Hopf Bifurcation Diagram with Vector Field (with audio)

With some BASS, check out
https://www.youtube.com/watch?v=EtjdjrKu9Jk

This video is a copy of http://www.youtube.com/watch?v=tTZmTbbLEps&list=TL0Xzl1jbbR6U except with the following audio explanation:

This video illustrates the formation of a hopf bifurcation for the parameter, mu. This example has two nullclines: the red curve shows no change in the x direction, and along the blue curve, there is no change in the y direction. As is with all bifurcation plots, the parameter is like a dial on a radio that we could tune while observing any fundamental changes to the vector field.

When mu is negative (as shown now), there is only 1 stable equilibrium, and it is the origin. Then when mu is greater than or equal to zero, the branch of equilibria bifurcates producing a stable limit cycle in the counter-clockwise direction. The origin has now become unstable, so trajectories near it spiral away from it and approach the circular limit cycle.

The example here is commonly used for instruction because one can change variables to polar coordinates and analyze the limit cycle in terms of its radial and angular variables without too much headache.

Видео Hopf Bifurcation Diagram with Vector Field (with audio) канала Jonathan Mitchell