Periodic orbits
[Under construction]
Apparently complex dynamics can originate from a small set of dynamically important recurring patterns. The image to the right shows two example recurring patterns (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).
As the state of a system changes in time its 'state-vector' traces out a trajectory, and if the pattern returns to one that has been seen before, then it traces out a closed 'periodic orbit'. Observe that the longer orbit (blue) in the example 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-vector, a=(a1,a2,a3,a4,...). As the grid could be refined indefinitely, the dimension (length) of a is formally infinite. In a dissipative system, however, the dynamics typically lives on a space of much smaller dimension (unrealistic flows are not visited by the system). To visualise this space, it is 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 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. The method makes possible permits visualisation of the state space via a wide range of projections. [ADD video/image showing this]
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: