1. CHAOS AND THE DOUBLE PENDULUM
By Nick Giffen (James Madison University) &
Laura Marafino (University of Mary Washington)

2. Overview What is the double pendulum and why are we studying it?
Governing equations and Numerical Methods
Use of the Immersive Visualization System

3. What is Chaos?
If we take two identical double pendulums and start them with almost identical (yet still different) initial conditions, what will happen?
When do we get chaos? When don’t we get chaos?

4. Introduction to Parameters
Gravitational acceleration
Mass of bob 1
Mass of bob 2
Length of arm 1
Length of arm 2
= 1+m1/m2

5. Initial Conditions Angle the first arm makes with vertical
Angle the second arm makes with vertical
Angular velocity of the first arm
Angular velocity of the second arm

6. Governing Equations Standard Equations Polynomial Equations
Use the Lagrangian
Polynomial Equations
Use elementary substitutions on standard equations

7. Standard Equations

8. A Basic Polynomial Substitution

9. Substitutions for Polynomial Equations

10. Polynomial Equations

11. Numerical Methods Runge-Kutta 4th order (RK4)
Runge-Kutta-Fehlberg 4th/5th order (RKF45)
Modified Picard method

12. Polynomial or Standard?
Energy conserved better in both RK4 and RKF45 with polynomial equations
Adaptive time steps larger in RKF45 with polynomial equations
Polynomial equations used for the Modified Picard method
Why are the Polynomial equations more useful in the Runge-Kutta methods?
Avoids needless approximate function evaluations (sine, cosine, exponents, etc.) after the first iteration
Computer only adds or multiplies every iteration thereafter which it can do exactly

13. Fortran Data Files STANDARD EQUATIONS POLYNOMIAL EQUATIONS
RKF45 METHOD
INITIAL ENERGY =
for time = 0.930: x1 = y1 = x2 = y2 =
Energy =
for time = 0.940: x1 = y1 = x2 = y2 =
Energy =
for time = 0.950: x1 = y1 = x2 = y2 =
Energy =
for time = 0.960: x1 = y1 = x2 = y2 =
Energy =
for time = 0.970: x1 = y1 = x2 = y2 =
Energy =
for time = 0.980: x1 = y1 = x2 = y2 =
Energy =
for time = 0.990: x1 = y1 = x2 = y2 =
Energy =
for time = 1.000: x1 = y1 = x2 = y2 =
FINAL Energy =
POLYNOMIAL EQUATIONS
RKF45 METHOD
INITIAL ENERGY =
for time = : x1 = y1 = x2 = y2 =
Energy =
for time = : x1 = y1 = x2 = y2 =
Energy =
for time = : x1 = y1 = x2 = y2 =
Energy =
for time = : x1 = y1 = x2 = y2 =
for time = : x1 = y1 = x2 = y2 =
Energy =
for time = : x1 = y1 = x2 = y2 =
Energy =
for time = : x1 = y1 = x2 = y2 =
Energy =
for time = : x1 = y1 = x2 = y2 =
FINAL Energy =

14. How Accurate is the Simulation?
Is chaos really observed in the simulation actually due to numerical error? …NO!
Choosing a time step and showing any subsequent smaller time steps will produce the same result
Total energy display

15. Poincaré Maps and Energy Surfaces
Two products of the chaotic double pendulum

16. Double Pendulum on the Immersive Visualization System
Display up to 73 double pendulums at once
Slightly varied IC’s for each one
Computational power allows us to advance far into the “future”
Zooming capabilities
5X5 central display
Narrowing the range of IC’s
Current displays
IC’s
Energy

17. Acknowledgements Advisors Dr. James Sochacki & Dr. William Ingham
Supported by
James Madison University’s College of Science and Mathematics
Computer Programmers
Joshua Blake, Justin Creasy, Garrett Allen,
& John Suarez
Additional Assistance
Dr. David Pruett
Tina Liu