PDϑ Control Strategy for a Fractional-Order Chaotic Financial Model

In this article, based on the previous works, a new fractional-order financial model is put up. The chaotic behavior of the fractional-order financial model is suppressed by designing an appropriate controller. By choosing the delay as the bifurcation parameter, we establish the sufficient condition to guarantee the stability and the existence of Hopf bifurcation of fractional-order financial model. Also, the influence of the delay and the fractional order on the stability and the existence of Hopf bifurcation of fractional-order financial model is revealed. An example is given to confirm the effectiveness of the analysis results. The main findings of this article play an important role in maintaining economic stability.

[1]  Xiaodi Li,et al.  Stabilization of Delay Systems: Delay-Dependent Impulsive Control , 2017, IEEE Transactions on Automatic Control.

[2]  Baogui Xin,et al.  Projective Synchronization of N-Dimensional Chaotic Fractional-Order Systems via Linear State Error Feedback Control , 2012 .

[3]  Lishan Liu,et al.  Positive solutions for a class of fractional 3-point boundary value problems at resonance , 2017 .

[4]  Y. Chu,et al.  Stochastic Hopf Bifurcation of a novel finance chaotic system , 2016 .

[5]  Ahmad Hajipour,et al.  Analysis and circuit simulation of a novel nonlinear fractional incommensurate order financial system , 2016 .

[6]  Wenjing Zhu,et al.  Exact Traveling Wave Solutions and Bifurcations of the Time-Fractional Differential Equations with Applications , 2019, Int. J. Bifurc. Chaos.

[7]  Wei-Ching Chen,et al.  Dynamics and control of a financial system with time-delayed feedbacks , 2008 .

[8]  Jinde Cao,et al.  Linear control for synchronization of a fractional-order time-delayed chaotic financial system , 2018, Chaos, Solitons & Fractals.

[9]  Yushu Chen,et al.  Nonlinear and chaotic analysis of a financial complex system , 2010 .

[10]  Young-Jai Park,et al.  Delayed feedback on the dynamical model of a financial system , 2011 .

[11]  Heng Liu,et al.  Linear Control of Fractional-Order Financial Chaotic Systems with Input Saturation , 2014 .

[12]  Xianhua Tang,et al.  Improved results for Klein-Gordon-Maxwell systems with general nonlinearity , 2018 .

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

[14]  Junwei Wang,et al.  Chaos Control of a Fractional-Order Financial System , 2010 .

[15]  Rongyan Zhang,et al.  Bifurcation Analysis for a Kind of Nonlinear Finance System with Delayed Feedback and Its Application to Control of Chaos , 2012, J. Appl. Math..

[16]  Baogui Xin,et al.  0-1 Test for Chaos in a Fractional Order Financial System with Investment Incentive , 2013 .

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

[18]  Jiqiang Jiang,et al.  Existence and nonexistence of positive solutions for the fractional coupled system involving generalized p-Laplacian , 2017, Advances in Difference Equations.

[19]  Xin Bao-Gui,et al.  Complexity evolvement of a chaotic fractional-orderfinancial system , 2011 .

[20]  Xiu-li Chai,et al.  Impulsive Synchronization and Adaptive-Impulsive Synchronization of a Novel Financial Hyperchaotic System , 2013 .

[21]  Yonghong Wu,et al.  Nontrivial solutions for a fractional advection dispersion equation in anomalous diffusion , 2017, Appl. Math. Lett..

[22]  Yushu Chen,et al.  Study for the Bifurcation Topological Structure and the Global Complicated Character of a Kind of Nonlinear Finance System(I) , 2001 .

[23]  Sachin Bhalekar,et al.  Chaos in fractional ordered Liu system , 2010, Comput. Math. Appl..

[24]  Xianhua Tang,et al.  Ground state solutions of Nehari–Pohozaev type for Kirchhoff-type problems with general potentials , 2017, Calculus of Variations and Partial Differential Equations.

[25]  Jinde Cao,et al.  Synchronization of Time Delayed Fractional Order Chaotic Financial System , 2017 .

