Unbounded dynamics in dissipative flows: Rössler model.

Transient chaos and unbounded dynamics are two outstanding phenomena that dominate in chaotic systems with large regions of positive and negative divergences. Here, we investigate the mechanism that leads the unbounded dynamics to be the dominant behavior in a dissipative flow. We describe in detail the particular case of boundary crisis related to the generation of unbounded dynamics. The mechanism of the creation of this crisis in flows is related to the existence of an unstable focus-node (or a saddle-focus) equilibrium point and the crossing of a chaotic invariant set of the system with the weak-(un)stable manifold of the equilibrium point. This behavior is illustrated in the well-known Rössler model. The numerical analysis of the system combines different techniques as chaos indicators, the numerical computation of the bounded regions, and bifurcation analysis. For large values of the parameters, the system is studied by means of Fenichel's theory, providing formulas for computing the slow manifold which influences the evolution of the first stages of the orbit.

[1]  Yu. A. Kuznetsov,et al.  NUMERICAL DETECTION AND CONTINUATION OF CODIMENSION-TWO HOMOCLINIC BIFURCATIONS , 1994 .

[2]  Giacomo Innocenti,et al.  A global qualitative view of bifurcations and dynamics in the Rössler system , 2008 .

[3]  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.

[4]  Hopf bifurcations and period-doubling transitions in Rössler model , 1985 .

[5]  F. Verhulst.  Methods and applications of singular perturbations , 2005 .

[6]  M. Hénon,et al.  A two-dimensional mapping with a strange attractor , 1976 .

[7]  J. Yorke,et al.  CHAOTIC ATTRACTORS IN CRISIS , 1982 .

[8]  C. Simó.  On the Hénon-Pomeau attractor , 1979 .

[9]  L. A. Belyakov.  Bifurcation set in a system with homoclinic saddle curve , 1980 .

[10]  Marco Monti,et al.  Characterization of the Rössler System in Parameter Space , 2007, Int. J. Bifurc. Chaos.

[11]  J. Eckmann,et al.  Iterated maps on the interval as dynamical systems , 1980 .

[12]  J. Yorke,et al.  Crises, sudden changes in chaotic attractors, and transient chaos , 1983 .

[13]  John Guckenheimer,et al.  The Forced van der Pol Equation I: The Slow Flow and Its Bifurcations , 2003, SIAM J. Appl. Dyn. Syst..

[14]  Yoshisuke Ueda,et al.  Optimal escape from potential wells—patterns of regular and chaotic bifurcation , 1995 .

[15]  Christopher Jones,et al.  Geometric singular perturbation theory , 1995 .

[16]  Roberto Barrio,et al.  Topological changes in periodicity hubs of dissipative systems. , 2012, Physical review letters.

[17]  Leon O. Chua,et al.  Slow Invariant Manifolds as Curvature of the Flow of Dynamical Systems , 2014, Int. J. Bifurc. Chaos.

[18]  Hans G. Kaper,et al.  Asymptotic analysis of two reduction methods for systems of chemical reactions , 2002 .

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

[20]  Ying-Cheng Lai,et al.  Transient Chaos: Complex Dynamics on Finite Time Scales , 2011 .

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

[22]  Neil Fenichel.  Geometric singular perturbation theory for ordinary differential equations , 1979 .

[23]  Roberto Barrio,et al.  Sensitivity tools vs. Poincaré sections , 2005 .

[24]  Roberto Barrio,et al.  Painting Chaos: a Gallery of Sensitivity Plots of Classical Problems , 2006, Int. J. Bifurc. Chaos.

[25]  Grebogi,et al.  Double crises in two-parameter dynamical systems. , 1995, Physical review letters.

[26]  Miguel A F Sanjuán,et al.  New developments in classical chaotic scattering , 2013, Reports on progress in physics. Physical Society.

[27]  B. Eckhardt,et al.  Regular and irregular potential scattering , 1986 .