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Poincare Map

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.

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Poincare Map

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

  4. Oscillators can be simulated by RLC circuits. Inductance as mass Resistance as damping Capacitance as inverse spring constant Equivalent Circuit L vin v C R

  5. Devices can exhibit negative resistance. Negative slope current vs. voltage Examples: tunnel diode, vacuum tube These were described by Van der Pol. Negative Resistance 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 force • The phase portraits show convergence to a steady state. • Defines a limit cycle. Wolfram Mathworld

  7. 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 Stroboscope Effect E > 2 E = 2 E < 2

  8. 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. Damping Portrait 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 l q m l f m pf f

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