Fuzzy Cell Mapping

Chapter

Abstract

Bifurcations of a Duffing oscillator in the presence of fuzzy noise are studied by means of the fuzzy generalized cell mapping (FGCM) method. The FGCM method is first introduced. Two categories of bifurcations are investigated, namely catastrophic and explosive bifurcations. Fuzzy bifurcations are characterized by topological changes of the attractors of the system, represented by the persistent groups of cells in the context of the FGCM method, and by changes of the steady state membership distribution. The fuzzy noise-induced bifurcations studied herein are not commonly seen in the deterministic systems, and can be well described by the FGCM method. Furthermore, two conjectures are proposed regarding the condition under which the steady state membership distribution of a fuzzy attractor is invariant or not.

Keywords

Explosive 

Notes

Acknowledgements

This material is based upon work supported by the National Science Foundation under Grant Nos. CMS-0219217 and INT-0217453, and by the Natural Science Foundation of China under Grant Nos. 10772140 and 11172224.

References

  1. Abraham RH, Stewart HB (1986) A chaotic blue sky catastrophe in forced relaxation oscillations. Physica D 19:394–400MathSciNetCrossRefGoogle Scholar
  2. Bucolo M, Fazzino S, Rosa ML, Fortuna L (2003) Small-world networks of fuzzy chaotic oscillators. Chaos Soliton Fract 17:557–565MATHCrossRefGoogle Scholar
  3. Crespo LG, Sun JQ (2003) Stochastic optimal control of nonlinear dynamic systems via bellman’s principle and cell mapping. Automatica 39(12):2109–2114MathSciNetMATHCrossRefGoogle Scholar
  4. Cuesta F, Ponce E, Aracil J (2001) Local and global bifurcations in simple Takagi–Sugeno fuzzy systems. IEEE Trans Fuzzy Syst 9(2):355–368CrossRefGoogle Scholar
  5. Doi S, Inoue J, Kumagai S (1998) Spectral analysis of stochastic phase lockings and stochastic bifurcations in the sinusoidally forced van der Pol oscillator with additive noise. J Stat Phys 90(5–6):1107–1127MathSciNetMATHCrossRefGoogle Scholar
  6. Edwards D, Choi HT (1997) Use of fuzzy logic to calculate the statistical properties of strange attractors in chaotic systems. Fuzzy Sets Syst 88(2):205–217MathSciNetCrossRefGoogle Scholar
  7. Friedman Y, Sandler U (1996) Evolution of systems under fuzzy dynamic laws. Fuzzy Sets Syst 84:61–74MathSciNetMATHCrossRefGoogle Scholar
  8. Friedman Y, Sandler U (1999) Fuzzy dynamics as an altemative to statistical mechanics. Fuzzy Sets Syst 106:61–74MathSciNetMATHCrossRefGoogle Scholar
  9. Grebogi C, Ott E, Yorke JA (1983) Crises, sudden changes in chaotic attractors, and transient chaos. Physica D 7:181–200MathSciNetCrossRefGoogle Scholar
  10. Grebogi C, Ott E, Yorke JA (1986) Critical exponents of chaotic transients in nonlinear dynamical systems. Phys Rev Lett 57:1284–1287MathSciNetCrossRefGoogle Scholar
  11. Guckenheimer J, Holmes PJ (1983) Nonlinear oscillations, dynamical systems and bifurcations of vector fields. Springer, New YorkMATHGoogle Scholar
  12. Holmes P, Rand D (1980) Phase portraits and bifurcations of the non-linear oscillator. Int J Nonlinear Mech 15:449–458MATHCrossRefGoogle Scholar
  13. Hong L, Sun JQ (2006) Bifurcations of fuzzy nonlinear dynamical systems. Commun Nonlinear Sci Numer Simul 11(1):1–12MathSciNetMATHCrossRefGoogle Scholar
  14. Hong L, Xu JX (1999) Crises and chaotic transients studied by the generalized cell mapping digraph method. Phys Lett A 262:361–375MathSciNetMATHCrossRefGoogle Scholar
  15. Hong L, Xu JX (2001) Discontinuous bifurcations of chaotic attractors in forced oscillators by generalized cell mapping digraph (GCMD) method. Int J Bifurcat Chaos 11:723–736MathSciNetMATHCrossRefGoogle Scholar
  16. Hsu CS (1987) Cell-to-cell mapping: a method of global analysis for non-linear systems. Springer, New YorkGoogle Scholar
