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Ch4: FLOWS ON THE CIRCLE

Ch4: FLOWS ON THE CIRCLE. Presented by Dayi Zhou 2/1/2006. Vector Field on the Circle. Vector field on the circle : a point on the circle the velocity vector at that point. Characteristic of this type of system.

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Ch4: FLOWS ON THE CIRCLE

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  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

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