Numerical investigation concerning the dynamics in parameter planes of the Ehrhard–Müller system

Abstract In this paper we investigate the nonlinear dynamics of the Ehrhard–Muller system, which is modeled by a set of three-parameter, three autonomous first-order nonlinear ordinary differential equations. More specifically, here we report on numerically computed parameter plane diagrams for this three-parameter system. The dynamical behavior of each point, in each parameter plane, was characterized by using Lyapunov exponents spectra, or independently by counting the number of local maxima of one of the variables, in one complete trajectory in the phase-space. Each of these diagrams indicates parameter values for which chaos or periodicity may be found. In other words, each of these diagrams displays delimited regions of both behaviors, chaos and periodicity. We show that these parameter planes contain self-organized typical periodic structures embedded in a chaotic region. We also show that multistability is present in the Ehrhard–Muller system.

[1]  Paulo C. Rech,et al.  Chaos, Periodicity, and Quasiperiodicity in a Radio-Physical Oscillator , 2017, Int. J. Bifurc. Chaos.

[2]  Paulo C. Rech.  How to embed shrimps in parameter planes of the Lorenz system , 2017 .

[3]  J. Gallas,et al.  Accumulation horizons and period adding in optically injected semiconductor lasers. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  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.

[5]  P. Ehrhard,et al.  Dynamical behaviour of natural convection in a single-phase loop , 1990, Journal of Fluid Mechanics.

[6]  Paulo C. Rech,et al.  Multistability and organization of periodicity in a Van der Pol–Duffing oscillator , 2017 .

[7]  Paulo C. Rech,et al.  Organization of periodic structures in a damped-forced oscillator , 2014 .

[8]  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.

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

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

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

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

[13]  Paulo C. Rech.  Period-Adding Structures in the Parameter-Space of a Driven Josephson Junction , 2015, Int. J. Bifurc. Chaos.

[14]  Paulo C. Rech,et al.  Spiral periodic structures in a parameter plane of an ecological model , 2015, Appl. Math. Comput..

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

[16]  Paulo C. Rech,et al.  Chaos and periodicity in Vallis model for El Niño , 2017 .

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

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

[19]  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.

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

[21]  I. L. Caldas,et al.  Characterization in bi-parameter space of a non-ideal oscillator , 2015, 1507.06791.

[22]  Paulo C. Rech,et al.  Dynamics of an erbium-doped fiber dual-ring laser , 2012 .

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

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

[25]  Junho Park,et al.  Periodic and Chaotic Dynamics of the Ehrhard-Müller System , 2016, Int. J. Bifurc. Chaos.

[26]  Paulo C. Rech,et al.  Self-similar structures in a 2D parameter-space of an inductorless Chua's circuit , 2008 .

[27]  Paulo C. Rech.  Spiral organization of periodic structures in the Lorenz–Stenflo system , 2016 .