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Poincare Map Oscillator Motion Harmonic motion has both a mathematical and geometric description. • Equations of motion • Phase portrait Plane pendulum q 2 E 1 2 q sin 2 E 2 02 l 12 T 2 ( ) (1 ) g 16 q E>2 The motion is characterized by a natural period. E=2 E<2 q Convergence The damped driven oscillator has both transient and steady-state behavior. q • Transient dies out • Converges to steady state q 2q 02 q f cos t 2 f cos t arctan 2 2 0 q ae t cos 02 2 t 02 2 2 4 2 2 q Equivalent Circuit L vin C • Inductance as mass • Resistance as damping • Capacitance as inverse spring constant v R q C dq vR Ri R dt vout vC vL L Oscillators can be simulated by RLC circuits. d 2 v R dv 1 2 v V0 sin t 2 dt L dt LC 2 di d q L 2 dt dt vin vL vC vR 0 R 2L 02 1 LC Negative Resistance Devices can exhibit negative resistance. • • Negative slope current vs. voltage Examples: tunnel diode, vacuum tube These were described by Van der Pol. R. V. Jones, Harvard University d 2v d 3 2 2 v v v V0 sin t 0 2 dt dt Relaxation Oscillator The Van der Pol oscillator shows slow charge build up followed by a sudden discharge. y 1 y 2 y y 0 • Self sustaining without a driving force The phase portraits show convergence to a steady state. • Defines a limit cycle. Wolfram Mathworld Stroboscope Effect q E>2 • Exact period maps to a point. E=2 E<2 The values of the motion may be sampled with each period. q The point depends on the starting point for the system. • Same energy, different point on E curve. This is a Poincare map 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. E exp[ 2 ( 4mk 2 Damped harmonic mq kq q motion k 2 12 T 2 ( ) 2 m 4m q q 1 2 1) ] Undamped curves Energetic Pendulum A driven double pendulum exhibits chaotic behavior. The Poincare map consists of points and orbits. l m l f m pf f • Orbits correspond to different energies • Motion stays on an orbit • Fixed points are non-chaotic