Is it all about Chaos?Is it all about Chaos?

MATH 6514: Industrial Mathematics IMATH 6514: Industrial Mathematics IFinal Project PresentationFinal Project Presentation

-Manas Bajaj (ME)Manas Bajaj (ME)- Qingguo Zhang (AE)Qingguo Zhang (AE)- Sripathi Mohan (AE)Sripathi Mohan (AE)

- Thao Tran (AE)Thao Tran (AE)

What we imagine is order is merely the prevailing form of chaos.- Kerry Thornley, Principia Discordia, 5th edition

ContentContent

Introduction to ChaosIntroduction to ChaosExamples of ChaosExamples of ChaosExperimental & Numerical ResultsExperimental & Numerical Results

Double PendulumDouble PendulumMagnetic PendulumMagnetic PendulumVibrating StringVibrating StringSwinging SpringSwinging Spring

ConclusionConclusionThe Road AheadThe Road AheadQuestions?Questions?

Deterministic theoryDeterministic theoryAdvocated by NewtonAdvocated by Newton

Example : Laws of Motion Example : Laws of Motion The exact behavior of any dynamical system can be simulated and The exact behavior of any dynamical system can be simulated and accurate predictions can be made about the behavior of a dynamical accurate predictions can be made about the behavior of a dynamical system at a future point in time with system at a future point in time with the given initial conditionsthe given initial conditionsThe dynamical system could be anything from the planets in the solar The dynamical system could be anything from the planets in the solar system to ocean currents.system to ocean currents.

Real World problems?Real World problems?How accurate can one be with measuring the initial conditions of How accurate can one be with measuring the initial conditions of systems like the heavenly bodies and ocean currents? systems like the heavenly bodies and ocean currents? Can never Can never achieve infinite accuracyachieve infinite accuracy

Erroneous notion amongst the faculty of scientists. Erroneous notion amongst the faculty of scientists. (“Shrink-Shrink” assumption)(“Shrink-Shrink” assumption)

Almost same Initial Conditions Almost same Initial Conditions Almost same behavioral predictionAlmost same behavioral prediction??

Loopholes in the beliefLoopholes in the beliefAssumptions taken for granted with the deterministic theoryAssumptions taken for granted with the deterministic theory

Chaos is a challenge to the “Shrink-Shrink” Chaos is a challenge to the “Shrink-Shrink” assumptionassumption

HenrHenri Poincare challenged this (1900)i Poincare challenged this (1900)The predictions can be grossly different for systems like Planets since an The predictions can be grossly different for systems like Planets since an accurate measurement of the initial conditions is not possible.accurate measurement of the initial conditions is not possible.

The world didn’t realize the problems yet.The world didn’t realize the problems yet.

Edward Lorenz’s weather prediction model Edward Lorenz’s weather prediction model (1960)(1960)

12 equations. Starts a simulation run from somewhere in the middle to check 12 equations. Starts a simulation run from somewhere in the middle to check a solution pattern :a solution pattern : Enters the value with less precision (Enters the value with less precision (0.506 0.506 VsVs 0.5061270.506127))““Butterfly Effect”Butterfly Effect”

What is CHAOS?What is CHAOS?

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

Examples of CHAOSExamples of CHAOS

1. Solution of Lorenz’s weather prediction model.

2. Waterwheel experiment (Lorenz’s waterwheel)

• mathematical model : serendipity - Lorenz

Divergent behavior with almost

the same set of initial conditions

Examples of CHAOS (cont.)Examples of CHAOS (cont.)3. The Double Pendulum Experiment

• We were able to perform this

4. The Magnetic Pendulum Experiment• We were able to perform this (Live demo)

Examples of CHAOS (cont.)Examples of CHAOS (cont.)- Mathematical Model- Mathematical Model - Bifurcation diagram- Bifurcation diagram

Solving:Solving:– y = xy = x22 + c + c (1)(1)

– y = xy = x (2)(2)

Studying the behavior Studying the behavior for different values of for different values of “c” “c”

Convergence with Convergence with different values of ‘”c”different values of ‘”c”

5. The Bifurcation diagram for this problem

Chaotic regime

Schematic Diagram Experimental Setup

Double PendulumDouble Pendulum

Governing EquationsGoverning Equations

02

sinsin*sin*2

2cos*cos

2423

221214212132

1

224

2

2322121212131

glmglm

llmllm

lml

mllllm

011

L

dt

dL0

22

L

dt

dL

L[Lagrangian] = KE - PE

Solved numerically using 4th order Runge-Kutta method in MATLAB

11stst Initial Condition 1 Initial Condition 1stst Trial: Trial: 1 & 1 & 2 vs. Time2 vs. Time

-16

-12

-8

-4

0

4

8

12

16

0 1 2 3 4 5 6 7

Time (s)

An

gle

(ra

d)

Exp_Theta 1 Exp_Theta 2 Num_Theta 1 Num_Theta 2

11stst Initial Condition 2 Initial Condition 2ndnd Trial Trial:: 1 & 1 & 2 vs. Time2 vs. Time

-40

-35

-30

-25

-20

-15

-10

-5

0

5

0 1 2 3 4 5 6 7 8

Time (s)

An

gle

(ra

d)

Exp_Theta1 Exp_Theta2 Num_Theta 1 Num_Theta 2

22ndnd Initial Condition 1 Initial Condition 1stst Trial Trial:: 1 & 1 & 2 vs. Time2 vs. Time

-15

-10

-5

0

5

0 2 4 6 8 10 12

Time (s)

Ang

le (

rad)

Exp_Theta 1 Exp_Theta 2 Num_Theta 2 Num_Theta 1

22ndnd Initial Condition 2 Initial Condition 2ndnd Trial Trial:: 1 & 1 & 2 vs. Time2 vs. Time

-16

-12

-8

-4

0

4

0 2 4 6 8 10 12

Time (s)

An

gle

(ra

d)

Exp_Theta 1 Exp_Theta 2 Num_Theta 1 Num_Theta 2

DiscussionDiscussion

Two sets of initial conditions, different Two sets of initial conditions, different results for each trial within each set. results for each trial within each set. – Chaotic behaviorChaotic behavior

Cannot sustain a-periodic motion due to Cannot sustain a-periodic motion due to high damping effects.high damping effects.

