In this experiment designed for use with PASCO Capstone software, the chaotic behavior of a driven nonlinear pendulum is explored by graphing its motion in phase space and by making a Poincare plot. These plots are compared to the motion of the pendulum when it is not chaotic.
The oscillator consists of an aluminum disk connected to 2 springs. A point mass on the edge of the aluminum disk makes the oscillator nonlinear. The frequency of the sinusoidal driver can be varied to investigate the progression from predictable motion to chaotic motion. Magnetic damping can be adjusted to change the character of the chaotic motion. The angular position and velocity of the disk are recorded as a function of time using a Rotary Motion Sensor. A real-time phase plot is made by graphing the angular velocity versus the displacement angle of the oscillation.
The Poincare plot is also graphed in real-time, superimposed on the phase plot. This is achieved by recording the point on the phase plot once every cycle of the driver arm as the driver arm blocks a photogate.
PASCO Capstone can graph the motion in phase space and superimpose the Poincare plot in real-time, showing students how the motion in phase space relates to actual motion of the oscillator.
The Poincare plot (in orange) shows the pendulum's velocity vs. position once per revolution of the driver. The purple background is the phase plot. This phase plot (angular velocity vs. angle) is graphed in PASCO Capstone using partial opacity, so the race gets darker as it traces back over itself.