Quantifying sensitive dependence
So far 'sensitive dependence' has been a picture. To do physics we need a number. Take two trajectories that start a tiny distance |\delta\mathbf{x}_0| apart and watch how the gap evolves. In a chaotic system it grows, on average, exponentially.
The separation of nearby trajectories, averaged along the motion, grows (or shrinks) exponentially at a rate λ.
The rate \lambda is the Lyapunov exponent. Defined carefully, we take the initial separation to zero (to stay in the linear regime) and the time to infinity (to average over the whole attractor).
The (largest) Lyapunov exponent: the long-time average exponential rate at which infinitesimally close trajectories separate.
What the sign means
The sign of \lambda classifies the motion. \lambda < 0: nearby trajectories converge — the system heads for a stable fixed point. \lambda = 0: neighbours neither separate nor merge — a stable limit cycle or quasiperiodic motion. \lambda > 0: neighbours diverge exponentially — chaos. An n-dimensional system actually has a spectrum of n exponents; a single positive one is enough to make the system chaotic.
The Lyapunov exponent of a map
For a one-dimensional map the exponent takes a clean, computable form: each step multiplies a separation by |f'(x_n)|, so the average growth rate is the long-run average of \ln|f'| along the orbit.
For an iterated map, the Lyapunov exponent is the orbit-average of the log-slope. For the logistic map at r = 4 it equals exactly ln 2 > 0 — provably chaotic.
A positive exponent has a vivid meaning: at r = 4, \lambda = \ln 2 means the map loses one bit of information about the initial condition every iteration. Know x_0 to 30 binary digits and after 30 steps you know nothing. That is the arithmetic behind the predictability horizon.
A working definition of chaos
The Poincaré section: from flow to map
A chaotic trajectory in three dimensions is a tangled ball of yarn — hard to see anything in. Poincaré's trick: place a plane through the phase space and record only the points where the trajectory pierces it (crossing one chosen way). The continuous flow collapses to a discrete Poincaré map in one lower dimension — turning a 3D flow into a 2D map we can actually study.
The pattern on the section is a diagnosis. A periodic orbit leaves a finite set of dots. A quasiperiodic orbit fills a smooth closed curve. A chaotic orbit sprinkles a structured, fractal cloud of points — order and chaos become visible at a glance.
Back to the double pendulum
Return to our opening example and put the tools to work. Track two nearly identical starts and measure how fast the gap grows: the slope of \ln|\delta\mathbf{x}| against time estimates \lambda. Crucially, the double pendulum is not chaotic everywhere: swing it gently and it is nearly regular; give it enough energy to flip over the top and it becomes chaotic. The same system is regular or chaotic depending on where you are in phase space.
Hamiltonian chaos and the KAM theorem
Conservative chaos has its own beautiful theory. An integrable system — one with as many conserved quantities as degrees of freedom — has all its motion wound neatly onto invariant tori (think nested doughnuts) in phase space; nothing chaotic. Perturb it, and the question is whether those tori survive.
The KAM theorem (Kolmogorov–Arnold–Moser) answers: under a small enough perturbation most tori survive, slightly deformed, while the resonant ones shatter into thin chaotic layers that thicken as the perturbation grows. This governs the long-term stability of the solar system, the confinement of particles in accelerators and fusion plasmas, and the onset of chaos in nearly-integrable systems everywhere.