1. Ch4: FLOWS ON THE CIRCLE Presented by Dayi Zhou 2/1/2006

2. Vector Field on the Circle Vector field on the circle  : a point on the circle the velocity vector at that point

3. Characteristic of this type of system A particle can eventually return to its starting place (flowing in one direction) Periodic solutions become possible

4. Definition A vector field on the circle is a rule that assigns a unique velocity vector to each point on the circle. Example: Cannot be regarded as a vector field on the circle f(  ) is a 2  -periodic function.

5. Uniform Oscillator

6. Nonuniform Oscillator

7. Oscillation Period

8. Square-root Scaling Law General feature of systems that are close to a saddle-node bifurcation