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Forecasting and Chaos

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Predictability of Chaotic Dynamics

Part of the book series: Springer Series in Synergetics ((SSSYN))

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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.

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Notes

  1. 1.

    In 1843, the British mathematician and astronomer John Couch Adams also began to work on the orbit of Uranus using the data he had, and he has been sometimes credited by the discovery of Neptune.

References

  1. Alligood, K.T., Sauer, T.D., Yorke, J.A.: Chaos. An Introduction to Dynamical Systems. Springer, New York (1996)

    Google Scholar 

  2. Bailey, D.H., Barrio, R., Borwein, J.M.: High-precision computation: mathematical physics and dynamics. Appl. Math. Comput. 218, 10106 (2012)

    MathSciNet  MATH  Google Scholar 

  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. Boffetta, G., Cencini, M., Falcioni, M., Vulpiani, A.: Predictability: a way to characterize complexity. Phys. Rep. 356, 367 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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. Carpintero, D.D., Aguilar, L.A.: Orbit classification in arbitrary 2D and 3D potentials. Mon. Not. R. Astron. Soc. 298, 21 (1998)

    Article  ADS  Google Scholar 

  7. Chandre, C., Wiggins, S., Uzer, T.: Time- frequency analysis of chaotic systems. Phys. D Nonlinear Phenom. 181, 171 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Cincotta, P.M., Simo, C.: Simple tools to study global dynamics in non-axisymmetric galactic potentials. Astron. Astrophys. 147, 205 (2000)

    ADS  Google Scholar 

  9. Flaschka, H.: The toda lattice. II. Existence of integrals. Phys. Rev. B 9, 1924 (1974)

    Google Scholar 

  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. 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. Gustavson, F.G.: Oil constructing formal integrals of a Hamiltonian system near ail equilibrium point. Astronom. J. 71, 670 (1966)

    Article  ADS  Google Scholar 

  13. Hairer, E.: A Runge-Kutta methods of order 10. J. Ins. Math. Appl. 21, 47 (1978)

    Article  MathSciNet  Google Scholar 

  14. Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, 2nd edn. Springer, New York (1996)

    Book  MATH  Google Scholar 

  15. Hairer, E., Wanner, G.: Analysis by Its History. Springer, New York (1997)

    MATH  Google Scholar 

  16. Hairer, E., Norsett, S.P., Wanner, G.: Solving ordinary differential equations, I, Nonstiff problems, 2nd edn. Springer, Berlin (1993)

    MATH  Google Scholar 

  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. Heisenberg, W.: Non linear problems in physics. Phys. Today 20, 27 (1967)

    Article  ADS  Google Scholar 

  19. Hénon, M., Heiles, C.: The applicability of the third integral of motion: some numerical experiments. Astron. J. 69, 73 (1964)

    Article  ADS  MathSciNet  Google Scholar 

  20. Herbst, B.M., Ablowitz, M.J.: Numerically induced chaos in the nonlinear schrodinger equation. Phys. Rev. Lett. 62, 2065 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  21. Heun, K.: Neue Methode zur approximativen Integration der Differentialgleichungen einer unabhangigen Veranderlichen. Z. Math Phys. 45, 23 (1900)

    MATH  Google Scholar 

  22. Iserles, A.: A First Course in the Numerical Analysis of Differential Equations. Cambridge University Press, Cambridge (1996)

    MATH  Google Scholar 

  23. Julyan, H.E.C., Oreste, P.: The dynamics of Runge–Kutta methods. Int. J. Bifurcation Chaos 2, 427 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  25. Kutta, W.: Beitrag zur naherungweisen Integration totaler Differenialgleichungen. Zeitschr. fur Math. und Phys. 46, 435 (1901)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Lambert, J.D.: Numerical Methods for Ordinary Differential Systems. Wiley, New York (1992)

    Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Laplace, P.S.: Marquis de, a Philosophical Essay on Probabilities. Wiley, Chapman and Hall Ltd., London (1902)

    Google Scholar 

  31. Lasagni, F.M.: Canonical Runge-Kutta methods. ZAMP 39, 952 (1988)

    ADS  MathSciNet  MATH  Google Scholar 

  32. Li, T., Yorke, J.A.: Period three implies chaos. Am. Math. Mon. 82(10), 985 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  33. Lorenz, E.: Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130 (1963)

    Article  ADS  MATH  Google Scholar 

  34. Milani, A., Nobili, A.M., Knezevic, Z.: Stable chaos in the asteroid belt. Icarus 125, 13 (1997)

    Article  ADS  Google Scholar 

  35. Press, W.H.: Numerical Recipes: The Art of Scientific Computing, 3rd edn. Cambridge University Press, Cambridge (2007)

