Kneadings, Symbolic Dynamics and Painting Lorenz Chaos. A Tutorial

TL;DR

This paper introduces a new computational method using kneading invariants for analyzing chaos in Lorenz systems, revealing complex structures and organizing centers across various physical models.

Contribution

It proposes and verifies a symbolic technique based on kneading invariants for exploring chaos and bifurcations in Lorenz attractors across different scientific fields.

Findings

Uncovered complex bi-parametric structures in Lorenz systems.

Detected organizing centers like T-points and saddles.

Validated the method on models from hydrodynamics, mathematics, and optics.

Abstract

A new computational technique based on the symbolic description utilizing kneading invariants is proposed and verified for explorations of dynamical and parametric chaos in a few exemplary systems with the Lorenz attractor. The technique allows for uncovering the stunning complexity and universality of bi-parametric structures and detect their organizing centers - codimension-two T-points and separating saddles in the kneading-based scans of the iconic Lorenz equation from hydrodynamics, a normal model from mathematics, and a laser model from nonlinear optics.