Development of a mathematical method for predicting the states of chaotic dynamical systems

Abstract

This paper presents a mathematical method for predicting the states of chaotic dynamic systems, taking into account local non-linearity, the spectral properties of the transition operator, and the limited predictability of chaotic processes. The relevance of the research stems from the high sensitivity of chaotic systems to initial conditions, as a result of which minor errors in estimating the current state cause discrepancies between the predicted and actual trajectories as the prediction horizon increases. The proposed method involves reconstructing the multidimensional state space along a time sequence, defining a local attractor region and constructing a quadratic nonlinear evolution operator. The scientific novelty lies in the combination of local state prediction, analysis of tangential dynamics and estimation of prediction uncertainty within a single mathematical procedure. The Jacobian matrix was used for the local characterisation of the transition, whilst the accumulated finite-time deformation was estimated using the singular spectrum of the resulting transition operator. The propagation of forecast uncertainty was carried out taking into account the quadratic curvature of the local operator. The permissible forecast horizon was determined adaptively based on a combined condition of not exceeding the permissible size of the uncertainty region and the limit intensity of local divergence of trajectories. To verify the effectiveness of the method, a simulation of the Lorenz system was carried out in Python. The results showed that, over the initial interval, the predicted trajectory remained close to the actual one, whilst a systematic increase in error was observed as the forecast horizon increased. It was established that the proposed approach allows not only for the generation of a forecast of the future state, but also for the determination of the time domain within which the forecast remains practically informative for the system under study. Keywords: chaotic dynamical system, state prediction, nonlinear operator, state-space reconstruction, local instability, spectral analysis, forecast horizon, simulation modelling.