7700.
Bourdeau-Brien, M., Kryzanowski, L. (2017). The impact of natural disasters on the stock returns and volatilities of local firms. The Quarterly Review of Economics and Finance, 63, 259. https://doi.10.1016/j.qref.2016.05.003.
Bowang, S., Kakmeni, F. M. M., Siewev, M. S. (2007). Secure communication via parameter modulation in a class of chaotic system. Journal of Commun. Nonlinear Sci. Numer. Simulation 12(03), 397. https://doi.10.1016/j.cnsns.2005.03.002.
Dousseh, Y. P., Monwanou, A. V., Koukpemedji, A. A., Miwadinou, C., Chabi Orou, J. B. (2022). Dynamics analysis, adaptive control, synchronization and anti-synchronization of a novel modified chaotic financial system. International Journal of Dynamics and Control, 11, 1. https://doi.10.1007/s40435-022-01003-6.
Eisencraft, M., Fanganiello, R. D.,Grzybowski, J. M. V., Soriano, D. C., Attux, R., Batista, A. M., Macau, E. E. N., Monteiro, L. H. A., Romano, J. M. T., Suyama, R., Yoneyama, T. (2012). Chaos-based communication systems in non-ideal channels. American Jornal of Physics, 17(12), 4707. https://doi.10.1016/j.cnsns.2011.05.030.
Filali, M. B., Pierre, B. (2014). On observer-based secure communication design using discrete-time hyperchaotic systems. Communications in Nonlinear Science and Numerical Simulation, 19(05), 1424. https://doi.10.1016/j.cnsns.2013.09.005.
Freeman, W. J. (1992). Tutorial on neurobiology: from single neurons to brain chaos. International Journal of Bifurcation and Chaos, 2(03), 451. https://doi.10.1142/S0218127492000653.
Fujisaka, H., Yamada, T. (1983). Stability theory of synchronized motion in coupled oscillator systems. Progress of Theoretical Physics, 69(01), 32. https://doi.10.1143/PTP.69.32.
Gabriel, P., Hilda, A. C. (1995). Extracting messages masked by chaos. Phys. Rev. Lett., 74, 1970. https://doi.10.1103/PhysRevLett.74.1970.
Gan, X. Wang, H., Yuan, R., Ao, P. (2021). A new criterion beyond divergence for judging the dissipation of a system: dissipative power. Frontiers in Physics, 9, 458. https://doi.10.3389/fphy.2021.695489.
Hsieh, D. A. (1991). Chaos and nonlinear dynamics: Application to financial markets. The Journal of Finance, 46, 1839. https://doi.10.2307/2328575.
Juarez, F. (2015). Chaos and Complexity in Financial Statements. Information Resources Management Association, IGI Global, 1399--1430. https://doi.10.4018/978-1-4666-8468-3.
Kilikevicius, A., Jurevicius, M., Bureika, G., Turia, V. (2015). Effect of external excitation on dynamic characteristics of vibration isolating table. Maintenance and Reliability, 17(2), 260. https://doi.10.17531/ein.2015.2.13.
Klioutchnikov, I., Sigova, M., Beizerov, N. (2017). Chaos Theory in Finance. Procedia Computer Science, 119, 368. https://doi.10.1016/j.procs.2017.11.196.
Liao, Y., Zhou, Y., Xu, F., Shu, X. (2020). A Study on the Complexity of a New Chaotic Financial System. Complexity, 1, 8821156. https://doi.10.1155/2020/8821156.
Lu, X. (2020). A financial chaotic system control method based on intermittent controller. Mathematical Problems in Engineering, 5810707. https://doi.10.1155/2020/5810707.
Metescu, A. (2022). Fractal market hypothesis vs. efficient market hypothesis: Applying the R/S analysis on the Romanian market. Journal of Public Administration, Finance and Law, 11, 199. https://doi.10.47743/jopafl-2022-23-17.
Olusola, O. I., Oyeleke, K. S., Vincent, U. E., Njah, A. N. (2020). Secure Communication Scheme Based on Synchronization of Non-Identical Hyperchaotic Systems. Journal of Applied Nonlinear Dynamics, 9 (2), 273. https://doi.10.5890/JAND.2020.06.009.
Pecora, L. M., Carroll, T. L. (1990). Synchronization in chaotic Systems. Phys. Rev. Lett. 64, 821. https://doi.10.1063/1.4917383.
Pikovsky, A. S., Rosemblum, M., Kurths, J. (2002). Synchronization: A universal concept in nonlinear science. American Jornal of Physics 70(6). https://doi.10.1119/1.1475332.
Rameika, R. (2007). New results from accelerator neutrino experiments. International Journal of Modern Physics A, 22(30), 5544. https://doi.10.1142/S0217751X07038803.
Ren, H. P., Baptista, M. S., Grebogi, C. (2013). Wireless communication with chaos. Phys. Rev. Lett., 110, 18410. https://doi.10.1103/PhysRevLett.110.184101.
Tusset, A., Fuziki, M., Balthazar, J., Andrade, D., Lenzi, G. G. (2023). Dynamic analysis and control of a financial system with chaotic behavior including fractional order. Fractal and Fractional, 7, 535. https://doi.10.3390/fractalfract7070535.
Wen, S., Shen, Y., Yang S., Wang, J. (2017). Dynamical response of Mathieu-Duffing oscillator with fractional-order delayed feedback. Chaos, Solitons and Fractals, 94, 54. https://doi.10.1016/j.chaos.2016.11.008.
Wu, Z., Xie, J., Fang, Y., Xu, Z. (2007). Controlling chaos with periodic parametric perturbations in Lorenz system. Chaos, Solitons and Fractals, 32, 104. https://doi.10.1016/j.chaos.2005.10.060.
Woolf, P. J. (2009). Chemical Process Dynamics and Controls, Open textbook library, University of Michigan Engineering Controls Group, Ann Arbor, United States, 447--458. https://eng.libretexts.org.
Xiao-Dan, Z., Liu, X.,Yuan, Z., Cheng, L. (2013). Chaotic dynamic behaviour analysis and control for a financial risk system. Chinese Physics B, 22, 030509. https://doi.10.1088/1674-1056/22/3/030509.
