On the adaptive sliding mode controller for a hyperchaotic fractional-order financial system

Abstract This manuscript mainly focuses on the construction, dynamic analysis and control of a new fractional-order financial system. The basic dynamical behaviors of the proposed system are studied such as the equilibrium points and their stability, Lyapunov exponents, bifurcation diagrams, phase portraits of state variables and the intervals of system parameters. It is shown that the system exhibits hyperchaotic behavior for a number of system parameters and fractional-order values. To stabilize the proposed hyperchaotic fractional system with uncertain dynamics and disturbances, an efficient adaptive sliding mode controller technique is developed. Using the proposed technique, two hyperchaotic fractional-order financial systems are also synchronized. Numerical simulations are presented to verify the successful performance of the designed controllers.

[1]  Xuehong Zhu,et al.  Long memory of price−volume correlation in metal futures market based on fractal features , 2013 .

[2]  O. Rössler.  An equation for hyperchaos , 1979 .

[3]  Ahmad Hajipour,et al.  Synchronization of chaotic Arneodo system of incommensurate fractional order with unknown parameters using adaptive method , 2016 .

[4]  Juan Núñez,et al.  Lie Theory: Applications to problems in Mathematical Finance and Economics , 2009, Appl. Math. Comput..

[5]  N. Ford,et al.  A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations , 2013 .

[6]  Dumitru Baleanu,et al.  An Efficient Non-standard Finite Difference Scheme for a Class of Fractional Chaotic Systems , 2018 .

[7]  Ahmed Sadek Hegazi,et al.  Dynamical behaviors and synchronization in the fractional order hyperchaotic Chen system , 2011, Appl. Math. Lett..

[8]  Sergio Adriani David,et al.  Partial chaos suppression in a fractional order macroeconomic model , 2016, Math. Comput. Simul..

[9]  Yigang He,et al.  Generation and circuit implementation of fractional-order multi-scroll attractors , 2016 .

[10]  Sara Dadras,et al.  Control of a fractional-order economical system via sliding mode , 2010 .

[11]  M. Caputo,et al.  A new Definition of Fractional Derivative without Singular Kernel , 2015 .

[12]  Aria Alasty,et al.  Delayed feedback control via minimum entropy strategy in an economic model , 2008 .

[13]  Shantanu Das,et al.  Multi-objective Active Control Policy Design for Commensurate and Incommensurate Fractional Order Chaotic Financial Systems , 2015, ArXiv.

[14]  Guoliang Cai,et al.  Dynamic analysis and control of a new hyperchaotic finance system , 2012 .

[15]  Zhenbo Li,et al.  Synchronization of a chaotic finance system , 2011, Appl. Math. Comput..

[16]  Saptarshi Das,et al.  Chaos suppression in a fractional order financial system using intelligent regrouping PSO based fractional fuzzy control policy in the presence of fractional Gaussian noise , 2012 .

[17]  Yibei Nian,et al.  Controlling fractional order chaotic systems based on Takagi-Sugeno fuzzy model and adaptive adjustment mechanism , 2010 .

[18]  R. Gorenflo,et al.  Fractional calculus and continuous-time finance , 2000, cond-mat/0001120.

[19]  Yong Xu,et al.  Chaos in diffusionless Lorenz System with a fractional Order and its Control , 2012, Int. J. Bifurc. Chaos.

[20]  Janusz A. Hołyst,et al.  Chaos control in economical model by time-delayed feedback method , 2000 .

[21]  Xingyuan Wang,et al.  Hopf bifurcation and topological horseshoe of a novel finance chaotic system , 2012 .

[22]  Jinhu Lü,et al.  Stability analysis of linear fractional differential system with multiple time delays , 2007 .

[23]  M. Haeri,et al.  Synchronization of chaotic fractional-order systems via active sliding mode controller , 2008 .

[24]  Wei-Ching Chen,et al.  Nonlinear dynamics and chaos in a fractional-order financial system , 2008 .

[25]  Chunguang Li,et al.  Chaos and hyperchaos in the fractional-order Rössler equations , 2004 .

[26]  Zhen Wang,et al.  Control of an uncertain fractional order economic system via adaptive sliding mode , 2012, Neurocomputing.

[27]  J. A. Tenreiro Machado,et al.  Pseudo Phase Plane and Fractional Calculus modeling of western global economic downturn , 2015, Commun. Nonlinear Sci. Numer. Simul..

[28]  D. Baleanu,et al.  New aspects of the adaptive synchronization and hyperchaos suppression of a financial model , 2017 .

[29]  Sundarapandian Vaidyanathan,et al.  Analysis, adaptive control and circuit simulation of a novel nonlinear finance system , 2016, Appl. Math. Comput..

[30]  Zhen Wang,et al.  Analysis of nonlinear dynamics and chaos in a fractional order financial system with time delay , 2011, Comput. Math. Appl..

[31]  Ali Khaki Sedigh,et al.  Stabilization of fractional order systems using a finite number of state feedback laws , 2011 .

[32]  A. Atangana,et al.  New Fractional Derivatives with Nonlocal and Non-Singular Kernel: Theory and Application to Heat Transfer Model , 2016, 1602.03408.

[33]  Marius-F. Danca,et al.  Suppressing chaos in discontinuous systems of fractional order by active control , 2014, Appl. Math. Comput..

[34]  Yi Chai,et al.  Control and Synchronization of Fractional-Order Financial System Based on Linear Control , 2011 .

[35]  Mojtaba Sharifi,et al.  Nonlinear robust adaptive sliding mode control of influenza epidemic in the presence of uncertainty , 2017 .

[36]  H. Yao,et al.  Chaos control in an investment model with straight-line stabilization method , 2008 .

[37]  Vasily E. Tarasov,et al.  Economic interpretation of fractional derivatives , 2017, 1712.09575.

[38]  Junhai Ma,et al.  Chaos and Hopf bifurcation of a finance system , 2009 .