Dual views of periodic orbits and transient turbulenceEquilibria and their stable/unstable manifolds teach us a lot about the geometry of the state space, but as they are stationary, no turbulence takes place there. As the following dual-view movies show, the time dependence of typical unstable structures seen in turbulence is better captured by unstable periodic orbits embedded in the turbulent regions of the state space. |
T=35.77 periodic orbit: | T=97.08 periodic orbit |
The period T=97.08 orbit (purple line) is a periodic orbit close to a two-period repeat of the period 35.77 orbit (dotted line), with an extra wiggle; it is the beginning of the infinite hierarchy of unstable periodic orbits embedded in the turbulent flow. From a 3D video it would be hard to tell that these are not segments of turbulent orbit - they capture well the typical coherent structures observed in the earlier videos.