🌀

Chaos & Dynamics

Deterministic systems with unpredictable behaviour. The Lorenz butterfly and double pendulum — the butterfly effect in action.

Chaos and dynamics explores how simple, fully deterministic rules can produce behaviour so sensitive to starting conditions that it becomes effectively unpredictable. This category covers strange attractors, fractals, bifurcations, limit cycles and the Lyapunov exponent that measures how fast nearby trajectories diverge. By running each interactive Chaos & Dynamics model in your browser, you learn to read phase portraits, recognise the period-doubling route to chaos and see the butterfly effect emerge in real time. These ideas matter far beyond mathematics: they underpin weather forecasting limits, cardiac rhythm analysis, population dynamics, fluid turbulence and the design of secure, chaos-based communication. Exploring them visually turns abstract nonlinear theory into something concrete and intuitive.

8+ simulations Three.js Strange Attractor · RK4

🌀 See also: Chaos Theory Topic Hub

Category Simulations

Open a simulation — it runs right in your browser

Sensitivity to initial conditions — a microscopic difference in starting parameters leads to completely divergent trajectories. That is chaos: not disorder, but extraordinary complexity from deterministic rules.

🌀
★★★ Advanced New
Elastic Pendulum
A mass on a spring that also swings — a 2-DOF coupled oscillator. When the spring frequency is about twice the swing frequency, energy sloshes between bounce and swing (2:1 resonance) and the motion turns chaotic.
Elastic Pendulum Coupled Oscillator Resonance Canvas 2D
🔌
★★★ Advanced
Chua's Circuit — Double-Scroll Attractor
The simplest electronic circuit proven chaotic. Watch the double-scroll strange attractor emerge from Chua's three ODEs with the piecewise-linear diode, and see two near-identical trajectories diverge exponentially.
Canvas 2D Strange Attractor RK4 Double Scroll
📉
★★★ Advanced
Logistic Map — Route to Chaos
The famous bifurcation diagram of x→rx(1−x): stable → period-2 → period-4 → chaos. Zoom for self-similar Feigenbaum structure, animate the cobweb, watch the Lyapunov exponent flip sign.
Canvas 2D Bifurcation Feigenbaum Lyapunov
🧲
★★★ Advanced
Magnetic Pendulum — Fractal Basins
A pendulum over magnets settles onto one — but tiny start changes flip the outcome, painting a fractal basin of attraction. Drop the bob or compute the full basin map.
Canvas 2D Fractal Basins Chaos RK4
🦋
Popular ★★☆ Moderate
Lorenz Attractor
Hundreds of trajectories simultaneously forming the “butterfly wings” — a strange attractor in 3D space. RK4 integration of the 1963 Lorenz system.
Three.js Strange Attractor RK4
🌀
★★☆ Moderate
Double Pendulum
120 pendulums with nearly identical initial conditions — within seconds the trajectories diverge completely. A classic demonstration of chaos.
Three.js RK4 Chaos
🔬
★★☆ Moderate
Fractal Explorer
Zoom infinitely into the Mandelbrot set and Julia sets. Complex quadratic maps z → z² + c iterated in GLSL, with smooth colouring for boundary detail.
GLSL Mandelbrot Complex Plane
🔸
★★★ Advanced
Reaction-Diffusion
Gray-Scott model: two chemical species A and B react and diffuse. Infinitely varied Turing patterns emerge from f and k parameters — spots, stripes, worms and mazes — all from two PDEs.
Gray-Scott Turing Patterns WebGL
🔺
★☆☆ Easy
Sierpiński Triangle
Chaos Game: pick a random corner and move half-way toward it — 20,000 points later the fractal emerges. Fractal dimension ≈ 1.585 via the Hausdorff measure.
Fractal Chaos Game Canvas 2D
🟩
Popular ★☆☆ Easy
Conway's Game of Life
Four local rules on a 2D grid produce gliders, oscillators and Turing-complete computation. The canonical example of emergence in complex systems.
Cellular Automata Emergence Canvas 2D
🌀
New ★★☆ Moderate
Rössler Attractor
3D chaotic spiral attractor with adjustable parameters a, b, c. Multiple simultaneous trails reveal the fractal structure via Three.js with OrbitControls.
Three.js Attractor 3D
🌿
New ★★☆ Moderate
Bifurcation Diagram
Period-doubling cascade of the logistic map r·x·(1−x). Drag to zoom into any region of the diagram and explore the onset of chaos at r ≈ 3.57.
Logistic Map Period Doubling Canvas 2D
〰️
New ★★☆ Moderate
Van der Pol Oscillator
Nonlinear oscillator with self-sustaining limit cycle. Phase portrait and time series side by side — watch the system settle from any initial condition as μ drives relaxation oscillations.
Limit Cycle Phase Portrait RK4
🔮
New ★★☆ Moderate
Thomas Attractor
3D symmetric dissipative strange attractor with cubic symmetry. Single parameter b controls the transition from fixed points through limit cycles to fully chaotic orbits via Three.js and RK4.
Three.js Strange Attractor 3D RK4
🔗
★★☆ Moderate
Double Pendulum
Two coupled pendulums reveal extreme sensitivity to initial conditions. Tiny differences in starting angle produce completely different trajectories — chaos in its purest form.
Canvas 2D ODE Chaos
🌀
★★★ Advanced
Kármán Vortex Street
Fluid flow past a cylinder generates an alternating vortex street — a classic example of periodic instability leading to chaotic behaviour at higher Reynolds numbers.
LBM WebGL Fluid
🌀
New ★★☆ Moderate
Strange Attractor
Explore Lorenz, Rössler, Thomas and Halvorsen attractors with RK4 integration. Drag to rotate the 3D trajectory, switch colour modes and watch the positive Lyapunov exponent drive sensitive dependence.
RK4 Lyapunov 3D Projection
🌀
New ★★☆ Moderate
Duffing Oscillator
Driven nonlinear oscillator ẍ + δẋ + αx + βx³ = γcos(ωt). Explore chaotic phase portraits and Poincaré sections in real time by tuning damping, cubic stiffness and forcing amplitude.
RK4 Phase Portrait Poincaré
⚖️
New ★★☆ Moderate
Double Pendulum Ensemble
Launch 30 double pendulums with nearly identical initial conditions and watch chaos separate them exponentially. Lyapunov exponent estimated from log-separation regression.
RK4 Lyapunov Butterfly Effect

