Manas Bajaj (ME) Qingguo Zhang (AE) Sripathi Mohan (AE) Thao Tran (AE) Is it all about Chaos? MATH 6514: Industrial Mathematics I Final Project Presentation What we imagine is order is merely the prevailing form of chaos. - Kerry Thornley, Principia Discordia, 5th edition Manas Bajaj (ME) Qingguo Zhang (AE) Sripathi Mohan (AE) Thao Tran (AE)

Content Introduction to Chaos Examples of Chaos Experimental & Numerical Results Double Pendulum Magnetic Pendulum Vibrating String Swinging Spring Conclusion The Road Ahead Questions?

Deterministic theory Advocated by Newton Example : Laws of Motion The exact behavior of any dynamical system can be simulated and accurate predictions can be made about the behavior of a dynamical system at a future point in time with the given initial conditions The dynamical system could be anything from the planets in the solar system to ocean currents. Real World problems? How accurate can one be with measuring the initial conditions of systems like the heavenly bodies and ocean currents?  Can never achieve infinite accuracy Erroneous notion amongst the faculty of scientists. (“Shrink-Shrink” assumption) Almost same Initial Conditions Almost same behavioral prediction?

Loopholes in the belief Assumptions taken for granted with the deterministic theory Chaos is a challenge to the “Shrink-Shrink” assumption Henri Poincaré challenged this (1900) The predictions can be grossly different for systems like Planets since an accurate measurement of the initial conditions is not possible. The world didn’t realize the problems yet. Edward Lorenz’s weather prediction model (1960) 12 equations. Starts a simulation run from somewhere in the middle to check a solution pattern : Enters the value with less precision (0.506 Vs 0.506127) “Butterfly Effect”

What is CHAOS? Chaos is a behavior exhibited by systems that are highly sensitive to Initial Conditions. Under certain system characteristics, one can witness the “Chaotic Regime” The behavior of a dynamical system in the “chaotic regime” can be completely different with the slightest change in the initial conditions. Irregular and highly complex behavior in time that follows deterministic equations. Predictions can be made about the spectrum. (Differs from “randomness”) - Roulette wheel This phenomenon is known as “Chaos”

Examples of CHAOS 1. Solution of Lorenz’s weather prediction model. Divergent behavior with almost the same set of initial conditions 1. Solution of Lorenz’s weather prediction model. Waterwheel experiment (Lorenz’s waterwheel) mathematical model : serendipity - Lorenz

Examples of CHAOS (cont.) The Double Pendulum Experiment We were able to perform this The Magnetic Pendulum Experiment We were able to perform this (Live demo)

Examples of CHAOS (cont.) - Mathematical Model - Bifurcation diagram Solving: y = x2 + c (1) y = x (2) Studying the behavior for different values of “c” Convergence with different values of ‘”c” Chaotic regime 5. The Bifurcation diagram for this problem

Double Pendulum Experimental Setup Schematic Diagram

Governing Equations L[Lagrangian] =  KE -  PE Solved numerically using 4th order Runge-Kutta method in MATLAB

1st Initial Condition 1st Trial: 1 & 2 vs. Time

1st Initial Condition 2nd Trial: 1 & 2 vs. Time

2nd Initial Condition 1st Trial: 1 & 2 vs. Time

2nd Initial Condition 2nd Trial: 1 & 2 vs. Time

Discussion Two sets of initial conditions, different results for each trial within each set.  Chaotic behavior Cannot sustain a-periodic motion due to high damping effects. Experimental setup challenges Pendulum has to be constrained in 2-D Plane of motion of Pendulum has to be on two different planes (parallel)

Magnetic Pendulum Magnet 1 Pendulum bob Magnet 3 Magnet 2

Governing Equations d (distance between the plane of the magnetic bob and the underline plane) R = Damping Coefficient << L C = Magnetic Coefficient Solved numerically using 4th order Runge-Kutta method in MATLAB

1st Trial: Trajectory of Pendulum Numerical Solution

2nd Trial: Trajectory of Pendulum Numerical Solution

3rd Trial: Trajectory of Pendulum Numerical Solution

1st Trial: Trajectory of Pendulum Experimental Solution

2nd Trial: Trajectory of Pendulum Experimental Solution

3rd Trial: Trajectory of Pendulum Experimental Solution

Discussion Experimental and Numerical simulations demonstrate that the system is highly sensitive to initial condition Hence the system is chaotic No experimental setup challenges as such 

Vibrating String Experimental Setup

Discussion Unable to capture the chaotic regime of the system.  The motion of the string is periodic Limitation on the amplitude of the voltage Restriction on the amplitude of the vibrating wire Point light source use to magnify the motion of wire (on a backdrop)

Swinging Spring

Discussion No mathematical model available for chaotic regime  Experimentally, spring tends to follow different trajectories when started with similar initial conditions Experimental challenge Lack of freedom to bounce above the plane of suspension (due to suspension point) Limitation on measuring the amplitude of spring zoom restriction due to the lack of a “real grid” (we were using graph sheets)

Conclusion Chaos Double Pendulum Magnetic Pendulum unstable dynamical system - approaches it in a very regular manner sensitivity to the initial conditions Double Pendulum Sensitivity to initial conditions demonstrated A-periodic motion of pendulums System can extend to “n” pendulums Magnetic Pendulum Show system sensitivity to initial condition

The Road Ahead Double Pendulum Magnetic Pendulum Vibrating String Develop better fixture to reduce damping factor on system Induce forcing function to system Magnetic Pendulum Solve for the system with attracting and repelling underlying magnets Vibrating String Stronger electromagnet Function generator with higher voltage output Swinging Spring Better experimental fixture Can bounce above the plane of fixture

Questions?