On calculating the dimension of invariant sets of dynamic systems

This work investigates numerical approaches for estimating the dimension of invariant sets onto which the trajectories of dynamic systems ``wind'', with a focus on fractal and correlation dimensions. While the classical fractal dimension becomes computationally challenging in spaces of dimension greater than two, the correlation dimension offers a more efficient and scalable alternative. We develop and implement a computational method for evaluating the correlation dimension of large discrete point sets generated by numerical integration of differential equations. An analogy is noted between this approach and the Richardson--Kalitkin method for estimating the error of a numerical method. The method is tested on two representative systems: a conservative system whose orbit lies on a two-dimensional torus, and the Lorenz system, a canonical example of a chaotic flow with a non-integer attractor dimension. In both cases, the estimated correlation dimensions agree with theoretical predictions and previously reported results. The developed software provides an effective tool for analysing invariant manifolds of dynamical systems and is suitable for further studies, including those involving reversible difference schemes and high-dimensional systems.