Apparently complex dynamics can originate from a small set of dynamically important recurring patterns - periodic orbits.
The image to the right shows two example orbits (green,red) that are the time-trace of the lower pendulum in the double pendulum setup. The longer orbit (blue) can be considered to be constructed from 'shadowing' of the two simpler orbits.
From this we can see that it is not essential to find all orbits for a system to describe it well (there will be infinitely many!). It is sufficient to find the short orbits, that we could call our 'alphabet', out of which the longer orbits are constructed.
An excellent (graduate level) free text on periodic orbits can be found at chaosbook.org.
Periodic orbits in pipe flow
Pipe flow, Re=2500. Regions of flow, slower than the mean flow is indicated in blue, called 'streaks'. Rotating 'vortex structures' are indicated in yellow. To simplify analysis a 4-fold rotational symmetry has been imposed. This does not significantly affect statistical properties of the flow, e.g. the turbulent friction factor:
Viewed in a moving frame, recurring patterns (periodic orbits) are observed:
A spatially localised patch of turbulence:
Periodic orbits in pipe flow, visualised in terms of their input (I), dissipation (D) and kinetic energy (E), relative to their laminar values: