Is it all about Chaos? - ACE Mathematics Laboratory · Is it all about Chaos? Final Project Report...

Is it all about Chaos? * 1 *

MATH 6514: Industrial Mathematics I

Fall 2002, Georgia Tech

Is it all about Chaos?

Final Project Report

MATH 6514: Industrial Mathematics I Fall 2002

Georgia Tech

Authored by:

Manas Bajaj Voice: 404-385-1674

[email protected]

Zhang, Qingguo Voice: 404-463-0886

[email protected]

Sr ipathi Mohan Voice: 404-463-0886

[email protected]

Thao Tran Voice: 404-463-0886

[email protected]

Evaluated by:

Dr. John McCuan Professor

Department of Mathematics Skiles 265, Georgia Tech

Atlanta, GA 30332 USA

Voice: (404) 894 4752 [email protected]




Table of Contents

1. Introduction………………………………………………………..3 2. Some examples of “Chaos”………………………………………5. 3. Trying our hands at “Chaos”…………………………………...13 4. Conclusion………………………………………………………..25 5. The Road ahead………………………….…...……………….…26 6. References……………………………………………………….. 27




1. Introduction

What we imagine is order is merely the prevailing form of chaos.

Kerry Thornley, Principia Discordia, 5th edition

Reflecting on the quote above and investigating the true nature of how things behave around us, it might not be bad idea to understand what Chaos is. Is it something that we see each day and disregard? Does it have any explanation? Is it a science? Is it something merely supernatural that provokes one’s intellect to unravel its mystery? Is it a theory, completely antipodal to Deterministic Sciences? By the way, if you take a careful look at the paragraph above, you would see a bunch of question marks (?). That’s Chaos! � With its presence, there shall always be those queries as to how will a system behave.

a.) What is Chaos? Let’s ask a simple question first? What does a chaotic system do? A chaotic system is not stable (its behavior changes with time) and it’s not periodic (it doesn’ t trace its path back and forth and hence it’s not periodic). So is it irregular motion? NO! A Chaotic system is an unpredictable system, whose behavior is highly dependent on its initial conditions and there is order in which it evolves to its unpredictable disordered behavior. You might think that it’s somewhere on the boundary between order and disorder but there is more to it.

Chaos is "orderly disorder created by simple processes." The genesis of this theory started with a study into instabilities in dynamical systems by physicist Henri Poincaré in 1900. He was interested in the mathematical equations that would describe the motion of planets around the sun. The philosophy of determinism speaks about the fact that every action in this universe is causal and can be predicted. The universe, in its behavior, unfolds with time in a predefined manner, according to the laws of physics. Every action in future or retrospect can accurately be predicted form the current happenings. Newton’s Laws a glaring example of the deterministic theory. The three laws of motion talk about the behavior of systems in a manner described by mathematical equations. The laws of motion, combined with the body of the knowledge concerning a specific science, were believed to provide




an accurate insight into the future behavior of a system. Newton believed that this could be applied to even study the motion of planets and described their paths in the future. For predicting the exact behavior of any system, whether it the convection currents in the atmosphere, the motion of a heavenly body or even a falling object, one needs to know the initial conditions of the system. The Newtonian theory says that the behavior can completely be determined and is exactly the same for the same initial conditions. The question is can we have exactly the same initial conditions? Real world systems behave based on measurements taken in reality. The initial conditions of a system can never be infinitely accurate. The scientific faculty has been lead to the belief that an almost accurate initial condition will lead to an almost accurate prediction of the behavior of the system, which is not the true case and this is what Chaos defies. We can never measure the initial conditions of a system accurately and some systems are so sensitive to these initial conditions that over a period of time, they completely diverge from what their behavior would been had they started with an initial condition seemingly indistinguishable from the current one. A good example of this is the tossing of a coin. No matter how hard one may try to let the system have the same initial conditions, it shall always be different and hence the result would vary from “head” or “ tail” from seemingly same initial conditions. From the above, we can infer that Uncertainty in dynamical systems doesn’ t arise due to the equations of behavior of the system since they are completely deterministic but due to the shift in the initial conditions of the system. The investigations of Henri Poincaré challenged the tacit assumption that an almost accurate measurement of the initial conditions was enough to predict their behavior correctly. When used for the case of measuring planetary movements, this can have a large impact on the predictions made, based on some initial conditions, since there will always be a greater degree of inaccuracy in measuring the initial conditions of such systems.




2. Some examples of “Chaos”

Real life is abuzz with examples from this realm and we would definitely like to touch upon a few, including the ones that were simple enough for us to try out within the time frame of the project.

(Example: 1) The most striking revelations about the existence of chaotic systems started with Edward Lorenz’s simulations in 1960 wherein he was trying to model a problem on weather prediction and he had a system of twelve equations that his computer setup was trying to solve. Based upon the results of one day, he wanted to check a particular pattern again and to save time, he decided to start from somewhere in the middle of the iteration rather than the beginning by looking at the value from a computer printout of the iteration. When he saw the new results, he was astonished since the predictions completely diverged from what they had been the day before. He noticed that the memory of the computer stored the numbers (solution of the equations) to six digits of decimal and he had, in order to save paper, printed only the first three digits. He had entered 0.506 instead of 0.506127 and hence, over a period of iteration, the current simulation had completely diverged from the results that he had the day before. A figure is included herewith, taken from Ian Stewart’s book on The Mathematics of Chaos.

