Bounds for the chaotic region in the Lorenz model

Abstract In a previous paper, the authors made an extensive numerical study of the Lorenz model, changing all three parameters of the system. We conjectured that the region of parameters where the Lorenz model is chaotic is bounded for fixed r . In this paper, we give a theoretical proof of the conjecture by obtaining theoretical bounds for the chaotic region and by using Fenichel theory. The theoretical bounds are complemented with numerical studies performed using the Maximum Lyapunov Exponent and OFLI2 techniques, and a comparison of both sets of results is shown. Finally, we provide a complete three-dimensional model of the chaotic regime depending on the three parameters.

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

[2]  Roberto Barrio,et al.  VSVO formulation of the taylor method for the numerical solution of ODEs , 2005 .

[3]  R. A. Smith,et al.  Some applications of Hausdorff dimension inequalities for ordinary differential equations , 1986, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[4]  R. F. Williams,et al.  Structural stability of Lorenz attractors , 1979 .

[5]  Holger R. Dullin,et al.  Extended Phase Diagram of the Lorenz Model , 2005, Int. J. Bifurc. Chaos.

[6]  Ferdinand Verhulst,et al.  Invariant Manifolds in Dissipative Dynamical Systems , 2005 .

[7]  Roberto Barrio,et al.  A three-parametric study of the Lorenz model , 2007 .

[8]  L. Sanchez,et al.  Convergence to equilibria in the Lorenz system via monotone methods , 2005 .

[9]  Knut H. Alfsen,et al.  Systematics of the Lorenz Model at σ = 10 , 1985 .

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

[11]  Roberto Barrio,et al.  Spurious structures in chaos indicators maps , 2009 .

[12]  Barry Saltzman,et al.  Finite Amplitude Free Convection as an Initial Value Problem—I , 1962 .

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

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

[15]  Bernd Krauskopf,et al.  Crocheting the Lorenz Manifold , 2004 .

[16]  Xinfu Chen.  Lorenz Equations Part I: Existence and Nonexistence of Homoclinic Orbits , 1996 .

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

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

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

[20]  James A. Yorke,et al.  Preturbulence: A regime observed in a fluid flow model of Lorenz , 1979 .

[21]  L. Shilnikov,et al.  NORMAL FORMS AND LORENZ ATTRACTORS , 1993 .

[22]  L. Chua,et al.  Methods of qualitative theory in nonlinear dynamics , 1998 .

[23]  Roberto Barrio,et al.  Sensitivity Analysis of ODES/DAES Using the Taylor Series Method , 2005, SIAM J. Sci. Comput..

[24]  J. Craggs.  Applied Mathematical Sciences , 1973 .

[25]  Valentin Afraimovich,et al.  Origin and structure of the Lorenz attractor , 1977 .

[26]  Peter Swinnerton-Dyer,et al.  Bounds for trajectories of the Lorenz equations: an illustration of how to choose Liapunov functions , 2001 .

[27]  John Guckenheimer,et al.  Periodic Orbit Continuation in Multiple Time Scale Systems , 2007 .

[28]  David Aubin,et al.  Writing the History of Dynamical Systems and Chaos: Longue Durée and Revolution, Disciplines and Cultures , 2002 .

[29]  William C. Troy,et al.  A Proof That the Lorenz Equations Have a Homoclinic Orbit , 1994 .

[30]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[31]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[32]  A. Rauh,et al.  Analytical investigation of the hopf bifurcation in the Lorenz model , 1986 .

[33]  M. Viana.  What’s new on lorenz strange attractors? , 2000 .

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

[35]  Roberto Barrio,et al.  Performance of the Taylor series method for ODEs/DAEs , 2005, Appl. Math. Comput..

[36]  Guanrong Chen,et al.  Estimating the bounds for the Lorenz family of chaotic systems , 2005 .

[37]  V. V. Bykov,et al.  The bifurcations of separatrix contours and chaos , 1993 .

[38]  Bernd Krauskopf,et al.  Global bifurcations of the Lorenz manifold , 2006 .

[39]  K. Robbins,et al.  Periodic solutions and bifurcation structure at high R in the Lorenz model , 1979 .

[40]  Martin Corless,et al.  Slow and fast invariant manifolds, and normal modes in a two degree-of-freedom structural dynamical system with multiple equilibrium states , 1998 .

[41]  Warwick Tucker,et al.  Foundations of Computational Mathematics a Rigorous Ode Solver and Smale's 14th Problem , 2022 .