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Response analysis of fuzzy nonlinear dynamical systems

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Abstract

The transient and steady-state membership distribution functions (MDFs) of fuzzy response of a Duffing–Van der Pol oscillator with fuzzy uncertainty are studied by means of the fuzzy generalized cell mapping (FGCM) method. A rigorous mathematical foundation of the FGCM is established with a discrete representation of the fuzzy master equation for the possibility transition of continuous fuzzy processes. Fuzzy response is characterized by its topology in the state space and its possibility measure of MDFs. The evolutionary orientation of MDFs is in accordance with invariant manifolds toward invariant sets. In the evolutionary process of a steady-state fuzzy response with an increase of the intensity of fuzzy noise, a merging bifurcation is observed in a sudden change of MDFs from two sharp peaks of maximum possibility to one peak band around unstable manifolds.

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References

  1. Moss, F., McClintock, P.V.E.: Noise in Nonlinear Dynamical Systems. Cambridge University Press, Cambridge (2007)

    Google Scholar 

  2. Klir, G.J., Folger, T.A.: Fuzzy Sets, Uncertainty, and Information. Prentice-Hall, Englewood Cliffs (1988)

    MATH  Google Scholar 

  3. Bucolo, M., Fazzino, S., Rosa, M.L., Fortuna, L.: Small-world networks of fuzzy chaotic oscillators. Chaos Solitons Fractals 17, 557–565 (2003)

    Article  MATH  Google Scholar 

  4. Gao, J.B., Hwang, S.K., Liu, J.M.: When can noise induce chaos? Phys. Rev. Lett. 82(6), 1132–1135 (1999)

    Article  Google Scholar 

  5. Kraut, S., Feudel, U.: Multistability, noise, and attractor hopping: the crucial role of chaotic saddles. Phys. Rev. E 66(1), 015207 (2002)

    Article  MathSciNet  Google Scholar 

  6. Santitissadeekorn, N., Bollt, E.M.: Identifying stochastic basin hopping by partitioning with graph modularity. Phys. D 231(2), 95–107 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Zaks, M.A., Sailer, X., Schimansky-Geier, L., Neiman, A.B.: Noise induced complexity: from subthreshold oscillations to spiking in coupled excitable systems. Chaos 15(2), 26117 (2005)

    Article  MathSciNet  Google Scholar 

  8. Hong, L., Sun, J.Q.: Bifurcations of fuzzy nonlinear dynamical systems. Commun. Nonlinear Sci. Numer. Simul. 11(1), 1–12 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Xu, W., He, Q., Fang, T., Rong, H.: Stochastic bifurcation in duffing system subject to harmonic excitation and in presence of random noise. Int. J. Non-Linear Mech. 39, 1473–1479 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Sun, J.Q.: Stochastic Dynamics and Control. Elsevier, Amsterdam (2006)

    Google Scholar 

  11. Friedman, Y., Sandler, U.: Fuzzy dynamics as an alternative to statistical mechanics. Fuzzy Sets Syst. 106, 61–74 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  12. Friedman, Y., Sandler, U.: Evolution of systems under fuzzy dynamic laws. Fuzzy Sets Syst. 84, 61–74 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  13. Buckley, J.J., Feuring, T.: Fuzzy differential equations. Fuzzy Sets Syst. 110, 43–54 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  14. Park, J.Y., Han, H.K.: Fuzzy differential equations. Fuzzy Sets Syst. 110, 69–77 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Yoshida, Y.: A continuous-time dynamic fuzzy system. (I) a limit theorem. Fuzzy Sets Syst. 113, 453–460 (2000)

    Article  MATH  Google Scholar 

  16. Buckley, J.J., Hayashi, Y.: Applications of fuzzy chaos to fuzzy simulation. Fuzzy Sets Syst. 99, 151–157 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lin, Y.K., Cai, G.Q.: Probabilistic Structural Dynamics: Advanced Theory and Applications. McGraw-Hill, New York (1995)

    Google Scholar 

  18. Nazaroff, G.J.: Fuzzy topological polysystems. J. Math. Anal. Appl. 41, 478–485 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kloeden, P.E.: Fuzzy dynamical systems. Fuzzy Sets Syst. 7, 275–296 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  20. Aubin, J.-P.: Fuzzy differential inclusions. Probl. Control Inf. Theory 19, 55–67 (1990)

    MathSciNet  MATH  Google Scholar 

  21. Kaleva, O.: The Cauchy problem for fuzzy differential equations. Fuzzy Sets Syst. 35, 389–396 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ma, M., Friedman, M., Kandel, A.: Numerical solutions of fuzzy differential equations. Fuzzy Sets Syst. 105, 133–138 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhang, Y., Qiao, Z., Wang, G.: Solving processes for a system of first-order fuzzy differential equations. Fuzzy Sets Syst. 95, 333–347 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  24. Chen, Y.-Y., Tsao, T.-C.: New approach for the global analysis of fuzzy dynamical systems. In Proceedings of the 27th IEEE Conference on Decision and Control, Austin, TX, USA, pp. 1415–1420 (1988)

  25. Chen, Y.Y., Tsao, T.C.: Description of the dynamical behavior of fuzzy systems. IEEE Trans. Syst. Man Cybern. 19(4), 745–755 (1989)

    Article  MathSciNet  Google Scholar 

  26. Smith, S.M., Comer, D.J.: Self-tuning of a fuzzy logic controller using a cell state space algorithm. In: Proceedings of the IEEE International Conference on Systems, Man and Cybernetics vol. 6, pp. 445–450 (1999)

  27. Song, F., Smith, S.M., Rizk, C.G.: Optimized fuzzy logic controller design for 4D systems using cell state space technique with reducedmapping error. In: Proceedings of the IEEE International Fuzzy Systems Conference, Seoul, South Korea, vol. 2, pp. 691–696 (1999)

  28. Hsu, C.S.: Cell-to-Cell Mapping: A Method of Global Analysis for Non-linear Systems. Springer-Verlag, New York (1987)

    Book  Google Scholar 

  29. Edwards, D., Choi, H.T.: Use of fuzzy logic to calculate the statistical properties of strange attractors in chaotic systems. Fuzzy Sets Syst. 88(2), 205–217 (1997)

    Article  MathSciNet  Google Scholar 

  30. Sun, J.Q., Hsu, C.S.: Global analysis of nonlinear dynamical systems with fuzzy uncertainties by the cell mapping method. Comput. Methods Appl. Mech. Eng. 83(2), 109–120 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  31. Sun, J.Q., Hsu, C.S.: The generalized cell mapping method in nonlinear random vibration based upon short-time Gaussian approximation. J. Appl. Mech. 57, 1018–1025 (1990)

    Article  MathSciNet  Google Scholar 

  32. Crespo, L.G., Sun, J.Q.: Stochastic optimal control of nonlinear systems via short-time Gaussian approximation and cell mapping. Nonlinear Dyn. 28, 323–342 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  33. Crespo, L.G., Sun, J.Q.: Solution of fixed final state optimal control problems via simple cell mapping. Nonlinear Dyn. 23(4), 391–403 (2000)

    Article  MATH  Google Scholar 

  34. Hong, L., Xu, J.: Crises and chaotic transients studied by the generalized cell mapping digraph method. Phys. Lett. A 262, 361–375 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  35. Hong, L., Xu, J.: Chaotic saddles in Wada basin boundaries and their bifurcations by the generalized cell-mapping digraph (GCMD) method. Nonlinear Dyn. 32(4), 371–385 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  36. Hong, L., Sun, J.Q.: Codimension two bifurcations of nonlinear systems driven by fuzzy noise. Phys. D 213, 181–189 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  37. Hong, L., Sun, J.Q.: A fuzzy blue sky catastrophe. Nonlinear Dyn. 55(3), 261–267 (2009)

    Article  MATH  Google Scholar 

  38. Guckenheimer, J., Holmes, P.J.: Nonlinear Oscillations. Dynamical Systems and Bifurcations of Vector Fields. Springer-Verlag, New York (1983)

    Book  MATH  Google Scholar 

  39. Jiang, J., Xu, J.X.: A method of point mapping under cell reference for global analysis of nonlinear dynamical systems. Phys. Lett. A 188, 137–145 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  40. Jiang, J., Xu, J.X.: An iterative method of point mapping under cell reference for the global analysis: theory and a multiscale reference technique. Nonlinear Dyn. 15(2), 103–114 (1998)

    Article  MATH  Google Scholar 

Download references

Acknowledgments

This work was supported by the Natural Science Foundation of China through the Grants 11332008 and 11172224.

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Authors and Affiliations

  1. State Key Lab for Strength and Vibration, Xi’an Jiaotong University, Xi’an, 710049, China

    Ling Hong & Jun Jiang

  2. School of Engineering, University of California at Merced, Merced, CA, 95344, USA

    Jian-Qiao Sun

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  1. Ling Hong
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  2. Jun Jiang
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  3. Jian-Qiao Sun
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Correspondence to Ling Hong.

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Hong, L., Jiang, J. & Sun, JQ. Response analysis of fuzzy nonlinear dynamical systems. Nonlinear Dyn 78, 1221–1232 (2014). https://doi.org/10.1007/s11071-014-1509-8

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  • DOI: https://doi.org/10.1007/s11071-014-1509-8

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