Experimental setup challengesExperimental setup challengesPendulum has to be constrained in 2-DPendulum has to be constrained in 2-D

Plane of motion of Pendulum has to be on two Plane of motion of Pendulum has to be on two different planes (parallel) different planes (parallel)

Magnetic PendulumMagnetic Pendulum

Magnet 3

Magnet 2

Pendulum bob

Magnet 1

Governing EquationsGoverning Equations

3i

32 2 2i 1

i i

3i

32 2 2i 1

i i

(x x(t))x (t) Rx (t) Cx(t) 0

(x x(t)) (y y(t)) d

(y y(t))y (t) Ry (t) Cy(t) 0

(x x(t)) (y y(t)) d

Solved numerically using 4th order Runge-Kutta method in MATLAB

d (distance between the plane of the magnetic bob and the underline plane)

R = Damping Coefficient

C = Magnetic Coefficient<< L

11stst Trial: Trajectory of Pendulum Trial: Trajectory of PendulumNumerical SolutionNumerical Solution

-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5

X Position

Y P

osi

tion

22ndnd Trial: Trajectory of Pendulum Trial: Trajectory of PendulumNumerical SolutionNumerical Solution

-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5

X Position

Y P

osit

ion

33rdrd Trial: Trajectory of Pendulum Trial: Trajectory of PendulumNumerical SolutionNumerical Solution

-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5

X Position

Y P

osit

ion

11stst Trial: Trajectory of Pendulum Trial: Trajectory of PendulumExperimental SolutionExperimental Solution

-1.5

-1

-0.5

0

0.5

1

1.5

-1 -0.5 0 0.5 1 1.5

X Position

Y P

os

itio

n

22ndnd Trial: Trajectory of Pendulum Trial: Trajectory of PendulumExperimental SolutionExperimental Solution

-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1

X Position

Y P

ositi

on

33rdrd Trial: Trajectory of Pendulum Trial: Trajectory of PendulumExperimental SolutionExperimental Solution

-1.5

-1

-0.5

0

0.5

1

1.5

-1 -0.5 0 0.5 1 1.5

X Position

Y Po

sitio

n

DiscussionDiscussion

Experimental and Numerical simulations Experimental and Numerical simulations demonstrate that the system is highly demonstrate that the system is highly sensitive to initial conditionsensitive to initial condition

Hence the system is chaoticHence the system is chaotic

No experimental setup challenges as such No experimental setup challenges as such

Vibrating StringVibrating StringExperimental SetupExperimental Setup

DiscussionDiscussion

Unable to capture the chaotic regime of Unable to capture the chaotic regime of the system. the system.

The motion of the string is periodicThe motion of the string is periodic

Limitation on the amplitude of the voltageLimitation on the amplitude of the voltageRestriction on the amplitude of the vibrating wireRestriction on the amplitude of the vibrating wire

Point light source use to magnify the Point light source use to magnify the motion of wire (on a backdrop)motion of wire (on a backdrop)

Swinging SpringSwinging Spring

DiscussionDiscussion

No mathematical model available for chaotic No mathematical model available for chaotic regime regime Experimentally, spring tends to follow different Experimentally, spring tends to follow different trajectories when started with similar initial trajectories when started with similar initial conditionsconditions

Experimental challengeExperimental challengeLack of freedom to bounce above the plane of suspension Lack of freedom to bounce above the plane of suspension (due to suspension point)(due to suspension point)

Limitation on measuring the amplitude of spring Limitation on measuring the amplitude of spring – zoom restriction due to the lack of a “real grid” (we were using zoom restriction due to the lack of a “real grid” (we were using

graph sheets)graph sheets)

ConclusionConclusion

Chaos Chaos unstable dynamical system - approaches it in a very regular unstable dynamical system - approaches it in a very regular mannermanner

sensitivity to the initial conditionssensitivity to the initial conditions

Double PendulumDouble PendulumSensitivity to initial conditions demonstratedSensitivity to initial conditions demonstrated

A-periodic motion of pendulumsA-periodic motion of pendulums

System can extend to “n” pendulumsSystem can extend to “n” pendulums

Magnetic PendulumMagnetic PendulumShow system sensitivity to initial conditionShow system sensitivity to initial condition

The Road AheadThe Road Ahead

Double PendulumDouble PendulumDevelop better fixture to reduce damping factor on systemDevelop better fixture to reduce damping factor on system

Induce forcing function to systemInduce forcing function to system

Magnetic PendulumMagnetic PendulumSolve for the system with attracting and repelling underlying Solve for the system with attracting and repelling underlying magnetsmagnets

Vibrating StringVibrating StringStronger electromagnetStronger electromagnet

Function generator with higher voltage outputFunction generator with higher voltage output

Swinging SpringSwinging SpringBetter experimental fixtureBetter experimental fixture

Can bounce above the plane of fixtureCan bounce above the plane of fixture

Questions?Questions?