Advertisement

Forecasting and Chaos

  • Juan C. Vallejo
  • Miguel A. F. Sanjuan
Chapter
Part of the Springer Series in Synergetics book series (SSSYN)

Abstract

Mankind has always been concerned with the desire of understanding the universe, knowing the ultimate reasons behind past events and having the ability of forecasting the future ones. From the earliest times, the study of natural cycles has been needed for a successful harvest. Astronomy, as one of the oldest sciences, was born with the main task of compiling the several observed phenomena in the skies. It attempted to understand the underlying mechanisms of the observations to figure out what was going to be observed in the future.

References

  1. 1.
    Alligood, K.T., Sauer, T.D., Yorke, J.A.: Chaos. An Introduction to Dynamical Systems. Springer, New York (1996)Google Scholar
  2. 2.
    Bailey, D.H., Barrio, R., Borwein, J.M.: High-precision computation: mathematical physics and dynamics. Appl. Math. Comput. 218, 10106 (2012)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bashford, F.: An Attempt to Test the Theories of Capillary Action by Comparing the Theoretical and Measured Forms of Drops of Fluid with an Explanation of the Method of Integration Employed in the Tables Which Give the Theoretical Form of Such Drops, by J.C. Adams. Cambridge University Press, Cambridge (1883)Google Scholar
  4. 4.
    Boffetta, G., Cencini, M., Falcioni, M., Vulpiani, A.: Predictability: a way to characterize complexity. Phys. Rep. 356, 367 (2002)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Carmona, R., Hwang, W., Torresani, B.: Wavelet analysis and its applications. In: Practical Time-Frequency Analysis: Continuous Wavelet and Gabor Transforms, with an Implementation in S, vol. 9. Academic, San Diego (1998)Google Scholar
  6. 6.
    Carpintero, D.D., Aguilar, L.A.: Orbit classification in arbitrary 2D and 3D potentials. Mon. Not. R. Astron. Soc. 298, 21 (1998)ADSCrossRefGoogle Scholar
  7. 7.
    Chandre, C., Wiggins, S., Uzer, T.: Time- frequency analysis of chaotic systems. Phys. D Nonlinear Phenom. 181, 171 (2003)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Cincotta, P.M., Simo, C.: Simple tools to study global dynamics in non-axisymmetric galactic potentials. Astron. Astrophys. 147, 205 (2000)ADSGoogle Scholar
  9. 9.
    Flaschka, H.: The toda lattice. II. Existence of integrals. Phys. Rev. B 9, 1924 (1974)Google Scholar
  10. 10.
    Froeschlé, C., Lega, E.: On the structure of symplectic mappings. The fast Lyapunov indicator: a very sensitivity tool. Celest. Mech. Dyn. Astron. 78, 167 (2000)Google Scholar
  11. 11.
    Gerlach, E., Skokos, C.: Comparing the efficiency of numerical techniques for the integration of variational equations. Discrete Contin. Dyn. Syst. 475 (2011)Google Scholar
  12. 12.
    Gustavson, F.G.: Oil constructing formal integrals of a Hamiltonian system near ail equilibrium point. Astronom. J. 71, 670 (1966)ADSCrossRefGoogle Scholar
  13. 13.
    Hairer, E.: A Runge-Kutta methods of order 10. J. Ins. Math. Appl. 21, 47 (1978)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, 2nd edn. Springer, New York (1996)zbMATHCrossRefGoogle Scholar
  15. 15.
    Hairer, E., Wanner, G.: Analysis by Its History. Springer, New York (1997)zbMATHGoogle Scholar
  16. 16.
    Hairer, E., Norsett, S.P., Wanner, G.: Solving ordinary differential equations, I, Nonstiff problems, 2nd edn. Springer, Berlin (1993)zbMATHGoogle Scholar
  17. 17.
    Heggie, D.C.: Chaos in the N-body problem of stellar dynamics. In: Roy, A.E. (ed.) Predictability, Stability and Chaos in N-Body Dynamical Systems. Plenum Press, New York (1991)Google Scholar
  18. 18.
    Heisenberg, W.: Non linear problems in physics. Phys. Today 20, 27 (1967)ADSCrossRefGoogle Scholar
  19. 19.
