# Numerical analysis of dynamical systems: unstable periodic orbits, hidden transient chaotic sets, hidden attractors, and finite-time Lyapunov dimension

## TL;DR

This paper discusses the challenges of numerically analyzing chaotic dynamical systems, focusing on hidden attractors, transient chaos, and finite-time Lyapunov dimensions using classic models like Lorenz, Rossler, and Vallis.

## Contribution

It introduces methods for detecting hidden attractors and accurately computing finite-time Lyapunov exponents and dimensions in low-order chaotic models.

## Key findings

- Identification of hidden chaotic attractors in Lorenz system.
- Demonstration of finite-time Lyapunov dimension computation.
- Analytical localization of attractors in Vallis system.

## Abstract

In this article, on the example of the known low-order dynamical models, namely Lorenz, Rossler and Vallis systems, the difficulties of reliable numerical analysis of chaotic dynamical systems are discussed. For the Lorenz system, the problems of existence of hidden chaotic attractors and hidden transient chaotic sets and their numerical investigation are considered. The problems of the numerical characterization of a chaotic attractor by calculating finite-time time Lyapunov exponents and finite-time Lyapunov dimension along one trajectory are demonstrated using the example of computing unstable periodic orbits in the Rossler system. Using the example of the Vallis system describing the El Nino-Southern Oscillation it is demonstrated an analytical approach for localization of self-excited and hidden attractors, which allows to obtain the exact formulas or estimates of their Lyapunov dimensions.