Sensitivity tools vs. Poincaré sections

In this paper we introduce a modification of the fast Lyapunov indicator (FLI) denominated OFLITT2 indicator that may provide a global picture of the evolution of a dynamical system. Therefore, it gives an alternative or a complement to the pictures given by the classical Poincare sections and, besides, it may be used for any dimension. We present several examples comparing with the Poincare sections in two classical problems, the Henon–Heiles and the extensible-pendulum problems. Besides, we show the application to Hamiltonians of three degrees of freedom as an isotropic harmonic oscillator in three dimensions perturbed by a cubic potential and non-Hamiltonian problems as a four-dimensional chaotic system. Finally, a numerical method especially designed for its computation is presented in the appendix.

[1]  Jacques Laskar,et al.  The chaotic motion of the solar system: A numerical estimate of the size of the chaotic zones , 1990 .

[2]  M. Hénon,et al.  The applicability of the third integral of motion: Some numerical experiments , 1964 .

[3]  Tony Shardlow,et al.  Frontiers in numerical analysis , 2003 .

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

[5]  Elena Lega,et al.  On the Structure of Symplectic Mappings. The Fast Lyapunov Indicator: a Very Sensitive Tool , 2000 .

[6]  Elena Lega,et al.  On the Relationship Between Fast Lyapunov Indicator and Periodic Orbits for Continuous Flows , 2002 .

[7]  F. Potra,et al.  Sensitivity analysis for atmospheric chemistry models via automatic differentiation , 1997 .

[8]  Froeschle,et al.  Graphical evolution of the arnold web: from order to chaos , 2000, Science.

[9]  G. Contopoulos,et al.  A fast method for distinguishing between ordered and chaotic orbits. , 1997 .

[10]  Christian H. Bischof,et al.  E cient Derivative Codes through AutomaticDi erentiation and Interface Contraction : AnApplication in Biostatistics , 1995 .

[11]  Z. Galias,et al.  Computer assisted proof of chaos in the Lorenz equations , 1998 .

[12]  Elena Lega,et al.  On the numerical detection of the effective stability of chaotic motions in quasi-integrable systems , 2002 .

[13]  Cristel Chandre,et al.  Time–frequency analysis of chaotic systems , 2002, nlin/0209015.

[14]  Jesús F. Palacián,et al.  On perturbed oscillators in 1-1-1 resonance: the case of axially symmetric cubic potentials , 2002 .

[15]  H. N. Núñez-Yépez,et al.  Regular and chaotic behaviour in an extensible pendulum , 1994 .

[16]  Guanrong Chen,et al.  On a four-dimensional chaotic system , 2005 .

[17]  G. Contopoulos,et al.  Spectra of stretching numbers and helicity angles in dynamical systems , 1996 .

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

[19]  Peter Eberhard,et al.  Automatic differentiation of numerical integration algorithms , 1999, Math. Comput..

[20]  M. N. Vrahatis,et al.  Detecting order and chaos in Hamiltonian systems by the SALI method , 2004, nlin/0404058.

[21]  Carles Simó,et al.  A Formal Approximation of the Splitting of Separatrices in the Classical Arnold's Example of Diffusi , 1999 .

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

[23]  Heinz Hanßmann,et al.  On the Hamiltonian Hopf Bifurcations in the 3D Hénon–Heiles Family , 2001 .

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

[25]  John Guckenheimer,et al.  Computing Periodic Orbits and their Bifurcations with Automatic Differentiation , 2000, SIAM J. Sci. Comput..

[26]  P. Zgliczynski.  Computer assisted proof of chaos in the Rössler equations and in the Hénon map , 1997 .

[27]  Martin Hairer,et al.  GniCodes - Matlab programs for geometric numerical integration , 2003 .

[28]  Ch. Skokos,et al.  Alignment indices: a new, simple method for determining the ordered or chaotic nature of orbits , 2001 .

[29]  Carles Simó,et al.  Phase space structure of multi-dimensional systems by means of the mean exponential growth factor of nearby orbits , 2003 .

[30]  George F. Corliss,et al.  Solving Ordinary Differential Equations Using Taylor Series , 1982, TOMS.

[31]  Jacques Laskar,et al.  Frequency analysis for multi-dimensional systems: global dynamics and diffusion , 1993 .