    Hénon, M., Heiles, C.: The applicability of the third integral of motion: some numerical experiments. Astron. J. 69, 73 (1964)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Herbst, B.M., Ablowitz, M.J.: Numerically induced chaos in the nonlinear schrodinger equation. Phys. Rev. Lett. 62, 2065 (1989)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Heun, K.: Neue Methode zur approximativen Integration der Differentialgleichungen einer unabhangigen Veranderlichen. Z. Math Phys. 45, 23 (1900)zbMATHGoogle Scholar
  22. 22.
    Iserles, A.: A First Course in the Numerical Analysis of Differential Equations. Cambridge University Press, Cambridge (1996)zbMATHGoogle Scholar
  23. 23.
    Julyan, H.E.C., Oreste, P.: The dynamics of Runge–Kutta methods. Int. J. Bifurcation Chaos 2, 427 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Kantz, H., Grebogi, C., Prasad, A., Lai, Y.C., Sinde, E.: Unexpected robustness-against-noise of a class of nonhyperbolic chaotic attractors. Phys. Rev. E 65, 026209 (2002)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    Kutta, W.: Beitrag zur naherungweisen Integration totaler Differenialgleichungen. Zeitschr. fur Math. und Phys. 46, 435 (1901)zbMATHGoogle Scholar
  26. 26.
    Kostelich, E.J., Kan, I., Grebogi, C., Ott, E., Yorke, J.A.: Unstable dimension variability: a source of nonhyperbolicity in chaotic systems. Phys. D 109, 81 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Lambert, J.D.: The initial value problem for ordinary differential equations. In: Jacobs, D. (ed.) The State of the Art in Numerical Analysis. Academic, New York (1977)Google Scholar
  28. 28.
    Lambert, J.D.: Numerical Methods for Ordinary Differential Systems. Wiley, New York (1992)Google Scholar
  29. 29.
    Larsson, S., Sanz-Serna, J.M.: A shadowing result with applications to finite element approximation of reaction-diffusion equations. Math. Compt. 68, 55 (1999)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Laplace, P.S.: Marquis de, a Philosophical Essay on Probabilities. Wiley, Chapman and Hall Ltd., London (1902)Google Scholar
  31. 31.
    Lasagni, F.M.: Canonical Runge-Kutta methods. ZAMP 39, 952 (1988)ADSMathSciNetzbMATHGoogle Scholar
  32. 32.
    Li, T., Yorke, J.A.: Period three implies chaos. Am. Math. Mon. 82(10), 985 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Lorenz, E.: Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130 (1963)ADSzbMATHCrossRefGoogle Scholar
  34. 34.
    Milani, A., Nobili, A.M., Knezevic, Z.: Stable chaos in the asteroid belt. Icarus 125, 13 (1997)ADSCrossRefGoogle Scholar
  35. 35.
    Press, W.H.: Numerical Recipes: The Art of Scientific Computing, 3rd edn. Cambridge University Press, Cambridge (2007)zbMATHGoogle Scholar
  36. 36.
    Ott, W., Yorke, J.A.: When Lyapunov exponents fail to exist. Phys. Rev. E 78, 056203 (2008)ADSMathSciNetCrossRefGoogle Scholar
  37. 37.
    Papaphilippou, Y., Laskar, J.: Global dynamics of triaxial galactic models through frequency map analysis. Astron. Astrophys. 329, 451 (1998)ADSGoogle Scholar
  38. 38.
    Pathak, J., Hunt, B., Girvan, M., Lu, Z., Ott, E.: Model-free prediction of large spatiotemporally chaotic systems from data: a reservoir computing approach. Phys. Rev. Lett. 120, 024102 (2018)ADSCrossRefGoogle Scholar
  39. 39.
    Pavani, R.: A two degrees-of-freedom hamiltonian model: an analytical and numerical study. In: Agarwal, R.P., Perera, K. (eds.) Proceedings of the Conference on Differential and Difference Equations and Applications, vol. 905. Hindawi Publishing Corporation, New York (2006)Google Scholar
  40. 40.
    Penrose, R.: Quantum implications. Essays in Honour of David Bohm. Routledge and Keegan, London/New York (1987)zbMATHGoogle Scholar
  41. 41.
    Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds and the Laws of Physics. Oxford University Press, Oxford (1989)zbMATHGoogle Scholar
  42. 42.
    Poincaré, H.: On the three-body problem and the equations of dynamics. Acta Math. 13, 1 (1890)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Poincaré, H.: Les Méthodes nouvelles de la mécanique céleste. Gauthier-Villars et Fils, Paris (1892)zbMATHGoogle Scholar