[26]  Xianhua Tang,et al.  Geometrically distinct solutions for Klein-Gordon-Maxwell systems with super-linear nonlinearities , 2019, Appl. Math. Lett..

[27]  Ma Junhai,et al.  Study for the bifurcation topological structure and the global complicated character of a kind of nonlinear finance system (I) , 2001 .

[28]  Chenglin Xu,et al.  The new result on delayed finance system , 2014 .

[29]  Xuebing Zhang,et al.  Hopf Bifurcation and Chaos of a Delayed Finance System , 2019, Complex..

[30]  Gabriela Mircea,et al.  Uncertain and Stochastic Financial Models with Multiple delays , 2012, Int. J. Bifurc. Chaos.

[31]  Guanchun Liu,et al.  Modeling and Application of a New Nonlinear Fractional Financial Model , 2013, J. Appl. Math..

[32]  Apostolos Serletis,et al.  Is there chaos in economic time series , 1996 .

[33]  Lishan Liu,et al.  Existence results for impulsive fractional integro-differential equation of mixed type with constant coefficient and antiperiodic boundary conditions , 2017, Boundary Value Problems.

[34]  Yuzhen Bai,et al.  Stability and Hopf bifurcation for a stage-structured predator–prey model incorporating refuge for prey and additional food for predator , 2019 .

[35]  Yuxia Li,et al.  Stability and Hopf Bifurcation of Fractional-Order Complex-Valued Single Neuron Model with Time Delay , 2017, Int. J. Bifurc. Chaos.

[36]  Bijnan Bandyopadhyay,et al.  Sliding Mode Control of Fractional Order Systems , 2015 .

[37]  Fanwei Meng,et al.  Traveling wave solutions for fractional partial differential equations arising in mathematical physics by an improved fractional Jacobi elliptic equation method , 2017 .

[38]  Yonghong Wu,et al.  Existence and uniqueness of global mild solutions for a class of nonlinear fractional reaction-diffusion equations with delay , 2016, Comput. Math. Appl..

[39]  Junhai Ma,et al.  Complexity analysis research of financial and economic system under the condition of three parameters’ change circumstances , 2012 .

[40]  Ma Jun-hai,et al.  Study for the bifurcation topological structure and the global complicated character of a kind of nonlinear finance system(II) , 2001 .

[41]  Jinde Cao,et al.  An Impulsive Delay Inequality Involving Unbounded Time-Varying Delay and Applications , 2017, IEEE Transactions on Automatic Control.

[42]  Yuxia Li,et al.  Stability and Hopf Bifurcation of a Three-Neuron Network with Multiple Discrete and Distributed Delays , 2017, Neural Processing Letters.

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

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

[45]  Bo Zhu,et al.  Local and global existence of mild solutions for a class of nonlinear fractional reaction-diffusion equations with delay , 2016, Appl. Math. Lett..

[46]  I. Podlubny.  Fractional differential equations , 1998 .

[47]  Lan Yao,et al.  Adaptive full state hybrid function projective synchronization of financial hyperchaotic systems with uncertain parameters , 2013 .

[48]  Jun Cao,et al.  Bifurcation Analysis and Chaos Switchover Phenomenon in a Nonlinear Financial System with Delay Feedback , 2015, Int. J. Bifurc. Chaos.

[49]  JinRong Wang,et al.  Exploring delayed Mittag-Leffler type matrix functions to study finite time stability of fractional delay differential equations , 2018, Appl. Math. Comput..

[50]  Jian Zhang,et al.  Ground state of Kirchhoff type fractional Schrödinger equations with critical growth , 2018, Journal of Mathematical Analysis and Applications.

[51]  Lingling Zhang,et al.  Stability for a novel time-delay financial hyperchaotic system by adaptive periodically intermittent linear control , 2016 .

[52]  D. Matignon.  Stability results for fractional differential equations with applications to control processing , 1996 .

[53]  Xianhua Tang,et al.  Existence of ground state solutions of Nehari-Pankov type to Schrödinger systems , 2018, Science China Mathematics.

[54]  S. Bhalekar,et al.  Chaos in fractional order financial delay system , 2016 .