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.

Neural Network for Low-Memory IoT Devices and MNIST Image Recognition Using Kernels Based on Logistic Map

“…The current study presents a new architecture for a neural network, where complex dynamics are simulated by the application of logistic mapping to the multilayer weights of a feedforward network. Previously, logistic mapping was used in neural networks as an activation function [32,33] and as a model object [34,35]. The application of logistic mapping significantly reduces the amount of memory used by the network without losing functionality.…”