Poincare Map

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Poincare Map
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Oscillator Motion

• Harmonic motion has both a mathematical and
  geometric description.
• Equations of motion
• Phase portrait
• The motion is characterized by a natural period.

Plane pendulum
E gt 2
E 2
E lt 2
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Stroboscope Effect

• The motion in phase space is confined to a
  surface.
• 1 dimension for pendulum

• The values of the motion may be sampled with each
  period.
• Exact period maps to a point.
• The point depends on the starting point for the
  system.
• Same energy, different point on E curve.
• This is a Poincare map

E gt 2
E 2
E lt 2
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Damping Portrait

• Damped simple harmonic motion has a well-defined
  period.
• The phase portrait is a spiral.
• The Poincare map is a sequence of points
  converging on the origin.

Damped harmonic motion
Undamped curves
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Section Map

• The equations of motion can be made into a
  sequence.
• Natural frequency w
• Could be driving frequency
• The equations describe a map from TQ ? TQ.
• Map independent of n
• A section is based on a projection map p.

is a section if
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Two Pendulums

• The phase space of a double pendulum is four
  dimensional.
• Configuration Q is a torus S1 ? S1
• Select the motion of one pendulum at a specific
  point in the space of the other.

q
f
1
2
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Commensurate Oscillations

• The Poincare section map can show the relation
  between different periods in motion.
• Periods that are rational multiples are
  commensurate.
• T1 (m/n) T2
• Finite number of points in section
• Irrationally related periods are incommensurate.

Spinning magnet, S. Jolad, Penn State
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