Advertisement

An Efficient Method for Computing Periodic Orbits of Conservative Dynamical Systems

  • M. N. Vrahatis
  • T. C. Bountis
Part of the NATO ASI Series book series (NSSB, volume 331)

Abstract

The accurate computation of periodic orbits and the precise knowledge of their properties are very important for studying the behavior of many dynamical systems of physical interest. In this paper, we present an efficient numerical method for computing to any desired accuracy periodic orbits (stable, unstable and complex) of any period. This method always converges rapidly to a periodic orbit independently of the initial guess, which is important when the mapping has many periodic orbits, stable and unstable close to each other, as is the case with conservative systems. We illustrate this method first, on the 2-Dimensional quadratic Hénon’s mapping, by computing rapidly and accurately several periodic orbits of high period. We also apply our method here to a 3-D conservative mapping as well as a 4-D complex version of Hénon’s map.

Keywords

Periodic Orbit Rotation Number Topological Degree Bisection Method Unstable Periodic Orbit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. Bazzani, P. Mazzanti, G. Servizi, and G. Turchetti, Normal forms for Hamil-tonian maps and nonlinear effects in a particle accelerator, Il Nuovo Cimento 102, 51–80 (1988).MathSciNetGoogle Scholar
  2. 2.
    G. D. Birkhoff, Dynamical systems with two degrees of freedom, Trans. Amer. Math. Soc. 18, 199–300 (1917).CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    T. C. Bountis, and R. H. G. Helleman, On the stability of periodic orbits of two-dimensional mappings, J. Math. Phys. 22, 1867–1877 (1981).ADSCrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    T. C. Bountis, and S. Tompaidis, Strong and weak instabilities in a 4D mapping model of FODO cell dynamics, in Future Problems in Nonlinear Particle Accelerators, edited by G. Turchetti and W. Scandale (World Scientific, 1991).Google Scholar
  5. 5.
    J. E. Dennis Jr., and R. B. Schnabel, Numerical methods for unconstrained optimization and nonlinear equations (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1983).zbMATHGoogle Scholar
  6. 6.
    R. Devogelaere, On the structure of symmetric periodic solutions of conservative systems with applications, in Contributions to the theory of nonlinear oscillations, pp. 53–84, edited by S. Lefschetz (Princeton U.P., Princeton, New Jersey, 1958).Google Scholar
  7. 7.
    A. Eiger, K. Sikorski, and F. Stenger, A bisection method for systems of nonlinear equations, ACM Trans. Math. Softw. 10, 367–377 (1984).CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    M. Feingold, L. P. Kadanoff, and O. Piro, Passive scalars, three-dimensional volume-preserving, and chaos, J. Stat. Phys. 50, 529–565 (1988).ADSCrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    J. M. Greene, A method for determining a stochastic transition, J. Math. Phys. 20, 1183–1201 (1979).ADSCrossRefGoogle Scholar
  10. 10.
    J. M. Greene, Locating three-dimensional roots by a bisection method, J. Comp. Phys. 98, 194–198 (1992).ADSCrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    M. Hénon, Numerical study of quadratic area-preserving mappings, Quart. Appl. Math. 27, 291–312 (1969).zbMATHMathSciNetGoogle Scholar
  12. 12.
    S. Komineas, M. N. Vrahatis, and T. C. Bountis, A study of period-doubling bifurcations in a 3-D conservative mapping, preprint, University of Patras, (1993).Google Scholar
  13. 13.
    H. T. Kook, and J. D. Meiss, Periodic orbits for reversible, symplectic mappings, Physica D 35, 65–86 (1989).ADSCrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    R. S. Mackay, J. D. Meiss, and I. C. Percival, Transport in Hamiltonian systems, Physica D 13, 55–81 (1985).ADSCrossRefMathSciNetGoogle Scholar
  15. 15.
    C. Miranda, Un’ osservazione su un teorema di Brouwer, Bol. Un. Mat. Ital. 3, 5–7 (1940).Google Scholar
  16. 16.
    J. M. Ortega, and W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables (Academic Press, New York, 1970).zbMATHGoogle Scholar
  17. 17.
    V. G. Papageorgiou, F.W. Nijhoff, and H.W. Capel, Integrable mappings and nonlinear integrable lattice equations, Phys. Lett. A 147, 106–114, (1990).ADSCrossRefMathSciNetGoogle Scholar
  18. 18.
    G. R. W. Quispel, H.W. Capel, V.G. Papageorgiou, and F.W. Nijhoff, Integrable mappings derived from soliton equations, Physica A 173, 243–266, (1991).ADSCrossRefMathSciNetGoogle Scholar
  19. 19.
    J. A. G. Roberts, The dynamics of trace maps, in this volume.Google Scholar
  20. 20.
    J. A. G. Roberts, and G. R. W. Quispel, Chaos and time-reversal symmetry. Order and Chaos in reversible dynamical systems, Physics Reports 216, No 2&3, 63–117 (1992).ADSCrossRefMathSciNetGoogle Scholar
  21. 21.
    K. Sikorski, Bisection is optimal, Numer. Math. 40, 111–117 (1982).CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    M. N. Vrahatis, An error estimation for the method of bisection in IR n, Bull. Soc. Math. Grèce 27, 161–174 (1986).zbMATHMathSciNetGoogle Scholar
  23. 23.
    M. N. Vrahatis, Solving systems of nonlinear equations using the nonzero value of the topological degree, ACM Trans. Math. Software 14, 312–329 (1988).CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    M. N. Vrahatis, Chabis: A mathematical software package for locating and evaluating roots of systems of non-linear equations, ACM Trans. Math. Software 14, 330–336 (1988).CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    M. N. Vrahatis, A short proof and a generalization of Miranda’s existence theorem, Proc. Amer. Math. Soc. 107, 701–703 (1989).zbMATHMathSciNetGoogle Scholar
  26. 26.
    M. N. Vrahatis, An efficient method for locating and computing periodic orbits of nonlinear mappings, submitted.Google Scholar
  27. 27.
    M. N. Vrahatis, T. C. Bountis, and N. Budinsky, A convergence-improving iterative method for computing periodic orbits near bifurcation points, J. Comp. Phys. 88, 1–14 (1990).ADSCrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    M. N. Vrahatis, and K. I. Iordanidis, A rapid generalized method of bisection for solving systems of nonlinear equations, Numer. Math. 49, 123–138 (1986).CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    M. N. Vrahatis, G. Servizi, G. Turchetti, and T. C. Bountis, A procedure to compute the periodic orbits and visualize the orbits of a 2D map, CERN SL/93 (AP) (1993).Google Scholar
  30. 30.
    M. N. Vrahatis, G. Turchetti, G. Servizi, and T. C. Bountis, Periodic orbits and invariant surfaces in 4D mapping models of particle beams, in preparation.Google Scholar

Copyright information

© Springer Science+Business Media New York 1994

Authors and Affiliations

  • M. N. Vrahatis
    • 1
  • T. C. Bountis
    • 1
  1. 1.Department of MathematicsUniversity of PatrasPatrasGreece

Personalised recommendations