Learn chaos
by watching order emerge.
Build intuition for nonlinear systems by changing parameters and watching trajectories, bifurcations, attractors, and predictability evolve in real time.
Move from equations to geometry
Each lesson connects the rule of evolution to time series, phase space, parameter changes, and quantitative measures.
Evolve
Start from an initial state and follow how the system changes.
Visualize
See trajectories, cobwebs, bifurcations, and attractors.
Perturb
Change parameters or initial conditions and compare outcomes.
Measure
Connect qualitative behavior to Lyapunov growth and recurrence.
Your nonlinear dynamics learning path
Begin with state and stability, then progress through maps, chaos, attractors, and nonlinear analysis.
Dynamical Systems Foundations
Learn to think in states, flows, and stability.
Build the language of dynamical systems: state variables, trajectories, fixed points, stability, and phase space.
What Is a Dynamical System?
Learn how states evolve through time, why phase space matters, and how fixed points organize nonlinear behavior.
Fixed Points and Stability
Perturb an equilibrium and see the difference between attraction, repulsion, and local stability.
Maps & Bifurcations
See simple rules generate complex behavior.
Use the logistic map to move from fixed points to period doubling, bifurcation, and chaos.
The Logistic Map
Iterate xₙ₊₁ = rxₙ(1−xₙ), build a cobweb plot, and watch fixed points become cycles and chaos.
Bifurcations and the Route to Chaos
Explore the logistic-map bifurcation diagram and see period doubling build a route from stability to chaos.
Chaos & Predictability
Understand unpredictability without randomness.
Explore sensitive dependence, exponential separation, Lyapunov exponents, and predictability horizons.
Sensitivity to Initial Conditions
Launch two nearly identical trajectories and watch a tiny initial difference grow into macroscopic disagreement.
Lyapunov Exponents
Estimate the average exponential separation rate and connect the sign of λ to stability, neutrality, and chaos.
Attractors & Phase Space
Watch trajectories organize in phase space.
Study attractors, the Lorenz system, geometry of trajectories, and the structure hidden inside irregular motion.
Nonlinear Analysis Tools
Measure structure in nonlinear signals.
Use recurrence, Poincaré sections, and related tools to reveal dynamical organization beyond a raw time series.
Recurrence and Poincaré Analysis
Reveal repeated states and hidden organization using recurrence plots and the idea of Poincaré sections.
Fractal Dimension and Box Counting
Estimate how geometrically complex a strange attractor is by covering it with boxes at multiple scales.
Phase-Space Reconstruction and Takens Embedding
Reconstruct hidden state-space geometry from a single measured time series using time-delay coordinates.
Correlation Dimension
Estimate the effective dimension of a reconstructed attractor from pairwise neighborhood statistics.
False Nearest Neighbors
Estimate a sufficient embedding dimension by testing whether nearest neighbors remain close when another delayed coordinate is added.
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