Chaos Theory

Chaos theory studies deterministic systems whose long-term behaviour is nevertheless practically unpredictable, because tiny differences in starting conditions grow exponentially over time. Every simulation on this hub is governed by an equation with no randomness in it at all — the same starting position, run twice, produces exactly the same result — yet in practice two starting positions that differ by less than a rounding error diverge completely within a short time. This is the mathematical heart of the popularised "butterfly effect", and this hub gathers every chaos, strange-attractor and fractal-dynamics simulation on mysimulator.uk into one guided starting point.

The clearest entry point is the logistic map, a single equation x → rx(1−x) originally used to model bounded population growth. As the parameter r increases, the long-term behaviour goes from a stable fixed point, through a cascade of period-doubling bifurcations (period 2, then 4, then 8...), and finally into full chaos — all visible at once on the bifurcation diagram, which plots every long-term value against every value of r. The ratio between successive period-doubling thresholds converges to the Feigenbaum constant δ ≈ 4.669, a universal number that shows up in unrelated chaotic systems far beyond population models, from dripping taps to certain electronic circuits.

The double pendulum and triple pendulum simulations make the same sensitivity mechanical and visible: launch several pendulums with starting angles that differ by a fraction of a degree, and within a few seconds their trajectories look completely unrelated, even though the underlying physics (Lagrangian mechanics, solved with an RK4 integrator) is exactly deterministic. The Lorenz attractor is the most famous continuous-time example — three coupled differential equations, derived in 1963 from a simplified model of atmospheric convection, whose trajectories never repeat and trace out the iconic "butterfly wings" shape in three dimensions. The Rössler attractor and Chua's circuit demonstrate that the same qualitative behaviour — a strange attractor with a positive Lyapunov exponent — can arise from much simpler equations, including a real electronic circuit built from off-the-shelf components.

Chaos and fractal geometry are two sides of the same coin: the boundary of a chaotic system's basin of attraction, or the set of parameters that produce bounded orbits, is typically an infinitely detailed fractal. The Mandelbrot and Julia set explorers show this directly in the complex plane — the boundary between points that escape to infinity and points that stay bounded under repeated iteration of a simple quadratic formula is self-similar at every scale you zoom into, mirroring the same sensitive dependence found in the pendulum and Lorenz systems.

Follow the learning path below for a suggested route from the simplest possible chaotic system to real mechanical and fractal examples, or browse the full grid and dive into whichever attractor or oscillator interests you most.

Every simulation on this hub genuinely integrates the underlying differential equation frame by frame with a numerical method such as RK4, rather than replaying a fixed recording — which is exactly why sensitivity to initial conditions is visible at all. Launch two double pendulums a fraction of a degree apart and the integrator computes two genuinely different trajectories step by step; the moment they visibly diverge is the moment the accumulated numerical difference has grown past the point where the two motions look related. The Lorenz and Rössler attractor simulations let you change σ, ρ, β or a, b, c directly and watch the qualitative shape of the attractor change — a small parameter shift can turn a single-loop limit cycle into a full double-scroll chaotic attractor, exactly as the bifurcation theory predicts. That live responsiveness is what separates an interactive chaos simulation from a static picture of the Lorenz butterfly: you are watching the mathematics compute its own unpredictability in real time, not looking at someone else's screenshot of it.

Chaos theory is not a purely academic curiosity — it is the reason weather forecasts are only reliable for a week or two no matter how much computing power is thrown at them, and the same mathematics of sensitive dependence shows up in cardiac arrhythmia, turbulent fluid flow, population dynamics, and the orbits of asteroids in certain resonances with Jupiter. Edward Lorenz discovered the effect that bears his name by accident, in 1961, while re-running a weather model from rounded intermediate output and getting a wildly different long-term forecast from a difference of one part in a thousand. Understanding chaos theory means understanding a hard limit on prediction that applies across an enormous range of real systems, not a quirk of any one equation — which is exactly why the same handful of ideas (sensitive dependence, strange attractors, bifurcation, fractal boundaries) recur across the very different-looking simulations gathered on this page.