  17. Hsu CS (1995) Global analysis of dynamical systems using posets and digraphs. Int J Bifurcat Chaos 5(4):1085–1118MATHCrossRefGoogle Scholar
  18. Hüllermeier E (1997) An approach to modelling and simulation of uncertain dynamical systems. Int J Uncertain Fuzz 5(2):117–137Google Scholar
  19. Hüllermeier E (1999) Numerical methods for fuzzy initial value problems. Int J Uncertain Fuzz 7(5):439–461MATHCrossRefGoogle Scholar
  20. Klir GJ, Folger TA (1988) Fuzzy sets, uncertainty, and information. Prentice-Hall, Englewood Cliffs, NJMATHGoogle Scholar
  21. Meunier C, Verga AD (1988) Noise and bifurcations. J Stat Phys 50(1/2):345–375MathSciNetMATHCrossRefGoogle Scholar
  22. Moss F, McClintock PVE (1989) Noise in nonlinear dynamical systems. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  23. Namachchivaya NS (1990) Stochastic bifurcation. J Appl Math Comput 38:101–159MATHCrossRefGoogle Scholar
  24. Namachchivaya NS (1991) Co-dimension two bifurcations in the presence of noise. J Appl Mech 58:259–265MATHCrossRefGoogle Scholar
  25. Risken H (1996) The Fokker–Planck equation. Springer, New YorkMATHGoogle Scholar
  26. Sandler U, Tsitolovsky L (2001) Fuzzy dynamics of brain activity. Fuzzy Sets Syst 121:237–245MathSciNetMATHCrossRefGoogle Scholar
  27. Satpathy PK, Das D, Gupta PBD (2004) A fuzzy approach to handle parameter uncertainties in Hopf bifurcation analysis of electric power systems. Int J Electr Power Energy Syst 26(7):531–538CrossRefGoogle Scholar
  28. Schenk-Hoppe KR (1996) Bifurcation scenarios of the noisy Duffing–van der Pol oscillator. Nonlinear Dynam 11:255–274MathSciNetCrossRefGoogle Scholar
  29. Song F, Smith SM, Rizk CG (1999) Fuzzy logic controller design methodology for 4d systems with optimal global performance using enhanced cell state space based best estimate directed search method. In: Proceedings of the IEEE international conference on systems, man, and cybernetics, Tokyo, Japan, vol 6, pp 138–143Google Scholar
  30. Sun JQ, Hsu CS (1990a) The generalized cell mapping method in nonlinear random vibration based upon short-time Gaussian approximation. J Appl Mech 57:1018–1025MathSciNetCrossRefGoogle Scholar
  31. Sun JQ, Hsu CS (1990b) Global analysis of nonlinear dynamical systems with fuzzy uncertainties by the cell mapping method. Comput Meth Appl Mech Eng 83(2):109–120MathSciNetMATHCrossRefGoogle Scholar
  32. Thompson JMT, Stewart HB (1986) Nonlinear dynamics and chaos. Wiley, Chichester, New YorkMATHGoogle Scholar
  33. Thompson JMT, Stewart HB, Ueda Y (1994) Safe, explosive, and dangerous bifurcations in dissipative dynamical systems. Phys Rev E 49(2):1019–1027MathSciNetCrossRefGoogle Scholar
  34. Tomonaga Y, Takatsuka K (1998) Strange attractors of infinitesimal widths in the bifurcation diagram with an unusual mechanism of onset. Nonlinear dynamics in coupled fuzzy control systems. II. Physica D 111(1–4):51–80MathSciNetMATHGoogle Scholar
  35. Ueda Y (1980) Steady motions exhibited by duffing’s equation: a picture book of regular and chaotic motions. In: Proceedings of new approaches to nonlinear problems in dynamics, Philadelphia, PA, pp 311–322Google Scholar
  36. Xu W, He Q, Fang T, Rong H (2003) Global analysis of stochastic bifurcation in Duffing system. Int J Bifurcat Chaos 13(10):3115–3123MATHCrossRefGoogle Scholar
  37. Yoshida Y (2000) A continuous-time dynamic fuzzy system. (I) A limit theorem. Fuzzy Sets Syst 113:453–460MATHCrossRefGoogle Scholar
  38. Zeeman EC (1982) Bifurcation and catastrophe theory. In: Proceedings of papers in algebra, analysis and statistics, Providence, Rhode Island, pp 207–272Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.School of EngineeringUniversity of CaliforniaMercedUSA
  2. 2.State Key Lab for Strength and VibrationXi’an Jiaotong UniversityXi’anChina

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