Hopf bifurcation and topological horseshoe of a novel finance chaotic system

Abstract This paper deals with the existence of both Hopf bifurcation and topological horseshoe for a novel finance chaotic system. First, through rigorous mathematical analysis, we show that a Hopf bifurcation occurs at systems’ three equilibriums S0,1,2 and Hopf bifurcation at equilibrium S0 is non-degenerate and supercritical. Second, the computer-assisted verifications for horseshoe chaos in the system are given. Simulation results are presented to support the analysis.

[1]  Jinhu Lu,et al.  A New Chaotic Attractor Coined , 2002, Int. J. Bifurc. Chaos.

[2]  Jean-Michel Grandmont,et al.  On Endogenous Competitive Business Cycles , 1985, Cycles and Chaos in Economic Equilibrium.

[3]  James A. Yorke,et al.  A Chaos Lemma , 2001, The American mathematical monthly.

[4]  Chuandong Li,et al.  Hopf bifurcation and chaos in macroeconomic models with policy lag , 2005 .

[5]  G. Sell,et al.  The Hopf Bifurcation and Its Applications , 1976 .

[6]  Y. Kuznetsov.  Elements of Applied Bifurcation Theory , 2023, Applied Mathematical Sciences.

[7]  Qingdu Li,et al.  Horseshoe chaos in a class of simple Hopfield neural networks , 2009 .

[8]  Xiao-Song Yang,et al.  Complex dynamics in simple Hopfield neural networks. , 2006, Chaos.

[9]  L. Chua,et al.  The double scroll family , 1986 .

[10]  Guanrong Chen,et al.  From Chaos To Order Methodologies, Perspectives and Applications , 1998 .

[11]  Xiao-Song Yang,et al.  Horseshoes in piecewise continuous maps , 2004 .

[12]  S. Wiggins.  Introduction to Applied Nonlinear Dynamical Systems and Chaos , 1989 .

[13]  E. Lorenz.  Deterministic nonperiodic flow , 1963 .

[14]  L. O. Chua,et al.  The double scroll family. I: Rigorous of chaos. II: Rigorous analysis of bifurcation phenomena , 1986 .

[15]  H. I. Freedman,et al.  Hopf bifurcation in three-species food chain models with group defense. , 1992, Mathematical biosciences.

[16]  Guanrong Chen,et al.  Bifurcation Analysis of Chen's equation , 2000, Int. J. Bifurc. Chaos.

[17]  O. Rössler.  An equation for continuous chaos , 1976 .

[18]  C. Çelik,et al.  Hopf bifurcation of a ratio-dependent predator–prey system with time delay , 2009 .

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

[20]  Guoliang Cai,et al.  A New Finance Chaotic Attractor , 2007 .

[21]  Xiao-Song Yang,et al.  Topological Horseshoes and Computer Assisted Verification of Chaotic Dynamics , 2009, Int. J. Bifurc. Chaos.

[22]  Alexander Lipton-Lifschitz,et al.  Predictability and unpredictability in financial markets , 1999 .