An adaptive fuzzy system for modeling chaos

doi:10.1016/0020-0255(94)90097-3

Information Sciences

Volume 81, Issues 3–4, November 1994, Pages 193-212

https://doi.org/10.1016/0020-0255(94)90097-3 Get rights and content

Abstract

Most chaotic systems do not have even dynamical behavior over the whole phase space. In some regions, the system stretches and branches more violently than others. In the regions where the dynamics are violent, finer representation must be given. In this paper, we make use of this property to model chaos. We present an adaptive system based on fuzzy logic. It can refine its representation of a region in the phase space if that region requires it. It does so by adaptively generating more fuzzy rules to model a region only if that region has very violent dynamics. Experiments were performed to test the adaptive fuzzy system for capturing the dynamics of a normal dynamical system (the Van der Pol oscillator) as well as two chaotic systems (the Lorenz and Rossler attractors). Results indicate that the fuzzy system can produce an accurate model of the three dynamical systems. The adaptive rule generation algorithm allowed the fuzzy system to have an optimal number of rules.

References (18)

O.E. Rossler
An equation for continuous chaos
Phys. Rev. Lett.
(1976)
D. Filev
Fuzzy modelling of complex systems
Int. J. Approx. Reasoning
(1991)
A. Lapedes et al.
Nonlinear signal processing using neural networks: Prediction and system modelling
M. Sato
A real time learning algorithm for recurrent analog neural networks
Biol. Cybernet.
(1990)
M. Sato
A learning algorithm to teach spatiotemporal patterns to recurrent neural networks
Biol. Cybernet.
(1990)
Y. Murakami and M. Sato, A recurrent network, which learns chaotic dynamics, Proceedings of the Third Australian...
B.A. Pearlmutter
Dynamic recurrent neural networks
R.J. Williams et al.
A learning algorithm for continually running fully recurrent neural networks
Neural Comput.
(1989)
B.A. Pearlmutter
Learning state space trajectories in recurrent neural networks
Neural Comput.
(1989)

There are more references available in the full text version of this article.

Cited by (14)

On fuzzifications of discrete dynamical systems
2011, Information Sciences
Let X denote a locally compact metric space and φ : X → X be a continuous map. In the 1970s Zadeh presented an extension principle helping us to fuzzify the dynamical system (X, φ), i.e., to obtain a map Φ for the space of fuzzy sets on X. We extend an idea mentioned in [P. Diamond, A. Pokrovskii, Chaos, entropy and a generalized extension principle, Fuzzy Sets Syst. 61 (1994) 277–283] to generalize Zadeh’s original extension principle.
In this paper we study basic properties of so-called g-fuzzifications, such as their continuity properties. We also show that, for any g-fuzzification: (i) a uniformly convergent sequence of uniformly continuous maps on X induces a uniformly convergent sequence of fuzzifications on the space of fuzzy sets and (ii) a conjugacy (resp., a semi-conjugacy) between two discrete dynamical systems can be extended to a conjugacy (resp., a semi-conjugacy) between fuzzified dynamical systems.
Throughout this paper we consider different topological structures in the space of fuzzy sets, namely, the sendograph, endograph and levelwise topologies.
A case-based approach for modeling nonlinear systems
1996, Fuzzy Sets and Systems
We propose a case-based approach for modeling nonlinear systems. Our method is a model-free approach which does not depend on any specific model structure. Given an input to a system, our method predicts the output by interpolating the outputs in the “similar” cases whose inputs are close to the given input. The similarity degree and coefficients used in interpolation are determined from the data in a context-dependent manner. We also propose a method to extract cases from a database to meet the specification on the prediction accuracy. Finally, applications to nonlinear dynamical systems are discussed.
A chaotic iterative fuzzy modeling for circulating a simple sentence
2021, Scientia Iranica
Series-NonUniform Rational B-Spline (S-NURBS) model: A geometrical interpolation framework for chaotic data
2013, Chaos
Fuzzy chaotic logistic maps
2012, World Automation Congress Proceedings
Modeling nonlinear dynamics and chaos: A review
2009, Mathematical Problems in Engineering

View all citing articles on Scopus

View full text

An adaptive fuzzy system for modeling chaos

Abstract

Phys. Rev. Lett.

Int. J. Approx. Reasoning

Nonlinear signal processing using neural networks: Prediction and system modelling

A real time learning algorithm for recurrent analog neural networks

Biol. Cybernet.

A learning algorithm to teach spatiotemporal patterns to recurrent neural networks

Biol. Cybernet.

Dynamic recurrent neural networks

A learning algorithm for continually running fully recurrent neural networks

Neural Comput.

Learning state space trajectories in recurrent neural networks

Neural Comput.