Structure and breakdown of invariant tori in a 4-D mapping model of accelerator dynamics

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

Volumn 7, Issue 12, 1997, Pages 2707-2722

  Vrahatis, M N ^a   Isliker, H ^a   Bountis, T C ^a  

^a UNIVERSITY OF PATRAS   (Greece)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0031537156     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127497001825     Document Type: Article

References (39)

1. Badii, R. & Politi, A. [1984] "Intrinsic oscillations in measuring the fractal dimension," Phys. Lett. A104, 303-305.
2. Bernstein, L. [1971] The Jacobi-Perron Algorithm. Its Theory and Application, Lecture Notes in Mathematics 207.
3. Bollt, E. M. & Meiss, J. D. [1993] "Breakup of invariant tori for the four dimensional semi-standard map," Physica D66, 282-297.
4. Bountis, T. C. & Tompaidis, S. [1991] "Strong and weak instabilities in a 4D mapping model of FODO cell dynamics," in Future Problems in Nonlinear Particle Accelerators, eds. Turchetti, G. & Scandale, W. (World Scientific, Singapore), pp. 112-127.
5. Contopoulos, G., Magnenat, P. & Martinet, L. [1982] "Invariant surfaces and orbital behaviour in dynamical systems of 3 degrees of freedom," Physica D6, 126-136.
6. Contopoulos, G. & Magnenat, P. [1985] "Simple three- dimensional periodic orbits in a galactic type potential," Cel. Mech. 37, 387-414.
7. Corrigan, R., Huson, F. & Month, M. (eds.) [1982] See papers in Physics of High Energy Accelerators (A.I.P., New York), A.I.P. Conf. Proc. 87.
8. Drossos, L., Ragos, O., Vrahatis, M. N. & Bountis, T. [1996] "Method for computing long periodic orbits of dynamical systems," Phys. Rev. E53, 1206-1211.
9. Eckmann, J.-P. & Ruelle, D. [1985] "Ergodic theory of chaos and strange attractors," Rev. Mod. Phys. 57, 617-656.
10. Ellner, S. [1988] "Estimating attractor dimensions from limited data: A new method, with error estimates," Phys. Lett. 133, 128-133.
11. Grassberger, P. & Procaccia, I. [1983a] "Characterization of strange attractors," Phys. Rev. Lett. 50, 346-349.
12. Grassberger, P. & Procaccia, I. [1983b] "Measuring the strangeness of strange attractors," Physica D9, 189-208.
13. Greene, J. M. [1979] "A method for determining a stochastic transition," J. Math. Phys. 20, 1183-1201.
14. Hénon, M. & Heiles, C. [1964] "The applicability of the third integral of the motion: Some numerical experiments," Astron. J. 69, 73-79.
15. Isliker, H. [1992] "A scaling test for correlation dimensions," Phys. Lett. A169, 313-322.
16. Kim, S. & Ostlund, S. [1985] "Renormalization of mappings of the two-torus," Phys. Rev. Lett. 55, 1165-1168.
17. Kook, H. & Meiss, J. D. [1989] "Periodic orbits of reversible symplectic mappings," Physica D35, 65-86.
18. Laskar, J., Froeschlé C. & Celletti, A. [1992] "The measure of chaos by the numerical analysis of fundamental frequencies. Application to the standard mapping," Physica D56, 253-269.
19. Laskar, J. [1993] "Frequency analysis for multidimensional systems. Global dynamics and diffusion," Physica D67, 257-281.
20. Lichtenberg, A. J. & Lieberman, M. A. [1983] Regular and Stochastic Motion, Applied Mathematical Sciences 38 (Springer-Verlag, New York).
21. MacKay, R. S. [1983] "A renormalisation approach to invariant circles in area-preserving maps," Physica D7, 283-300.
22. MacKay, R. S. & Meiss, J. D. (eds.) [1986] See papers in Hamiltonian Dynamical Systems (Adams Hilger, Bristol).
23. MacKay, R. S., Meiss, J. D. & Stark, J. [1994] "An approximate renormalization for the break-up of invariant tori with three frequencies," Phys. Lett. A190, 417-424.
24. Papaphilippou, Y. [1996] (private communication).
25. Schweiger, F. [1973] The Metrical Theory of Jacobi- Perron Algorithm, Lecture Notes in Mathematics 334.
26. Smith, L. A. [1988] "Intrinsic limits on dimension calculations," Phys. Lett. A133, 283-288.
27. Takens, F. [1981] "Detecting strange attractors in turbulence," in Dynamical Systems and Turbulence,Lecture Notes in Mathematics 898 (Springer, Berlin), pp. 366-381.
28. Takens, F. [1984] "On the numerical determination of the dimension of an attractor," in Dynamical Systems and Turbulence, Lecture Notes in Mathematics 1125, (Springer, Berlin), pp. 99-106.
29. Theiler, J. [1986] "Spurious dimension from correlation algorithms applied to limited time-series data," Phys. Rev. A34, 2427-2432.
30. α noise," Phys. Lett. A155, 480-493.
31. Todesco, E. [1994] "Analysis of resonant structures of four-dimensional symplectic mappings, using normal forms," Phys. Rev. E50, 4298-4301.
32. Tompaidis, S. [1995] "Approximation of invariant surfaces by periodic orbits in high-dimensional maps. Some rigorous results," preprint submitted for publication.
33. Turchetti, G. & Scandale, W. (eds.) [1991] See papers in Future Problems in Nonlinear Particle Accelerators (World Scientific, Singapore).
34. Vrahatis, M. N. [1988] "Solving systems of nonlinear equations using the nonzero value of the topological degree," ACM Trans. Math. Software 14, 312-329.
35. Vrahatis, M. N. [1995] "An efficient method for locating and computing periodic orbits of nonlinear mappings," J. Comput. Phys. 119, 105-119.
36. Vrahatis, M. N. & Bountis, T. C. [1994] "An efficient method for computing periodic orbits of conservative dynamical systems," in Int. Conf. Hamiltonian Mechanics, Integrability and Chaotic Behavior, ed. Seimenis, J. (Plenum Press, New York), pp. 261-274.
37. Vrahatis, M. N., Bountis, T. C. & Kollmann, M. [1996] "Periodic orbits and invariant surfaces of 4D nonlinear mappings," Int. J. Bifurcation and Chaos 6(8), 1425-1437.
38. Vrahatis, M. N., Servizi, G., Turchetti, G. & Bountis, T. C. [1993] "A procedure to compute the periodic orbits and visualize the orbits of a 2D map," CERN SL/93-06 (AP).
39. Wolf, A., Swift, J. B., Swinney, H. L. & Vastano, J. A. [1985] "Determining Lyapunov exponents from a time series," Physica B16, 285-317.