$ show JuliaDynamics/ChaosTools.jl
ChaosTools.jl is a Julia library providing tools for analyzing nonlinear dynamics and chaotic behavior. It offers functionality for calculating Lyapunov exponents, detecting chaos via methods like GALI and the 0-1 test, analyzing predictability, finding fixed points and periodic orbits, estimating signal periods, analyzing rare events through return time statistics, and performing dimensionality reduction. The package is designed to work with DynamicalSystems.jl and can be used as a standalone package or as part of the broader DynamicalSystems ecosystem. The architecture follows a modular organization with functions grouped by topic (chaos detection, dimensionality reduction, periodicity, etc.), leveraging Julia's multiple dispatch and type system for clean, reusable code.
The LLM identified 5 files worth highlighting for reuse:
Implements the Generalized Alignment Index (GALI) algorithm for efficient chaos detection. The function computes GALI_k by tracking the product of singular values of deviation vectors in a tangent dynamical system. It's self-contained, depending only on LinearAlgebra and the TangentDynamicalSystem abstraction, making it easily reusable in other projects needing chaos detection capabilities.
Implements the Benettin algorithm for calculating the maximum Lyapunov exponent. The core algorithm evolves two neighboring trajectories with periodic rescaling to compute the average exponential divergence rate. It uses a ParallelDynamicalSystem for efficiency and contains well-encapsulated rescaling logic that could be extracted for use in other numerical analysis projects requiring Lyapunov exponent calculations.
Computes the full spectrum of Lyapunov exponents using QR-decomposition on the parallelepiped defined by deviation vectors. Features an optimized buffered QR decomposition implementation for performance, automatically handles both in-place and out-of-place systems, and cleanly separates the low-level algorithm from the high-level interface. The component demonstrates strong numerical engineering practices.
Implements Broomhead-King coordinates as an improvement over standard delay coordinates for dimensionality reduction. Performs SVD on a trajectory matrix constructed from the time series with optimal delay. The function is remarkably self-contained - taking only an abstract vector and dimension, returning SVD components with zero external dependencies beyond LinearAlgebra. This makes it highly portable to other projects needing preprocessing for dynamical systems analysis.
Provides sophisticated analysis of return times for continuous dynamical systems. Implements both linear intersection and accurate interpolation methods (using Optim.jl and Roots.jl) for detecting crossing times with sets in state space. The code cleanly separates concerns with distinct functions for trajectory stepping, crossing detection, and exit/entry time updating, demonstrating excellent modular design for a complex numerical algorithm.