Periodic orbits
[Under construction]
Apparently complex dynamics can originate from a small set of dynamically important recurring patterns. As the state of the system changes in time it traces out a trajectory, and if the pattern returns to one seen before, then it traces out a closed 'periodic orbit'.
The image to the right shows two example periodic orbits (green,red) that exist in the double pendulum setup. The lines are the time-trace of the lower pendulum, i.e. the state of the system has been projected onto (x,y)=(position of lower pendulum). Observe that a 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 the periodic orbits of a system to describe its dynamics well, which is a relief, as there are likely to be infinitely many! It is sufficient to find the short periodic orbits, that we may 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
Here, by 'pattern' we mean the speed and direction of flow at each point in a fluid. By listing the velocity at every point on a grid we form a 'state space', a=(a1,a2,a3,a4,...). As the grid could be refined, the dimension (length) of a is formally infinite. In order to visualise the state space, however, it is more usual to project onto a more meaningful vector, e.g. (x,y)=(energy,dissipation).
The following video shows a selection of periodic orbits in pipe flow, visualised in terms of their input (I), dissipation (D) and kinetic energy (E), 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, recurring patterns (periodic orbits) are observed:
The following is an example of a periodic orbit in the form of a spatially localised patch of turbulence: