Poincare Map

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

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

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

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

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

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 next