Poincare Map
Oscillator Motion Harmonic motion has both a mathematical and geometric description. Equations of motionEquations of motion Phase portraitPhase portrait The motion is characterized by a natural period. E < 2 E = 2 E > 2 Plane pendulum
Convergence The damped driven oscillator has both transient and steady-state behavior. Transient dies outTransient dies out Converges to steady stateConverges to steady state
Equivalent Circuit Oscillators can be simulated by RLC circuits. Inductance as mass Resistance as damping Capacitance as inverse spring constant v in v C L R
Negative Resistance Devices can exhibit negative resistance. Negative slope current vs. voltageNegative slope current vs. voltage Examples: tunnel diode, vacuum tubeExamples: tunnel diode, vacuum tube These were described by Van der Pol. R. V. Jones, Harvard University
Relaxation Oscillator The Van der Pol oscillator shows slow charge build up followed by a sudden discharge. Self sustaining without a driving forceSelf sustaining without a driving force The phase portraits show convergence to a steady state. Defines a limit cycle.Defines a limit cycle. Wolfram Mathworld
Stroboscope Effect E < 2 E = 2 E > 2 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
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
Energetic Pendulum A driven double pendulum exhibits chaotic behavior. The Poincare map consists of points and orbits. Orbits correspond to different energies Motion stays on an orbit Fixed points are non-chaotic pp l l m m