Key Concepts

The mathematical ideas underpinning chaotic dynamics

Sensitive Dependence
Tiny differences in initial conditions grow exponentially. Two trajectories starting ε apart diverge as ε·e^(λt) where λ is the Lyapunov exponent (positive for chaos).
Strange Attractor
A fractal set in phase space toward which trajectories are attracted yet never repeat. The Lorenz attractor has Hausdorff dimension ≈ 2.06 — between a 2D surface and a 3D volume.
Self-Similarity
Fractals look the same at every zoom level. The Mandelbrot set's boundary is self-similar at all scales; its area is finite but boundary length is infinite.
Emergence
Complex global behaviour arising from simple local rules. Conway's Game of Life is Turing-complete — universal computation from four rules applied to a binary grid.

Learning Resources

Articles and tutorials about the algorithms in this category

Article Lorenz Attractor and Chaotic Systems Three Lorenz ODEs. σ, ρ, β parameters. Lyapunov exponent. RK4 integration. → Article Double Pendulum & RK4 Lagrange equations. Sensitivity to initial conditions. 4th-order Runge-Kutta method. → Article Fractal Geometry & the Mandelbrot Set Iterative mapping z² + c in the complex plane. Smooth coloring in GLSL. →
All articles →

About Chaos & Dynamical Systems Simulations

Butterfly effects, strange attractors, and sensitive dependence — visible

Chaos theory studies deterministic systems whose long-term behaviour is exquisitely sensitive to initial conditions. The Lorenz attractor traces the trajectory of a simplified atmospheric convection model and never repeats, yet stays within a bounded butterfly-shaped region. The double pendulum shares the same set of governing equations as any coupled oscillator but becomes unpredictable within seconds for even tiny changes in starting angle.

These simulations use high-accuracy numerical integrators (RK4) to faithfully reproduce the chaotic divergence of trajectories. Bifurcation diagrams reveal the exact parameter values where order transitions to chaos. Strange attractors demonstrate fractal geometry in phase space. By running two nearly identical initial conditions side by side you can directly observe the exponential divergence that defines chaos — a phenomenon central to weather prediction and non-linear science.

Chaos theory has real-world implications far beyond mathematics: weather forecasting becomes unreliable beyond roughly two weeks precisely because the atmosphere is a chaotic system with positive Lyapunov exponents. The same mathematics appears in cardiac arrhythmia, population dynamics, laser physics, and financial markets. These simulations let you measure divergence between two trajectories directly — making the abstract concept of chaos viscerally concrete.

Key Concepts

Topics and algorithms you'll explore in this category

Strange AttractorsBounded aperiodic trajectories in phase space
Lyapunov ExponentMeasures the rate of divergence of nearby trajectories
BifurcationParameter values where system behaviour qualitatively changes
RK4 IntegrationHigh-accuracy solver for chaotic ODEs
Phase SpaceState-space representation of dynamical systems
Butterfly EffectSensitive dependence on initial conditions

Frequently Asked Questions

Common questions about this simulation category

What makes a system chaotic?
A system is chaotic when it is deterministic yet its long-term behaviour is extremely sensitive to initial conditions — quantified by a positive Lyapunov exponent. The Lorenz system, for example, diverges exponentially from almost-identical starting states within seconds.
What is a strange attractor?
A strange attractor is a fractal subset of phase space that a chaotic trajectory approaches but never exactly repeats. The Lorenz butterfly and Rössler band are classic examples: bounded, aperiodic, and with fractal dimension between 2 and 3.
How do bifurcation diagrams show the route to chaos?
A bifurcation diagram plots the long-term values a system settles into against a control parameter. As the parameter increases, period-doubling bifurcations accumulate until the system enters fully chaotic behaviour — a universal route described by Feigenbaum's constant.
What is the butterfly effect and can it be observed in the simulations?
The butterfly effect refers to sensitive dependence on initial conditions: a microscopic change in starting state produces dramatically different outcomes. In the Double Pendulum Ensemble simulation you can launch 30 pendulums with angles differing by 0.001° and watch them diverge completely within seconds, directly measuring the positive Lyapunov exponent.

Other Categories

Every Chaos & Dynamics simulation here runs free in your browser, letting you experiment with each interactive Chaos & Dynamics model — Lorenz and Rössler attractors, the double pendulum, logistic map and Duffing oscillator — without installing anything. Adjust parameters, perturb initial conditions and measure the Lyapunov exponent to learn Chaos & Dynamics online at your own pace, whether you are a student, educator or curious researcher. The same nonlinear mathematics drives real-world applications such as modern weather forecasting, where sensitive dependence on initial conditions sets the practical limit of how far ahead the atmosphere can reliably be predicted.