A Logistic Growth Model with Optimization Matching Scheme for Chaotic Systems

Abstract

A chaotic system is a deterministic nonlinear dynamic system that expresses unpredictable and random output. Furthermore, a chaotic system is sensitive to initial conditions. Even a slight change in input can produce significantly different outcomes. This report describes a logistic growth model based on an optimization matching scheme to predict the solution of chaotic systems. Similarities and contradictions between chaotic systems and nonlinear systems are observed. Then, a loss function, which measures the differences between the chaotic system and the logistic growth model, is defined. Using a gradient method, the parameter in the logistic growth model is updated iteratively until convergence is achieved. Therefore, the gradient method minimizes the differences in terms of mean square errors. With the optimal parameter, the solution of the logistic growth model approximates the solution of chaotic systems. For illustration, the one-dimension two-parameter sin-cos (1DTPSC) system in the encryption, two-dimension Van der Pol oscillator and three-dimension Chua’s circuits are studied. These three systems present chaotic behaviour for certain initial conditions and model parameters. The simulation results show the accuracy of the logistic growth model based on the optimization matching scheme, where their chaotic solutions are well predicted. In conclusion, the efficiency of the logistic growth model with an optimization matching scheme for handling chaotic systems is highly demonstrated.