    MATH  Google Scholar 

  36. Ott, W., Yorke, J.A.: When Lyapunov exponents fail to exist. Phys. Rev. E 78, 056203 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  37. Papaphilippou, Y., Laskar, J.: Global dynamics of triaxial galactic models through frequency map analysis. Astron. Astrophys. 329, 451 (1998)

    ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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. Penrose, R.: Quantum implications. Essays in Honour of David Bohm. Routledge and Keegan, London/New York (1987)

    MATH  Google Scholar 

  41. Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds and the Laws of Physics. Oxford University Press, Oxford (1989)

    MATH  Google Scholar 

  42. Poincaré, H.: On the three-body problem and the equations of dynamics. Acta Math. 13, 1 (1890)

    MathSciNet  MATH  Google Scholar 

  43. Poincaré, H.: Les Méthodes nouvelles de la mécanique céleste. Gauthier-Villars et Fils, Paris (1892)

    MATH  Google Scholar 

  44. Ruelle, D., Takens, F.: On the nature of turbulence. Commun. Math. Phys. 20, 167 (1971)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  45. Runge, C.: Ueber die numerische Aflosung von Differentialgleichungen. Math. Anal. 46, 167 (1895)

    Article  MATH  Google Scholar 

  46. Saiki, Y., Sanjuán, M.A.F.: Low-dimensional paradigms for high-dimensional hetero-chaos. Chaos 28, 103110 (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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. Sauer, T., Grebogi, C., Yorke, J.A.: How long do numerical chaotic solutions remain valid? Phys. Lett. A 79, 59 (1997)

    Article  Google Scholar 

  49. Sanz-Serna, J.M.: Runge Kutta schemes for Hamiltonian systems. BIT, 28, 877 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  50. Sanz-Serna, J.M., Larsson, S.: Shadows, chaos and saddles. Appl. Numer. Math. 13, 449 (1991)

    MathSciNet  MATH  Google Scholar 

  51. Skokos, C. Alignment indices: a new, simple method for determining the ordered or chaotic nature of orbits. J. Phys. A 34, 10029 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  52. Skokos, C.: The Lyapunov characteristic exponents and their computation. Lect. Notes Phys. 790, 63 (2010)

    Article  ADS  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  54. Stuchi, T.J.: Symplectic integrators revisited. Braz. Jour. Phys. 32, 4 (2002)

    Google Scholar 

  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. Szebeheley, V.G., Peters, C.F.: Complete solution of a general problem of three bodies. Astronom. J. 72, 876 (1967)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  58. Tailleur, J., Kurchan, J.: Probing Rare physical trajectories with Lyapunov weighted dynamics. Nature, 3, 203 (2007)

    Google Scholar 

  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. Valluri, M., Merrit, D.: Regular and chaotic dynamics of triaxial stellar systems. Astrophys. J. 506, 686 (1998)

    Article  ADS  Google Scholar 

  61. Viana, R.L., Grebogi, C.: Unstable dimension variability and synchronization of chaotic systems. Phys. Rev. E 62, 462 (2000)

    Article  ADS  Google Scholar 

  62. Voglis, N., Contopoulos, G., Efthymiopoulos, C.: Detection of ordered and chaotic orbits using the dynamical spectra. Celest. Mech. Dyn. Astron. 73, 211 (1999)

    Article  ADS  MATH  Google Scholar 

  63. Wellstead, P.E.: Introduction to Physical System Modelling. Academic, London (1979)

    Google Scholar 

  64. Wisdom, J., Holman, M.: Symplectic maps for the n-body problem, a stability analysis. Astron. J. 104, 2022 (1992)

    Article  ADS  Google Scholar 

  65. Zhong, G.: Marsden, Lie-Poisson Hamilton Jacobi theory and Lie-Poisson integrators. Phys. Lett. A 133, 134 (1988)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Departamento de Fisica, Universidad Rey Juan Carlos, Móstoles, Madrid, Spain

    Juan C. Vallejo & Miguel A. F. Sanjuan

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  1. Juan C. Vallejo
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  2. Miguel A. F. Sanjuan
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Vallejo, J.C., Sanjuan, M.A.F. (2019). Forecasting and Chaos. In: Predictability of Chaotic Dynamics . Springer Series in Synergetics. Springer, Cham. https://doi.org/10.1007/978-3-030-28630-9_1

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