1. Poincare Map

2. 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 > 2
E = 2
E < 2

3. 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 > 2
E = 2
E < 2

4. 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

5. 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

6. 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

7. 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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