Apparently complex dynamics can originate from just a small set of dynamically important recurring patterns. The image to the right shows two recurring patterns (green,red) that exist for the double pendulum. (A simple android application and screenshots of the double-pendulum can be found HERE;) As time progresses, the position of the lower pendulum traces out the lines shown, (x,y)=(position of lower pendulum). As the line is a closed loop, it traces out a repeating pattern - a 'periodic orbit'.
Observe that the longer orbit (blue) can be considered to be constructed from 'shadowing' of the two simpler orbits. It is possible to understand a lot about the dynamics by finding the short periodic orbits. These short orbits we might call our 'alphabet', out of which infinitely many 'words' may be constructed, i.e. an infinite number of longer orbits.
An excellent (graduate level) free text on periodic orbits can be found at chaosbook.org.
Periodic orbits in pipe flow
In this case, the 'state' of the system corresponds to the speed and direction of travel of the fluid at every position in the flow. By listing the velocity at every point we form a state-vector, a=(a1,a2,a3,a4,...). As there are infinitely many positions, the dimension (length) of a is formally infinite. However, for dissipative systems, the dynamics typically lives on a space of much smaller dimension (like a 2D-sheet within 3D space), because only realistic states (those on the sheet) occur. States initially off the sheet are attracted to it as time progresses. To visualise this space, it is necessary to project the state onto a shorter, more meaningful vector, e.g. (x,y)=(energy,dissipation).
The following video shows a selection of periodic orbits in pipe flow, visualised in terms input energy driving the flow (I), dissipation (D) and kinetic energy (E) of the motion (relative to their laminar (non-turbulent) values):
The setup for the previous video was pipe flow with flow rate measured by Re=DU/nu=2500 (D=diameter, U=mean axial speed, nu=kinematic viscosity). The next video shows flow in the lab frame. Regions of flow slower than the mean flow are 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.
When viewed in a moving frame it is possible to see recurring patterns, i.e. periodic orbits (next video). The speed at which the frame moves, however, is different for every periodic orbit. To determine this automatically we use the Method of slices.
Periodic orbits exist that encompass the broad range of observed flow patterns. The following video shows an example of a periodic orbit that is representative a spatially localised patch of turbulence: