Bifurcations of fuzzy nonlinear dynamical systems

Abstract We present some recent developments of the fuzzy generalized cell mapping method (FGCM) in this paper. The topological property of the FGCM and its finite convergence of membership distribution vector are discussed. Powerful algorithms of digraphs are adopted for the analysis of topological properties of the FGCM systems. Bifurcations of fuzzy nonlinear dynamical systems are studied by using the FGCM method. A backward algorithm is introduced to study the unstable equilibrium solutions and their bifurcation. We have found that near the deterministic bifurcation point, the fuzzy system undergoes a complex transition as the control parameter varies. In this transition region, the steady state membership distribution is dependent on the initial condition. If we use the measure and topology of the α-cut (α = 1) of the steady state membership function of the persistent group representing the stable fuzzy equilibrium solution to characterize the fuzzy bifurcation, assuming the uniform initial condition within the persistent group, the bifurcation of the fuzzy dynamical system is then completed within an interval of the control parameter, rather than at a point as is the case of deterministic systems.

[1]  K. Takatsuka,et al.  Strange attractors of infinitesimal widths in the bifurcation diagram with an unusual mechanism of onset , 1998 .

[2]  Jian-Qiao Sun,et al.  Global analysis of nonlinear dynamical systems with fuzzy uncertainties by the cell mapping method , 1990 .

[3]  C. Hsu.  GLOBAL ANALYSIS OF DYNAMICAL SYSTEMS USING POSETS AND DIGRAPHS , 1995 .

[4]  C. Meunier,et al.  Noise and bifurcations , 1988 .

[5]  Ueda,et al.  Safe, explosive, and dangerous bifurcations in dissipative dynamical systems. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[6]  Javier Aracil,et al.  Local and global bifurcations in simple Takagi-Sugeno fuzzy systems , 2001, IEEE Trans. Fuzzy Syst..

[7]  D. Edwards,et al.  Use of fuzzy logic to calculate the statistical properties of strange attractors in chaotic systems , 1997, Fuzzy Sets Syst..

[8]  Shinji Doi,et al.  Spectral Analysis of Stochastic Phase Lockings and Stochastic Bifurcations in the Sinusoidally Forced van der Pol Oscillator with Additive Noise , 1998 .

[9]  C. G. Rizk,et al.  A fuzzy logic controller design methodology for 4D systems with optimal global performance using enhanced cell state space based best estimate directed search method , 1999, IEEE SMC'99 Conference Proceedings. 1999 IEEE International Conference on Systems, Man, and Cybernetics (Cat. No.99CH37028).

[10]  C. Hsu,et al.  The Generalized Cell Mapping Method in Nonlinear Random Vibration Based Upon Short-Time Gaussian Approximation , 1990 .

[11]  Haiwu Rong,et al.  Stochastic bifurcation in duffing system subject to harmonic excitation and in presence of random noise , 2004 .

[12]  Ling Hong,et al.  Crises and chaotic transients studied by the generalized cell mapping digraph method , 1999 .

[13]  Luis G. Crespo,et al.  Stochastic Optimal Control via Bellman’s Principle , 2004 .

[14]  Ling Hong,et al.  Discontinuous bifurcations of Chaotic attractors in forced oscillators by generalized Cell Mapping digraph (GCMD) Method , 2001, Int. J. Bifurc. Chaos.

[15]  P. K. Satpathy,et al.  A fuzzy approach to handle parameter uncertainties in Hopf bifurcation analysis of electric power systems , 2004 .