Strange Attractors
A strange attractor is a set in phase space that a chaotic system approaches but never repeats. It has a fractal dimension — neither a surface (2D) nor a volume (3D), but something in between. The Lorenz attractor has dimension ≈ 2.06; most of phase space is forbidden.
Lorenz Attractor
The original butterfly: ẋ=σ(y−x), ẏ=x(ρ−z)−y, ż=xy−βz. Tune σ, ρ, β and watch the trajectory trace the double-wing attractor in 3D — two nearby trajectories diverge exponentially.
Rössler Attractor
O.E. Rössler's simpler three-variable chaotic system — one non-linearity (xz term) vs Lorenz's two. Observe period-doubling route to chaos as parameter c increases.
Thomas' Cyclically Symmetric Attractor
René Thomas' attractor with three-fold symmetry: ẋ=sin(y)−bx, ẏ=sin(z)−by, ż=sin(x)−bz. Transition between periodic and chaotic regimes as b decreases through b≈0.208.
Van der Pol Oscillator
ẍ − μ(1−x²)ẋ + x = 0. The canonical limit-cycle oscillator — tunable between near-harmonic (μ≪1) and relaxation oscillation (μ≫1). Phase portrait and time series.
Lorenz System — Standard Parameters
dx/dt = σ(y − x) [σ ≈ 10 → turbulent Prandtl number]
dy/dt = x(ρ − z) − y [ρ ≈ 28 → normalised Rayleigh number]
dz/dt = xy − βz [β ≈ 8/3 → geometric factor]
Lyapunov exponent λ₁ ≈ +0.906 → doubling time ≈ 0.764 s
Attractor dimension D₂ ≈ 2.06 → "thin fractal"
Bifurcations & Period Doubling
A bifurcation is a qualitative change in a dynamical system as a parameter crosses a threshold. The logistic map x_{n+1} = rx_n(1−x_n) exhibits every known route to chaos in a single equation — period-doubling cascades, intermittency, and windows of periodicity deep inside the chaotic regime.
Bifurcation Diagram
Full logistic map bifurcation diagram — magnify any window to reveal self-similar copies of the whole structure. Feigenbaum constant δ ≈ 4.6692 measured inline.
Double Pendulum
Two coupled pendula — integrable for small angles, chaotic for large. Launch two trajectories with 0.001° separation and watch them diverge within seconds.
Turing Patterns & Reaction-Diffusion
Alan Turing's 1952 paper "The Chemical Basis of Morphogenesis" predicted that a system of two chemicals — one an activator, one an inhibitor — diffusing at different rates and reacting would spontaneously form spatial patterns from a uniform state. These Turing patterns appear in animal coats, fish skin, and seashell pigmentation.
Reaction-Diffusion (Gray-Scott)
Gray-Scott two-species model: ∂u/∂t = Du∇²u − uv² + F(1−u), ∂v/∂t = Dv∇²v + uv² − (F+k)v. Tune feed rate F and kill rate k to explore spots, stripes, and solitons.
Sensitivity to Initial Conditions
Launch 20 nearby Lorenz trajectories simultaneously. The coloured trails diverge exponentially — visualising the Lyapunov exponent as a spreading cloud.
The Feigenbaum constant δ ≈ 4.6692. In any period-doubling route to chaos — logistic map, Hénon map, driven pendulum, dripping tap — the ratio between successive bifurcation parameter intervals converges to the same universal constant. Mitchell Feigenbaum discovered this in 1975 on a programmable calculator. It was the first demonstration that chaos has universal, quantitative structure.