Contents

The module DynamicalSystems re-exports all following functionality.

Core types

• Intuitive, consistent APIs for the definition of general dynamical systems under a unified struct DynamicalSystem. The following combinations are possible:
  • Continuous or Discrete systems. Continuous systems use DifferentialEquations.jl for solving the ODE problem.
  • In-place or out-of-place (large versus small systems).
  • Auto-differentiated or not (for the Jacobian function).

• Automatic "completion" of the dynamics of the system with numerically computed Jacobians, in case they are not provided by the user.
• Robust implementations of all kinds of integrators, that evolve the system, many states of the system, or even deviation vectors. See the Advanced documentation for this.
• Library of Predefined Dynamical Systems that have been used extensively in scientific research.
• Unified & dedicated interface for numerical data: Dataset.

Entropies

• An interface to estimate Entropies & Probabilities from trajectories or state space sets.
• Dozens of Probabilities Estimators for doing so, including standard binning, counting, permutations, nearest neighbor based, time-scale based, among others.
• Fast and cheap (memory-wise) method for computing histograms of large datasets: binhist.

Delay Coordinates Embedding

Performing delay coordinate embeddings and finding optimal parameters for doing so.

• Flexible, super-efficient and abstracted Delay Coordinates Embedding interface.
  • Supports multiple dimensions and multiple timescales.

• Methods that estimate optimal embedding parameters: Traditional Optimal Embedding.
• Unified Optimal Embedding approach (advanced algorithms).
• Fast calculation of mutual information: selfmutualinfo.
• Unified neighborhood interface.

ChaosTools

ChaosTools.jl is a collection of many algorithms for (chaotic or not) dynamical systems. All algorithms are independent of each other but they are also not expansive enough to be a standalone package.

Orbit Diagrams & PSOS

1. Orbit diagrams (aka bifurcation diagrams) of maps: orbitdiagram.
2. Poincaré surfaces of section for continuous systems: poincaresos.
3. Automated production of orbit diagrams for continuous systems: produce_orbitdiagram.

Lyapunov Exponents

The following treat systems where the equations of motion are known:

1. Maximum Lyapunov exponent for both discrete and continuous systems: lyapunov.
2. Lyapunov spectrum for both discrete and continuous systems: lyapunovspectrum.
3. Finite-time, finite-size lyapunov exponent (a.k.a. nonlinear Lyapunov exponent): local_growth_rates.

Detecting & Categorizing Chaos

1. The Generalized Alignment Index: $\text{GALI}_k$ : gali.
   • Implemented for both discrete and continuous systems.
2. A test to categorize strong chaos, partially predictable chaos and regular behavior: predictability.
   • Implemented for both discrete and continuous systems.
3. The 0-1 test for chaos: testchaos01
4. The expansion entropy: expansionentropy.

Fractal Dimension

1. Dozens of methods to calculate a fractal dimension
2. Entropy-based
3. Correlation-sum-based
4. Kaplan-Yorke dimension: kaplanyorke_dim.

And, in order to automatically deduce dimensions, we also offer methods for:

• Partitioning a function $y(x)$ vs. $x$ into regions where it is approximated by a straight line, using a flexible algorithm with a lot of control over the outcome. See linear_regions.
• Detection of largest linear region of a function $y(x)$ vs. $x$ and extraction of the slope of this region.

Nonlinear Timeseries Analysis

1. Broomhead-King coordinates: broomhead_king.
2. Numerically determining the maximum Lyapunov exponent of a (e.g. experimentally) measured timeseries: lyapunov_from_data.

Periodicity & Ergodicity

1. Numerical method to find unstable and stable fixed points of any order $n$ of a discrete map (of any dimensionality): periodicorbits.
   • Convenience functions for defining and realizing all possible combinations of $\mathbf{\Lambda}_k$ matrices required in the above method.
2. Estimating the period of a timeseries: estimate_period.
3. Return and transit time statistics for a subset of the state space: mean_return_times, exit_entry_times.

Attractor Basins, Tipping Points

• Efficient algorithms for calculating basins of attraction of an arbitrary dynamical system: basins_of_attraction.
• Final state sensitivity: uncertainty_exponent.
• Tipping points related functionality: basin_fractions, tipping_probabilities.

Recurrence Analysis

RecurrenceAnalysis.jl offers tools to compute and analyze Recurrence Plots, a field called Recurrence Quantification Analysis.

• Recurrence Plots, with cross-recurrence and joint-recurrence.
• Recurrence Quantification Analysis (RQA):
  • Recurrence rate, determinism, average/maximum diagonal length, divergence, laminarity, trend, entropy, trapping time, average/maximum vertical length.
  • Fine-tuning of the algorithms that compute the above (e.g. Theiler window and many more)
  • Windowed RQA of the above
• Recurrence Networks