Robust Stabilization of a Non-Linear Chaotic Financial System with Uncertain Parameters
Keywords:Nonlinear system, Chaotic dynamical system, Financial system, Adaptive control, Lyapunov stability theory, Robust stabilization
The paper investigates an analytical approach for robust stabilization of nonlinear chaotic financial system in the presence of uncertain parameters. The primary focus of this paper is to find a robust solution for quickly adjusting and controlling the interest rate, investment demand and price exponent when the chaotic phenomenon occurs in the financial system or economic crisis happens. The paper first demonstrates the non-linear dynamical model of the chaotic financial system and then it adopts Lyapunov stability theory based adaptive control scheme for robust stabilization of nonlinear chaotic financial system in the presence of uncertain parameters. Numerical simulations are demonstrated to verify the effectiveness of the proposed control scheme. The simulation results of this paper show that control scheme successfully eliminates the chaos of the nonlinear financial systems
 E. Ott, C. Grebogi, and J. A. Yorke, â€œControlling chaos,â€ Physical review letters, vol. 64, no. 11, p. 1196, 1990.
 Z. W.-H. Cai Guo-Liang, Tan Zhen-Mei and T. Wen-Tao, â€œDynamical analysis of a new chaotic system and its chaotic control,â€ Acta Physica Sinica, vol. 56, no. 11, p. 6230, 2007.
 J. Lu, G. Chen, and S. Zhang, â€œThe compound structure of a new chaotic attractor,â€ Chaos, Solitons and Fractals, vol. 14, no. 5, pp. 669â€“672, 2002.
 D. Guegan, â€œChaos in economics and finance,â€ Annual Reviews in Control, vol. 33, no. 1, pp. 89â€“93, 2009.
 G. Cai and J. Huang, â€œA new finance chaotic attractor,â€ International Journal of Nonlinear Science, vol. 3, no. 3, pp. 213â€“220, 2009.
 I. Evstigneev and M. Taksar, â€œDynamic interaction models of economic equilibrium,â€ Journal of Economic Dynamics and Control, vol. 33, no. 1, pp. 166â€“182, 2009.
 G. Chen and X. Dong, â€œFrom chaos to order-perspectives and methodologies in controlling chaotic nonlinear dynamical systems,â€ International Journal of Bifurcation and Chaos, vol. 3, no. 06, pp. 1363â€“1409, 1993.
 R. Genesio and A. Tesi, â€œControl techniques for chaotic dynamical systems,â€ Kybernetika, vol. 29, no. 5, pp. 469â€“478, 1993.
 Surendar, A. (n.d.). Short communication: Role of Microbiology in the Pharmaceutical &Medical Device. 433| International Journal of Pharmaceutical Research, 10(3).
 G. Chen and X.-N. Dong, â€œOn feedback control of chaotic continuous-time systems,â€ Circuits and Systems I: Fundamental Theory and Applications, IEEE Transactions on, vol. 40, no. 9, pp. 591â€“601, 1993.
 K. Kemih, M. Halimi, M. Ghanes, H. Fanit, and H. Salit, â€œControl and synchronization of chaotic attitude control of satellite with backstepping controller,â€ The European Physical Journal Special Topics, vol. 223, no. 8, pp. 1579â€“1589, 2014.
 C.-C. Fuh and H.-H. Tsai, â€œControl of discrete-time chaotic systems via feedback linearization,â€ Chaos, Solitons & Fractals, vol. 13, no. 2, pp. 285â€“294, 2002.
 B. Andrievskii and A. Fradkov, â€œControl of chaos: Methods and applications. i. methods,â€ Automation and remote control, vol. 64, no. 5, pp. 673â€“713, 2003.
 H. Saberi Nik, P. He, and S. T. Talebian, â€œOptimal, adaptive and single state feedback control for a 3d chaotic system with golden proportion equilibria,â€ Kybernetika, vol. 50, no. 4, pp. 596â€“615, 2014.
 A. Y. Leung, X.-F. Li, Y.-D. Chu, and X.-B. Rao, â€œA simple adaptive-feedback scheme for identical synchronizing chaotic systems with uncertain parameters,â€ Applied Mathematics and Computation, vol. 253, pp. 172â€“183, 2015.
 H. Yu, G. Cai, and Y. Li, â€œDynamic analysis and control of a new hyperchaotic finance system,â€ Nonlinear Dynamics, vol. 67, no. 3, pp. 2171â€“2182, 2012.
 S. K. Choudhary, â€œLqr based optimal control of chaotic dynamical systems,â€ International Journal of Modelling and Simulation, vol. 35, no. 3-4, pp. 104â€“112, 2016.
