N-dimensional dynamical systems exploiting instabilities in full

Chaos

Volumn 10, Issue 4, 2000, Pages 760-770

  Rius, J ^a   Figueras, M ^a   Herrero, R ^b   Farjas, J ^c   Pi, F ^a   Orriols, G ^a  

^a UNIVERSITAT AUTÒNOMA DE BARCELONA   (Spain)

^b UNIVERSITAT POLITÈCNICA DE CATALUNYA   (Spain)

^c UNIVERSITY OF GIRONA   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034366941     PISSN: 10541500     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1324650     Document Type: Article

References (28)

1. Dynamic Biological Networks, edited by R. M. Harris-Warrik, E. Marder, A. I. Selverton, and M. Moulins (MIT Press, Cambridge, 1992).
2. See, e.g., J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer, New York, 1983); S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos (Springer, New York, 1990).
3. See, e.g., J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer, New York, 1983); S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos (Springer, New York, 1990).
4. L. D. Landau, C. R. (Dokl.) Acad. Sci. USSR 44, 311 (1944). reproduced in Chaos II, edited by Hao Bai-Lin (World Scientific, Singapore, 1990), p. 115.
5. L. D. Landau, C. R. (Dokl.) Acad. Sci. USSR 44, 311 (1944). reproduced in Chaos II, edited by Hao Bai-Lin (World Scientific, Singapore, 1990), p. 115.
6. The torus bifurcation does not permit the superposition with arbitrary phase assumed by Landau in relation to the occurrence moment of the secondary instability.
7. See, e.g., Hydrodynamic Instabilities and the Transition to Turbulence, edited by H. L. Swinney and J. P. Gollub (Springer-Verlag, Berlin, 1981).
8. D. Ruelle and F. Takens, Commun. Math. Phys. 20, 167 (1971); S. Newhouse, D. Ruelle, and F. Takens, ibid. 64, 35 (1978).
9. D. Ruelle and F. Takens, Commun. Math. Phys. 20, 167 (1971); S. Newhouse, D. Ruelle, and F. Takens, ibid. 64, 35 (1978).
10. Half of the bifurcations occur on each point, if N is odd, while one bifurcation more happens on the initially stable point, if N is even.
11. A sequence of torus bifurcations initiated from a fixed point is intrinsically related to the successive Hopf bifurcations of that point and the corresponding oscillations are usually based on the same frequencies.
12. In fact, only one attractor can emerge from the variety of Hopf bifurcations of a saddle-node pair of fixed points but additional attractors may arise from secondary processes.
13. J. Rius, M. Figueras, R. Herrero, F. Pi, G. Orriols, and J. Farjas, Phys. Rev. E 62, 333 (2000).
14. G. Orriols, C. Schmidt, and F. Pi, Opt. Commun. 63, 66 (1987).
15. R. Herrero, R. Pons, J. Farjas, F. Pi, and G. Orriols, Phys. Rev. E 53, 5627 (1996).
16. J. I. Rosell, J. Farjas, R. Herrero, F. Pi, and G. Orriols, Physica D 85, 509 (1995).
17. J. Farjas, J. I. Rosell, R. Herrero, R. Pons, F. Pi, and G. Orriols, Physica D 95, 107 (1996).
18. S. L. McCall, Appl. Phys. Lett. 32, 284 (1978); H. M. Gibbs, S. S. Tang, J. L. Jewell, D. A. Weinberger, and K. Tai, ibid. 41, 221 (1982).
19. S. L. McCall, Appl. Phys. Lett. 32, 284 (1978); H. M. Gibbs, S. S. Tang, J. L. Jewell, D. A. Weinberger, and K. Tai, ibid. 41, 221 (1982).
20. H. J. Kong and W. Y. Hwang, J. Appl. Phys. 67, 6066 (1990).
21. J. I. Rosell, F. Pi, F. Boixader, R. Herrero, F. Farjas, and G. Orriols, Opt. Commun. 82, 162 (1991).
22. R. Herrero, F. Boixader, G. Orriols, J. I. Rosell, and F. Pi, Opt. Commun. 113, 324 (1994).
23. P. Glendinning and C. Sparrow, J. Stat. Phys. 35, 645 (1984).
24. The two-dimensional eigenspace and the oscillation frequency of the secondary Hopf bifurcation of the saddle cycle are directly related to those of the first Hopf bifurcation of the fixed point. This implies that the secondary bifurcation will be subcritical.
25. This has been verified by continuously following both periodic orbits.
26. L. P. Shil'nikov, Int. J. Bifurcation Chaos Appl. Sci. Eng. 4, 489 (1994).
27. The possibilities of invariant tori generation through sequences of bifurcations initiated from a fixed point are not well known. The theory of universal unfoldings developed for certain low-dimensional degeneracies (Ref. 2) gives precise information for N up to 4, but only some general ideas may be tentatively extrapolated to higher dimensional problems.
28. Eighty trajectories initiated from different points of the unstable eigenspace of the saddle cycle. The proximity of the initial point to the saddle orbit is chosen such that the error due to the separation between the eigenspace and the invariant manifold is of the same order as the error of the numerical integration. The initial points are taken in 10 temporally equidistant places along the orbit and with 8 different output angles within the two-dimensional eigenspace transverse to the orbit.