Concept

Synchronisation

The falling into step of coupled oscillators or chaotic systems, so that their phases or states come to agree. It switches on at a threshold of coupling set by how strongly the units differ or how fast chaos separates them.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Two chaotic maps, coupled weakly and strongly. The gap between two coupled copies of the logistic map at full chaos: at ε = 0.15 it stays large; at ε = 0.3 it falls to nothing.

Two chaotic orbits made to agree

Chaos means two copies of a system started a hair apart soon have nothing in common. Couple them — let each take a fraction ε of its next step from the other — and above a definite strength they agree to every digit for ever, while each stays exactly as chaotic as before. The strength needed is read off the same number that measured the chaos: the Lyapunov exponent.

dynamics · Sensitive dependence

Oscillators scattered below the threshold and gathered above it. N = 240, Lorentzian width 0.5: K = 0.5 gives r = 0.068; K = 2.5 gives r = 0.775.

A crowd of clocks that falls into step

Give a thousand oscillators a thousand different natural rhythms and let each be pulled towards the average phase. Below a definite pull nothing happens; above it a cluster forms and grows, and Kuramoto's 1975 calculation says exactly where: at twice the spread of the frequencies, with the order growing as the square root of one minus the threshold over the pull. A simulation lands on that curve to the third decimal.

dynamics · Sensitive dependence

Named alongside it

The objects these essays reach for when they reach for this one.

Simulation Stability Threshold Bifurcation Central limit theorem Chaos Eigenvalue Logistic map Lyapunov exponent Oscillation Periodic orbit Phase transition

All concepts