1. Poincare Map

2. 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

3. Convergence  The damped driven oscillator has both transient and steady-state behavior. Transient dies outTransient dies out Converges to steady stateConverges to steady state

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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  pp   l l m m