  44. 44.
    Ruelle, D., Takens, F.: On the nature of turbulence. Commun. Math. Phys. 20, 167 (1971)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Runge, C.: Ueber die numerische Aflosung von Differentialgleichungen. Math. Anal. 46, 167 (1895)zbMATHCrossRefGoogle Scholar
  46. 46.
    Saiki, Y., Sanjuán, M.A.F.: Low-dimensional paradigms for high-dimensional hetero-chaos. Chaos 28, 103110 (2018)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Sandor, Z., Erdi, B., Szell, A., Funk, B.: The relative Lyapunov indicator. An efficient method of chaos detection. Celest. Mech. Dyn. Astron. 90, 127 (2004)Google Scholar
  48. 48.
    Sauer, T., Grebogi, C., Yorke, J.A.: How long do numerical chaotic solutions remain valid? Phys. Lett. A 79, 59 (1997)CrossRefGoogle Scholar
  49. 49.
    Sanz-Serna, J.M.: Runge Kutta schemes for Hamiltonian systems. BIT, 28, 877 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Sanz-Serna, J.M., Larsson, S.: Shadows, chaos and saddles. Appl. Numer. Math. 13, 449 (1991)MathSciNetzbMATHGoogle Scholar
  51. 51.
    Skokos, C. Alignment indices: a new, simple method for determining the ordered or chaotic nature of orbits. J. Phys. A 34, 10029 (2001)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Skokos, C.: The Lyapunov characteristic exponents and their computation. Lect. Notes Phys. 790, 63 (2010)ADSCrossRefGoogle Scholar
  53. 53.
    Skokos, C., Bountis, T.C., Antonopoulos, C.: Geometrical properties of local dynamics in Hamiltonian systems: the Generalized Alignment Index (GALI) method. Phys. D 231, 30 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Stuchi, T.J.: Symplectic integrators revisited. Braz. Jour. Phys. 32, 4 (2002)Google Scholar
  55. 55.
    Suris, Y.B.: Preservation of sympletic structure in the numerical solution of Hamiltonian systems. In: Filippov, S.S. (ed.) Numerical Solution of Differential Equations. Akademii Nauk SSSR, Ins. Prikl. Mat., Moscow (1988)Google Scholar
  56. 56.
    Szebeheley, V.G., Peters, C.F.: Complete solution of a general problem of three bodies. Astronom. J. 72, 876 (1967)ADSCrossRefGoogle Scholar
  57. 57.
    Szezech, Jr J.D., Schelin, A.B., Caldas, I.L., Lopes, S.R., Morrison, P.J., Viana, R.L.: Finite-time rotation number: a fast indicator for chaotic dynamical structures. Phys. Lett. A 377, 452 (2013)ADSCrossRefGoogle Scholar
  58. 58.
    Tailleur, J., Kurchan, J.: Probing Rare physical trajectories with Lyapunov weighted dynamics. Nature, 3, 203 (2007)Google Scholar
  59. 59.
    Tsiganis, K., Anastasiadis, A., Varvoglis, H.: Dimensionality differences between sticky and non-sticky chaotic trajectory segments in a 3D Hamiltonian system. Chaos Solitons Fractals, 2281 (2000)Google Scholar
  60. 60.
    Valluri, M., Merrit, D.: Regular and chaotic dynamics of triaxial stellar systems. Astrophys. J. 506, 686 (1998)ADSCrossRefGoogle Scholar
  61. 61.
    Viana, R.L., Grebogi, C.: Unstable dimension variability and synchronization of chaotic systems. Phys. Rev. E 62, 462 (2000)ADSCrossRefGoogle Scholar
  62. 62.
    Voglis, N., Contopoulos, G., Efthymiopoulos, C.: Detection of ordered and chaotic orbits using the dynamical spectra. Celest. Mech. Dyn. Astron. 73, 211 (1999)ADSzbMATHCrossRefGoogle Scholar
  63. 63.
    Wellstead, P.E.: Introduction to Physical System Modelling. Academic, London (1979)Google Scholar
  64. 64.
    Wisdom, J., Holman, M.: Symplectic maps for the n-body problem, a stability analysis. Astron. J. 104, 2022 (1992)ADSCrossRefGoogle Scholar
  65. 65.
    Zhong, G.: Marsden, Lie-Poisson Hamilton Jacobi theory and Lie-Poisson integrators. Phys. Lett. A 133, 134 (1988)ADSMathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Juan C. Vallejo
    • 1
  • Miguel A. F. Sanjuan
    • 1
  1. 1.Departamento de FisicaUniversidad Rey Juan CarlosMóstolesSpain

Personalised recommendations