Suppression of periodic structures and the onset of hyperchaos in a parameter-space of the Baier–Sahle flow

Abstract We investigate a two-dimensional parameter-space of the Baier–Sahle flow, which is a mathematical model consisting of a set of n autonomous, four-parameter, first-order nonlinear ordinary differential equations. By using the Lyapunov exponents spectrum to numerically characterize the dynamics of the model in the chosen parameter-space, we show that for n = 3 it presents typical periodic structures embedded in a chaotic region, forming a spiral structure that coils up around a focal point while period-adding bifurcations take place. We also show that these structures are destroyed as n is increased, as well as we delimit hyperchaotic regions with two or more positive Lyapunov exponents in the investigated parameter-space, for n greater than 3.

[1]  Andrey Shilnikov,et al.  Global organization of spiral structures in biparameter space of dissipative systems with Shilnikov saddle-foci. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  Zhouchao Wei,et al.  Hidden Attractors and Dynamical Behaviors in an Extended Rikitake System , 2015, Int. J. Bifurc. Chaos.

[3]  P. Glendinning,et al.  Global structure of periodicity hubs in Lyapunov phase diagrams of dissipative flows. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  J. Gallas,et al.  Periodicity hub and nested spirals in the phase diagram of a simple resistive circuit. , 2008, Physical review letters.

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

[6]  Paulo C. Rech,et al.  Spiral periodic structure inside chaotic region in parameter-space of a Chua circuit , 2012, Int. J. Circuit Theory Appl..

[7]  M. Yao,et al.  Study of hidden attractors, multiple limit cycles from Hopf bifurcation and boundedness of motion in the generalized hyperchaotic Rabinovich system , 2015 .

[8]  Roberto Barrio,et al.  Qualitative analysis of the Rössler equations: Bifurcations of limit cycles and chaotic attractors , 2009 .

[9]  Christopher Essex,et al.  Competitive Modes and their Application , 2006, Int. J. Bifurc. Chaos.

[10]  Paulo C. Rech,et al.  Organization of the Dynamics in a Parameter Plane of a Tumor Growth Mathematical Model , 2014, Int. J. Bifurc. Chaos.

[11]  Julien Clinton Sprott,et al.  Elementary quadratic chaotic flows with a single non-hyperbolic equilibrium , 2015 .

[12]  Sohrab Effati,et al.  Hybrid projective synchronization and control of the Baier-Sahle hyperchaotic flow in arbitrary dimensions with unknown parameters , 2014, Appl. Math. Comput..

[13]  Zhouchao Wei,et al.  On the periodic orbit bifurcating from one single non-hyperbolic equilibrium in a chaotic jerk system , 2015 .

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

[15]  Jason A. C. Gallas,et al.  The Structure of Infinite Periodic and Chaotic Hub Cascades in Phase Diagrams of Simple Autonomous Flows , 2010, Int. J. Bifurc. Chaos.

[16]  Andrew Y. T. Leung,et al.  Symmetry and Period-Adding Windows in a Modified Optical Injection Semiconductor Laser Model , 2012 .

[17]  Nikolay V. Kuznetsov,et al.  Time-Varying Linearization and the Perron Effects , 2007, Int. J. Bifurc. Chaos.

[18]  R. A. Gorder,et al.  Competitive modes for the Baier–Sahle hyperchaotic flow in arbitrary dimensions , 2013 .

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

[20]  Baier,et al.  Design of hyperchaotic flows. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[21]  Ruedi Stoop,et al.  Real-world existence and origins of the spiral organization of shrimp-shaped domains. , 2010, Physical review letters.

[22]  Paulo C. Rech,et al.  Hopfield neural network: The hyperbolic tangent and the piecewise-linear activation functions , 2012, Neural Networks.

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