On numerical study of attractors of ODEs

TL;DR

This paper proposes a numerical method to study attractors of dissipative ODEs by simulating multiple trajectories to identify regions of high density, addressing issues caused by numerical errors.

Contribution

It introduces a new approach for modeling Milnor's attractor using trajectory bundles and density estimation, improving qualitative analysis of dynamical systems.

Findings

Demonstrated method on 2D and 3D examples

Successfully identified attractor regions with high trajectory density

Addressed discrepancies caused by numerical errors in simulations

Abstract

Due to errors of numerical methods, qualitative behavior of the numerically simulated solution may be completely different from qualitative behavior of the real trajectory. Therefor in study of attractors the more useful idea is to simulate a bundle of trajectories for a (relatively) short time (see Dellnitz etc.) We suggest a method of modelling Milnor's attractor of dissipative dynamical system. It is based on numerical simulation of bundle of trajectories, emanating from some set, and locating regions with high density of trajectories. We present 2D and 3D examples of application of our method.