Chaos in two-loop negative feedback systems

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 65, Issue 5, 2002, Pages

  Bastos De Figueiredo, J C ^a   Diambra, L ^a   Glass, Leon ^b   Malta, C P ^a  

^a UNIVERSITY OF SÃO PAULO   (Brazil)

^b MCGILL UNIVERSITY   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

CHAOS THEORY; DIFFERENTIAL EQUATIONS; DYNAMICS; FUNCTIONS; NEURAL NETWORKS; OPTICAL SYSTEMS; PARAMETER ESTIMATION; SET THEORY;

CHEMICAL SIGNALS; FEEDBACK LOOPS; TIME ARGUMENT; TIME DELAYS;

FEEDBACK;

EID: 41349109061     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.65.051905     Document Type: Article

References (45)

1. U. an der Heiden and M. C. Mackey, J. Math. Biol. 8, 345 (1982).
2. M. C. Mackey and L. Glass, Science 197, 287 (1977).
3. R. M. May, Ann. N.Y. Acad. Sci. 357, 267 (1980).
4. A. I. Mees and P. E. Rapp, J. Math. Biol. 5, 99 (1978).
5. L. Glass and M. C. Mackey, From Clocks to Chaos: The Rhythms of Life (Princeton University Press, Princeton, NJ, 1988).
6. H. T. Milhorn, The Application of Control Theory to Physiological Systems (Saunders, Philadelphia, PA, 1966).
7. K. Pakdaman and C. P. Malta, IEEE Trans. Neural Netw. 9, 231 (1998).
8. C. P. Malta and C. Grotta-Ragazzo, Int. J. Bifurcation Chaos Appl. Sci. Eng. 1, 657 (1991).
9. J. F. Perez, C. P. Malta, and F. A. B. Coutinho, J. Theor. Biol. 71, 505 (1978).
10. K. Ikeda, K. Kondo, and O. Akimoto, Phys. Rev. Lett. 49, 1467 (1982); K. Ikeda and K. Matsumoto, Physica D 29, 223 (1987).
11. K. Ikeda, K. Kondo, and O. Akimoto, Phys. Rev. Lett. 49, 1467 (1982); K. Ikeda and K. Matsumoto, Physica D 29, 223 (1987).
12. M. W. Derstine, H. M. Gibbs, F. A. Hopf, and D. L. Kaplan, Phys. Rev. A 27, 3200 (1983).
13. C. R. de Oliveira and C. P. Malta, Phys. Rev. A 36, 3997 (1987).
14. S. Akselrod, D. Gordon, A. Ubel, D. C. Shannon, A. C. Barger, and R. J. Cohen, Science 213, 220 (1981).
15. R. I. Kitney and O. Rompelman, The Study of Heart Rate Variability (Clarendon Press, Oxford, 1980).
16. M. Kobayashi and T. Musha, IEEE Trans. Biomed. Eng. 29, 456 (1982).
17. L. Goodman, IEEE Trans. Biomed. Eng. 11, 82 (1964).
18. A. W. Cowley, J. F. Liard, and C. Guyton, Circ. Res. 32, 564 (1973).
19. L. Glass, A. Beuter, and D. Larocque, Math. Biosci. 90, 111 (1988).
20. P. A. Merton, H. B. Morton, and C. Rashbass, Nature (London) 216, 583 (1967).
21. M. N. Oĝustötelli and R. B. Stein, J. Math. Biol. 3, 87 (1976).
22. A. Beuter, J. Bélair, and C. Labrie, Bull. Math. Biol. 55, 525 (1993).
23. A. Beuter and K. Vasilakos, Chaos 5, 35 (1995).
24. L. Glass and C. P. Malta, J. Theor. Biol. 145, 217 (1990).
25. J. K. Hale and W. Huang, J. Math. Anal. Appl. 178, 344 (1993).
26. J. Bélair and S. A. Campbell, SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 54, 1402 (1994).
27. J. P. Mahaffy, K. M. Joiner, and P. J. Zak, Int. J. Bifurcation Chaos Appl. Sci. Eng. 5, 779 (1995).
28. L. Olien and J. Bélair, Physica D 102, 349 (1997).
29. X. G. Li, S. G. Ruan, and J. J. Wei, J. Math. Anal. Appl. 236, 254 (1999).
30. L. P. Shayer and S. A. Campbell, SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 61, 673 (2000).
31. K. Gopalsamy and I. K. C. Leung, IEEE Trans. Neural Netw. 8, 341 (1997).
32. In general, this method is more efficient than the Runge-Kutta one because the convergence is achieved for a step that is greater or equal to the step required by the method of Runge-Kutta. See C. P. Malta and M. L. S. Teles, Int. J. Appl. Math. 3, 379 (2000).
33. J. C. Bastos de Figueiredo, Ph.D. thesis, IFUSP, 2000.
34. F. Takens, Lect. Notes Math. 898, 266 (1986).
35. M. J. Feigenbaum, J. Stat. Phys. 19, 25 (1978).
36. P. Grassberger and I. Procaccia, Physica D 9, 189 (1983); Phys. Rev. Lett. 50, 346 (1983).
37. P. Grassberger and I. Procaccia, Physica D 9, 189 (1983); Phys. Rev. Lett. 50, 346 (1983).
38. T. Sauer, J. Yorke, and M. Casdagli, J. Stat. Phys. 65, 579 (1991).
39. J. Theiler, Phys. Rev. A 34, 2427 (1986).
40. H. Kantz and T. Schreiber, Chaos 5, 143 (1995).
41. P. Grassberger, Phys. Lett. A 128, 369 (1988).
42. P. Eckmann and D. Ruelle, Rev. Mod. Phys. 57, 617 (1985).
43. R. Stoop and J. Parisi, Physica D 50, 89 (1991).
44. H. Kantz, Phys. Lett. A 185, 77 (1994).
45. M. T. Rosenstein, J. J. Collins, and C. J. De Luca, Physica D 65, 117 (1993).