Identification of the Hénon chaotic map by fuzzy modeling and Nelder–Mead simplex method

A nonlinear identification approach for describing the dynamical behavior of a Hénon chaotic map based on Nelder–Mead simplex method and Takagi–Sugeno (T–S) fuzzy model is proposed in this paper. Nonlinear dynamic systems exhibiting chaotic behavior arise in real world problems in many areas of science and technology. The investigation of such systems becomes further attractive since they are simple systems able to imitate the behavior of complex systems. An alternative to approximate a nonlinear system is to employ fuzzy models since they are universal approximators able to adequately approximate any continuous functions to an arbitrary precision. Due to that it became a powerful tool for the nonlinear identification and control. Another advantage of using T–S fuzzy modeling is its characteristic of representing a highly nonlinear functional relation with a small number of rules. Nelder–Mead simplex method is proposed here as the method for optimizing the premise part while least mean squares technique is employed for consequent part of production rules of a T–S fuzzy model. Numerical results indicate that the description of discrete chaotic dynamics of Hénon map can be accomplished by exploring the effectiveness of NMO approach combined with T–S fuzzy modeling in constructing an appropriate nonlinear identification.