Analytical and numerical investigations of the phase-locked loop with time delay

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 67, Issue 5, 2003, Pages 8-

  Schanz, Michael ^a   Pelster, Axel ^b  

^a UNIVERSITY OF STUTTGART   (Germany)

^b FREIE UNIVERSITÄT BERLIN   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84864472181     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.67.056205     Document Type: Article

References (43)

1. Y. Cho and T. Umeda, Opt. Commun. 59, 131 (1986).
2. E.V. Grigorieva, S.A. Kashchenko, N.A. Loiko, and A.M. Samson, Physica D 59, 297 (1992).
3. L. Glass and M.C. Mackey, Science 197, 287 (1977).
4. M.C. Mackey, Bull. Math. Biol. 41, 829 (1979).
5. J. Bélair and M.C. Mackey, J. Dyn. and Diff. Eq. 1, 299 (1989).
6. M.C. Mackey, J. Econ. Theory 48, 497 (1989).
7. J. M. Cushing, Integrodifferential Equations and Delay Models in Population Dynamics, Lecture Notes in Biomathematics, Vol. 20 (Springer, New York, 1977).
8. K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics (Kluwer Academic, Dordrecht, 1992).
9. A. Namajunas, K. Pyragas, and A. Tamasevicius, Phys. Lett. A 204, 255 (1995).
10. K. Pyragas, Phys. Lett. A 170, 421 (1992).
11. K. Pyragas and A. Tamasevicius, Phys. Lett. A 181, 99 (1993).
12. E. Ott, C. Grebogi, and J.A. Yorke, Phys. Rev. Lett. 64, 1196 (1990).
13. W. Wischert, A. Wunderlin, A. Pelster, M. Olivier, and J. Groslambert, Phys. Rev. E 49, 203 (1994).
14. P. Tass, A. Wunderlin, and M. Schanz, J. Biol. Phys. 21, 83 (1995).
15. J. K. Hale, Theory of Functional Differential Equations (Springer, New York, 1997).
16. N. Krasovskii, Stability of Motion (Stanford University Press, Stanford, 1963).
17. H. Haken, Synergetics–An Introduction, 3rd ed. (Springer, New York, 1983).
18. H. Haken, Advanced Synergetics (Springer, New York, 1987).
19. H. Haken, Information and Selforganization (Springer, New York, 1988).
20. H. Haken, Synergetic Computers and Cognition (Springer, New York, 1991).
21. H. Haken, Principles of Brain Functioning (Springer, New York, 1996).
22. J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Applied Mathematical Sciences, Vol. 19 (Springer, New York, 1976).
23. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Applied Mathematical Sciences, Vol. 42 (Springer, New York, 1993).
24. R. H. Rand and D. Armbruster, Perturbation Methods, Bifurcation Theory and Computer Algebra, Applied Mathematical Sciences, Vol. 65 (Springer, New York, 1987).
25. M. Schanz, Ph.D. thesis (in German), Universität Stuttgart, Shaker, 1997.
26. J. Kevorkian, in Space Mathematics III, Lectures in Applied Mathematics, Vol. 7 (American Mathematical Society, Providence, RI, 1966), p. 206.
27. W. Lick, SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 17, 815 (1969).
28. A. Wunderlin and H. Haken, Z. Phys. B 21, 393 (1975).
29. C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers–Asymptotic Methods and Perturbation Theory (McGraw-Hill, New York, 1978).
30. P. Manneville, Dissipative Structures and Weak Turbulence (Academic Press, San Diego, 1990).
31. E. Grigorieva, H. Haken, S.A. Kashchenko, and A. Pelster, Physica D 125, 123 (1999).
32. W.M. Wright, J. Reine Agnew Math 194, 66 (1955).
33. M. Schanz and A. Pelster, SIAM J. Appl. Dyn. Syst. (to be published), eprint nlin.CD/0203005.
34. M. Schanz and A. Pelster, in Proceedings of the 15th IMACS World Congress on Scientific Computation, Modelling and Applied Mathematics, Berlin, 1997, edited by A. Sydow (Wissenschaft und Technik Verlag, Berlin, 1997), Vol. 1, p. 215.
35. C. Simmendinger, Diploma thesis (in German), Universität Stuttgart, 1997.
36. A. Wolf, J.B. Swift, H.L. Swinney, and J.A. Vastano, Physica D 16, 285 (1985).
37. J.D. Farmer, Physica D 4, 366 (1982).
38. S. Grossmann and S. Thomae, Z. Naturforsch. A 32A, 1353 (1977).
39. M.J. Feigenbaum, J. Stat. Phys. 19, 25 (1978).
40. M. Le Berre, E. Ressayre, A. Tallet, H.M. Gibbs, D.L. Kaplan, and M.H. Rose, Phys. Rev. E 35, 4020 (1987).
41. V. Avrutin, R. Lammert, M. Schanz, and G. Wackenhut (unpublished).
42. V. Avrutin, R. Lammert, M. Schanz, and G. Wackenhut, Computer code ANT4.669 (University of Stuttgart, Stuttgart, 2002).
43. See http://www.AnT4669.de; the Poincaré sections of Fig. 33 in Sec. IV were obtained by using this software package, which is designed for the simulation and analysis of nonlinear dynamical systems belonging to various classes, for instance, maps, ordinary, delay, and functional differential equations, as well as many subclasses derived from these. ANT4.669 supports the investigation of their dynamic behavior with several provided investigation methods, for instance, period analysis, the Lyapunov exponent calculation, or the generalized Poincaré section analysis and much more. Another feature of ANT4.669 is the ability to investigate a dynamical system by varying some relevant influence quantities, such as the control parameters, initial values, or even some parameters of the investigation methods. The calculations which are necessary to perform such one-, two-, or even higher-dimensional scans can be easily distributed among several nodes of a cluster or grid using a client server architecture.