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

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

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?

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

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

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

Standard Equations

A Basic Polynomial Substitution

Substitutions for Polynomial Equations

Polynomial Equations

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

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 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 Computer only adds or multiplies every iteration thereafter which it can do exactly

Fortran Data Files 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 = STANDARD 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 =

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 Choosing a time step and showing any subsequent smaller time steps will produce the same result Total energy display Total energy display

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

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

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