|Week 6 overview|
- 11. Slice and dice
- 11. Qualitative dynamics for pedestrians
- Homework 6
Actions of a Lie group on a state trace out a manifold of equivalent states, or its group orbit. Symmetry reduction is the identification of a unique point on a group orbit as the representative of this equivalence class. Thus, if the symmetry is continuous, the interesting dynamics unfolds on a lower-dimensional `quotiented', or `reduced' state space M/G. In the method of slices the symmetry reduction is achieved by cutting the group orbits with a set of hyperplanes, one for each continuous group parameter, with each group orbit of symmetry-equivalent points represented by a single point, its intersection with the slice. Moving frames give us a great deal of freedom - we discuss how to choose a frame The most natural of all moving frames: the comoving frame, the frame for space cowboys.
Chapter Slice & dice
Read Sects. 13.4 and 13.5. The rest is optional.
|Symmetry reduction NBB|
|Symmetry reduced equations of motion|
|Sections and slices are local, good up to a border|
|A spatial Fourier expansion NBB|
|First Fourier mode slice NBB|
|In-slice time NBB|
As of 2 Mar 2015, exists only as a video, this is not yet written up in ChaosBook The most natural of all moving frames: the comoving frame, the frame for space cowboys.
Qualitative properties of a flow partition the state space in a topologically invariant way.
|Discussion forum for week 6|
|Master Slicer Certificate|
|Ring of Fire Visualize the O(2) equivariance of Kuramoto-Sivashinsky (AKA Ring of Fire)|