Tutorials · Overview

Tutorials¶

End-to-end journeys through the library. Where the analysis pages each isolate a single tool and the system catalogue each document a single model, a tutorial wires several together on one system — from the equations you write to a finished result you could put in a paper, every number reproducible and every claim traceable to the literature.

Each one is self-contained and runnable end to end: paste the blocks in order, keep the pinned initial conditions, and you get the numbers and figures shown. Read them in any order — but if you are new, the first three trace the arc most nonlinear-dynamics courses follow, from what is chaos to how do I find it in real data.

The journeys¶

Understanding chaos¶

Anatomy of a chaotic attractor — define the Lorenz system, integrate it, plot the butterfly, then certify the chaos: watch two neighbouring states diverge, compute the Lyapunov spectrum, read the Kaplan–Yorke dimension, and cross-check it against the correlation dimension. The vocabulary the rest of the toolkit is built on.
The road to chaos — where does chaos come from? Sweep the logistic map's growth rate and watch a fixed point period-double onto the textbook onsets \(r_1 = 3\) and \(r_2 = 1+\sqrt6\), cascade into chaos, and open a period-3 window. Then do the same for a flow — a Rössler bifurcation diagram through a Poincaré section.

Working from data¶

Reconstruction from one signal — you have a single recorded channel \(x(t)\) and no equations. Rebuild the attractor by delay embedding, choose the delay and dimension with principled heuristics, measure the fractal dimension and a data-driven Lyapunov exponent, then run a surrogate test to rule out coloured noise.
Poincaré sections & return maps — section a flow to turn it into a discrete map, then expose the one-dimensional map hiding inside a strange attractor — the famous Lorenz \(z\)-maxima cusp.

Global structure & noise¶

Basins & multistability — a bistable oscillator has two possible fates. Find both attractors, paint the basins that decide which initial conditions reach each one, quantify the boundary, then tilt the potential and watch a basin annihilate at a tipping point.
Noise-driven dynamics — add noise. Integrate a single stochastic path, gather ensemble statistics, watch noise drive Kramers escape between two wells, and check it against the analytic laws.

How to read them¶

Every tutorial follows the same discipline:

The system is defined once. A class with params, dim, and a symbolic right-hand side — the definition contract — or a built-in from the catalogue. Everything downstream composes over that one object.
Every number is reproducible. Snippets pass an explicit ic= (or seed= for anything stochastic), so the printed values are exactly what the code produces. Paste and run; you should see the same figures.
Every claim cites its source. The onsets, exponents and dimensions quoted are literature values, and the See-also links point to the analysis reference page where each tool is documented in full.

More journeys are added as the library grows.