Dynamic Analysis of a Lü Model in Six Dimensions and Its Projections

Abstract In this article, extended complex Lü models (ECLMs) are proposed. They are obtained by substituting the real variables of the classical Lü model by complex variables. These projections, spanning from five dimensions (5D) and six dimensions (6D), are studied in their dynamics, which include phase spaces, calculations of eigenvalues and Lyapunov’s exponents, Poincaré maps, bifurcation diagrams, and related analyses. It is shown that in the case of a 5D extension, we have obtained chaotic trajectories; meanwhile the 6D extension shows quasiperiodic and hyperchaotic behaviors and it exhibits strange nonchaotic attractor (SNA) features.

[1]  Zhong Liu,et al.  Hyperchaos from an Augmented Lü System , 2010, Int. J. Bifurc. Chaos.

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

[3]  Tomasz Kapitaniak,et al.  On strange nonchaotic attractors and their dimensions , 1991 .

[4]  Paulo C. Rech,et al.  Hyperchaotic states in the parameter-space , 2012, Appl. Math. Comput..

[5]  Guanrong Chen,et al.  YET ANOTHER CHAOTIC ATTRACTOR , 1999 .

[6]  Paulo C. Rech,et al.  The organization of periodicity and hyperchaos in an asymmetric coupling of two chaotic Rössler oscillators , 2013 .

[7]  A. Wolf,et al.  Determining Lyapunov exponents from a time series , 1985 .

[8]  Tudor Bînzar,et al.  On a new chaotic system , 2015 .

[9]  Chen Zengqiang,et al.  Local bifurcation analysis of a four-dimensional hyperchaotic system ⁄ , 2008 .

[10]  Jitao Sun,et al.  Stability analysis of complex-valued nonlinear delay differential systems , 2013, Syst. Control. Lett..

[11]  José-Job Flores-Godoy,et al.  Some attractors in the Extended Complex Lorenz Model , 2013, Int. J. Bifurc. Chaos.

[12]  Paulo C. Rech.  Chaos and hyperchaos in a Hopfield neural network , 2011, Neurocomputing.

[13]  Yang Tao,et al.  Impulsive stabilization for control and synchronization of chaotic systems: theory and application to secure communication , 1997 .

[14]  Paulo C. Rech,et al.  Suppression of periodic structures and the onset of hyperchaos in a parameter-space of the Baier–Sahle flow , 2016 .

[15]  Paulo C. Rech,et al.  A hyperchaotic Chua System , 2009, Int. J. Bifurc. Chaos.

[16]  Claudia Valls,et al.  Complex Analysis and Differential Equations , 2012 .

[17]  Emad E. Mahmoud,et al.  On the hyperchaotic complex Lü system , 2009 .

[18]  Yuming Chen,et al.  Dynamics of a hyperchaotic Lorenz-type system , 2014 .

[19]  Z. E. Musielak,et al.  High-Dimensional Chaos in dissipative and Driven Dynamical Systems , 2009, Int. J. Bifurc. Chaos.

[20]  Amin Zarei,et al.  Complex dynamics in a 5-D hyper-chaotic attractor with four-wing, one equilibrium and multiple chaotic attractors , 2015 .

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

[22]  Christophe Letellier,et al.  A nine-dimensional Lorenz system to study high-dimensional chaos , 1998 .

[23]  César Cruz-Hernández,et al.  Synchronization of Time-Delay Chua's Oscillator with Application to Secure Communication , 2004 .

[24]  Sergey P. Kuznetsov,et al.  Strange Nonchaotic Attractors: Dynamics Between Order And Chaos in Quasiperiodically Forced Systems , 2006 .

[25]  Emad E. Mahmoud,et al.  ANALYSIS OF HYPERCHAOTIC COMPLEX LORENZ SYSTEMS , 2008 .

[26]  Mark J. McGuinness,et al.  The real and complex Lorenz equations in rotating fluids and lasers , 1982 .

[27]  Jitao Sun,et al.  Stability Analysis of Complex-Valued Nonlinear Differential System , 2013, J. Appl. Math..

[28]  Gamal M. Mahmoud,et al.  On Autonomous and nonautonomous Modified hyperchaotic Complex Lü Systems , 2011, Int. J. Bifurc. Chaos.

[29]  Paulo C. Rech,et al.  Delimiting hyperchaotic regions in parameter planes of a 5D continuous-time dynamical system , 2014, Appl. Math. Comput..

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

[31]  Roberto Barrio,et al.  When chaos meets hyperchaos: 4D Rössler model , 2015 .

[32]  Paulo C. Rech,et al.  Hyperchaos in a New Four-Dimensional Autonomous System , 2010, Int. J. Bifurc. Chaos.