 S. Boccaletti, C. Grebogi, Y.-C. Lai, H. Mancini, and D. Maza, â€œThe control of chaos: theory and applications,â€ Physics reports, vol. 329, no. 3, pp. 103â€“197, 2000.
 H. Zhang, D. Liu, and Z. Wang, Controlling chaos: suppression, synchronization and chaotification. Springer London, 2009.
 T. Shinbrot, C. Grebogi, E. Ott, and J. A. Yorke, â€œUsing small perturbations to control chaos,â€ Nature, vol. 363, no. 6428, pp. 411â€“417, 1993.
 M. Yang and G. Cai, â€œChaos control of a non-linear finance system,â€ Journal of Uncertain Systems, vol. 5, no. 4, pp. 263â€“270, 2011.
 D. Kumar and S. Kumar, â€œUltimate numerical bound estimation of chaotic dynamical finance model,â€ in Modern Mathematical Methods and High-Performance Computing in Science and Technology, pp. 71â€“81, Springer, 2016.
 G. L. Cai and M. Z. Yang, â€œGlobally exponentially attractive set and synchronization of a novel three-dimensional chaotic finance system,â€ in 2010 Third International Conference on Information and Computing, vol. 2, pp. 70â€“73, June 2010.
 H. Salarieh and A. Alasty, â€œChaos control in an economic model via minimum entropy strategy,â€ Chaos, Solitons & Fractals, vol. 40, no. 2, pp. 839â€“847, 2009.
 J. Ding and H. Yao, â€œChaos control of a kind of non-linear finance system,â€ J. Jiangsu Univ, vol. 15, no. 6, pp. 500â€“504, 2004.
 M. Sun and L.-X. Tian, â€œAdaptive synchronization of non-linear chaotic finance system [j],â€ Journal of Jiangsu University (National Science Edition), vol. 6, p. 008, 2005.
 A. Jabbari and H. Kheiri, â€œAnti-synchronization of a modified three-dimensional chaotic finance system with uncertain parameters via adaptive control,â€ International Journal of Nonlinear Science, vol. 14, no. 2, pp. 178â€“185, 2012.
 J. Ding, W. Yang, and H. Yao, â€œA new modified hyperchaotic finance system and its control,â€ International Journal of Nonlinear Science, vol. 8, no. 1, pp. 59â€“66, 2009.
 W.-C. Chen, â€œDynamics and control of a financial system with time-delayed feedbacks,â€ Chaos, Solitons & Fractals, vol. 37, no. 4, pp. 1198â€“1207, 2008.
 P. He and Y. Li, â€œControl and synchronization of a hyperchaotic finance system via single controller scheme,â€ International Journal of Intelligent Computing and Cybernetics, vol. 8, no. 4, pp. 330â€“344, 2015.
 P. Wang, D. Li, X. Wu, J. LÃ¼, and X. Yu, â€œUltimate bound estimation of a class of high dimensional quadratic autonomous dynamical systems,â€ International Journal of Bifurcation and Chaos, vol. 21, no. 09, pp. 2679â€“2694, 2011.
 P. Wang, Y. Zhang, S. Tan, and L. Wan, â€œExplicit ultimate bound sets of a new hyperchaotic system and its application in estimating the hausdorff dimension,â€ Nonlinear Dynamics, vol. 74, no. 1-2, pp. 133â€“142, 2013.
 D. Li, J.-a. Lu, X. Wu, and G. Chen, â€œEstimating the ultimate bound and positively invariant set for the lorenz system and a unified chaotic system,â€ Journal of Mathematical Analysis and Applications, vol. 323, no. 2, pp. 844â€“853, 2006.
 H. K. Khalil and J. Grizzle, Nonlinear systems, vol. 3. Prentice hall New Jersey, 1996.
 G, Abikhanova, A Ahmetbekova, E Bayat, A Donbaeva, G Burkitbay (2018). International motifs and plots in the Kazakh epics in China (on the materials of the Kazakh epics in China), OpciÃ³n, AÃ±o 33, No. 85. 20-43.
 Akhpanov, S. Sabitov, R. Shaykhadenov (2018). Criminal pre-trial proceedings in the Republic of Kazakhstan: Trend of the institutional transformations. OpciÃ³n, AÃ±o 33. 107-125.
 G Cely Galindo (2017) Del Prometeo griego al de la era-biÃ³s de la tecnociencia. Reflexiones bioÃ©ticas OpciÃ³n, AÃ±o 33, No. 82 (2017):114-133