Concept

Poincare section

The set of points where an orbit crosses a fixed surface in its phase space, recorded crossing by crossing. It turns a continuous flow into a map one dimension lower, where regular motion draws curves and chaos scatters dots.

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

The flow, reduced to one dimension. A scatter of 2395 points: each successive maximum of the Lorenz trajectory's third coordinate against the one before it. The points lie along a single curve with a sharp peak, which is the one-dimensional map the flow induces.

The flow that is really a map

A trajectory wandering through three dimensions is hard to reason about. Record only the successive maxima of one coordinate and the wandering collapses onto a curve — a map of an interval to itself, with a corner in the middle, which is a thing the theory can handle.

dynamics · Strange attractor

Islands of order in a chaotic sea. Poincaré section of the Hénon–Heiles system at energy 1/8: orbits started on a line of the section, crossings coloured by a separation test, 16 regular and 4 chaotic.

A conserved quantity that is not there

In 1964 Michel Hénon and Carl Heiles asked whether a star in a galaxy obeys a third conservation law besides energy and angular momentum. They cut its orbit by a plane and looked. At low energy every orbit drew a clean curve, as if the law held; at higher energy the curves dissolved into dust. Integrating their model again shows the share of chaotic orbits climbing from nothing at energy 1/12 to almost all at 1/6.

dynamics · Sensitive dependence

Named alongside it

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

Chaos Conservation law Dimension Fixed point Hamiltonian system Iteration Kam theorem Lyapunov exponent Return map Strange attractor Tent map

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