Logistic map neural modelling: A theoretical foundation

Abstract

The aim of this paper is to establish a theoretical framework for the modelling and simulation of chaotic attractors using neural networks. The attractor paradigm in this paper is the logistic map, which is modelled via neural networks in the convergence, periodic and chaotic regions. It is proved that, under certain conditions, the function simulated by the neural model is actually the logistic map with a different value of the λ parameter from the theoretical value. A two-dimensional system is defined and studied, facilitating the generation of the theoretical time series and the associated simulation error. The fixed points of periods p = 1 and p = 2 are identified and studied with respect to their stability. For higher period values, a theorem concerning the periodicity of the simulation error is postulated and proved. The minimum simulation error value is calculated using analytical methods, and the chaotic nature of the system with respect to Lyapunov exponents is described. Conclusions are discussed with respect to the experimental results obtained by the simulation models.

Keywords:

• Chaotic attractor
• Neural network
• Logistic map