1. The PASCO Pendulum Weight attached to rotating disc Springs attached to either side of disc in pulley fashion One spring is driven by sinusoidal force Sensors take angular position, angular velocity and driving frequency data

2. PASCO Chaos Setup Driven, double-spring oscillator Necessary two-minima potential Variable: –Driving Amplitude –Driving Frequency –Magnetic Damping –Spring Tension The magnetic damping measurement The measurement of the amplitude

3. Mapping the Potential I.Let the weight rotate all the way around once, without driving force II.Take angular position vs. angular velocity data for the run III.Potential energy is defined by the equation Two “wells” represent equilibrium points. In the lexicon of chaos theory, these are “strange attractors”.

4. Mapping the Potential We notice that the potential curve is highly dependent on the position of the driving arm (Left and Right refer to directions when facing the apparatus) Right Well Left Well

5. Mapping the Potential II

6. Can we plot these on the same set of axes? Then we can compare more than just the general shape of the wells. Total Energy Solving for potential. Total energy of system depends on initial conditions. Θ is initial angular position Mapping the Potential II Resonance frequency Substitute for k Normalized potential energy in terms of things we know/ can measure: resonance frequency, initial position, and angular velocity.

7. Mapping the Potential II Three ingredients to measure adjusted potential: initial position, resonance frequency, and plot of angular velocity vs. angular position. 1. Apply small perturbation (tap it) and measure resonance period of well. 2. Pull pendulum back to initial position: θ 0 3. Use angular velocity vs. angular position graph in this region to obtain potential well as a function of angular position.

8. Mapping the Potential II

11. Mapping Chaos Region Procedure Adjust driving arm Raster across frequency spectrum, noting if each point is chaotic or not. Make initial conditions the same for each data point. Repeat. Distinguishing Chaotic from Non-Chaotic Non-chaotic: periodic motion in one of the wells or oscillations about highest point. See figure below. Chaotic: unpredictable how long it will stay in each well. Non chaotic Chaotic Note: When deciding if a set of parameters gave chaotic or non-chaotic motion, observation times varied greatly. We found that oftentimes motion that started off chaotic would settle down into non-chaotic motion after as long as six or seven minutes. So, there is some ambiguity in deciding if a set of parameters gives chaotic or non-chaotic motion.

12. Below Chaotic Region Frequency ~0.65 Hz Chaotic Region Frequency ~0.80 Hz Above Chaotic Region Frequency ~1.00 Hz Chaos Data

13. 4.7 Volts Chaotic Regions * Damping distance of 0.3 cm yielded no chaotic points Chaotic Regions Dependent on: Driving Frequency, Driving Amplitude, Magnetic Damping Larger Amplitude – Larger Region More Damping – Higher Amplitudes, and narrower range of Frequency Hysteresis – Dependent on direction of approach

14. Chaotic Regions II

16. Pendulum Maintenance Scratchy sound coming from behind pendulum wheel. Effectively added damping to the system. We oiled the oscillation axis. Apply oil to the channel. Rotate the axis back and forth We used this oil.