1. 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)

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

3. 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?

4. 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 )
“Butterfly Effect”

5. 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”

6. 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

7. 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)

8. 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

9. Double Pendulum
Experimental Setup
Schematic Diagram

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

11. 1st Initial Condition 1st Trial: 1 & 2 vs. Time

12. 1st Initial Condition 2nd Trial: 1 & 2 vs. Time

13. 2nd Initial Condition 1st Trial: 1 & 2 vs. Time

14. 2nd Initial Condition 2nd Trial: 1 & 2 vs. Time

15. 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)

16. Magnetic Pendulum
Magnet 1
Pendulum bob
Magnet 3
Magnet 2

17. 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

18. 1st Trial: Trajectory of Pendulum Numerical Solution

19. 2nd Trial: Trajectory of Pendulum Numerical Solution

20. 3rd Trial: Trajectory of Pendulum Numerical Solution

21. 1st Trial: Trajectory of Pendulum Experimental Solution

22. 2nd Trial: Trajectory of Pendulum Experimental Solution

23. 3rd Trial: Trajectory of Pendulum Experimental Solution

24. 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 

25. Vibrating String Experimental Setup

26. 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)

27. Swinging Spring

28. 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)

29. 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

30. 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

31. Questions?