Figure 1: Results from Lorenz’s atmospheric model. This has been taken from Ian Stewart’s book, “ Does God play Dice? The Mathematics of Chaos”




The difference between the starting points of the two curves is so small that it can be compared to flapping of the wings of a butterfly and rightly so, this effect came to be known as The Butterfly Effect and as Ian Stewart puts it in his book, The f l appi ng of a s i ngl e but t er f l y ' s wi ng t oday pr oduces a t i ny change i n t he st at e of t he at mospher e. Over a per i od of t i me, what t he at mospher e act ual l y does di ver ges f r om what i t woul d have done. So, i n a mont h' s t i me, a t or nado t hat woul d have devast at ed t he I ndonesi an coast doesn' t happen. Or maybe one t hat wasn' t goi ng t o happen does. (Ian Stewart, Does God Play Dice? The Mathematics of Chaos, pg. 141)

(Example: 2) A second set of example that sprung from Lorenz’s work thereafter was the Waterwheel experiment. He reduced down the set of twelve equations from the weather model and made a system of equation simpler to observe this behavior. Later on, it was found that he his new set of equation predicts the dynamics of a Waterwheel. Essentially, it consists of a set of cups and a stream of water flowing from some place at the top. Depending upon the speed of flow of water and the resulting motion and thereby the resulting angular velocity and the amount of water that collects in the cups, this system turns to show chaos and is highly dependent on the initial

. Figure 2: Lorenz’s Waterwheel experiment, the dynamical equations for which were unknowingly derived by Lorenz by stripping down hi model for atmospheric weather prediction




(Example 3)

Another example is regarding the prediction of biological populations. The equation would be simple if population just rises indefinitely, but the effect of predators and a limited food supply make this equation incorrect. The simplest equation that takes this into account is the following:

Next’s population = r * this year 's population * (1 - this year 's population)

In this equation, the population is a number between 0 and 1, where 1 represents the maximum possible population and 0 represents extinction. R is the growth rate. The question was, how does this parameter affect the equation? The obvious answer is that a high growth rate means that the population will settle down at a high population, while a low growth rate means that the population will settle down to a low number. This trend is true for some growth rates, but not for every one.

Robert May, a biologist, decided to see what would happen to the equation as the growth rate value changes. At low values of the growth rate, the population would settle down to a single number. For instance, if the growth rate value is 2.7, the population will settle down to .6292. As the growth rate increased, the final population would increase as well. Then, something weird happened. As soon as the growth rate passed 3, the line broke in two. Instead of settling down to a single population, it would jump between two different populations. It would be one value for one year, go to another value the next year, then repeat the cycle forever. Raising the growth rate a little more caused it to jump between four different values. As the parameter rose further, the line bifurcated (doubled) again. The bifurcations came faster and faster until suddenly, chaos appeared. Past a certain growth rate, it becomes impossible to predict the behavior of the equation. However, upon closer inspection, it is possible to see white strips. Looking closer at these strips reveals little windows of order, where the equation goes through the bifurcations again before returning to chaos. This self-similarity, the fact that the graph has an exact copy of itself hidden deep inside, came to be an important aspect of chaos.

The diagram below is the one being discussed above.




Figure 3: The Bifurcation Diagram for the population equation (James Gleich, “ Chaos - Making a new Science” , Pg 71)




(Example 4) This example is something for which we were able to set up an experiment and investigate the mathematical model too. The next chapter discussed about this example. The motion of a DOUBLE PENDULUM is a good example of a chaotic system. As shown below, the motion of such a system can be highly dependent upon the initial conditions (θ1 and θ2). From the plot below, we can see that the behavior for such a system can be highly non periodic.




Figure 4: The Double Pendulum Experiment and the response of θ1 and θ2 as function of time

(Example 5)

Another example of a chaotic system is the Magnetic Pendulum. The bob of the pendulum has a magnet attached to it and is under the effect of the underlying magnets, of reverse polarity, and is free to move. The motion of the pendulum is highly sensitive on the initial position from where this system is left to move. The pendulum will, after some time interval, hang exactly above one of the underlying magnets and which magnet the pendulum will hang on top of is very much sensitive to the initial conditions. We did perform this experiment too and were able to observe this behavior. We plan to demonstrate this behavior in our Final presentation. The next chapter, where we discuss our results, will talk about the same.




Figure 5: Setup for the Magnetic Pendulum Experiment

(Example 6) One of the most interesting examples that we sighted on Chaos was regarding the behavior of the solution of a system of two equations.

y = x2 + c (1) y = x (2)

We may start from some initial condition (initial value of x) and then get the corresponding value of “y” from the first equation. Now, this becomes our new “x” , as per the second equation and we keep on iterating hence. Basically, we have developed the following mapping:

f: x --> x2 + c.

The path followed by “y” is called the orbit of the system and the initial value of “x” with which we started is called the seed of the orbit. Hence { x, f(x), f2(x),….fn(x)} is the trajectory of the orbit.

Investigating the system for c=0,

will yield the following results:

All orbits approach either zero or infinity except for those with seed x = +/-1. The points zero and infinity are called attracting fixed points or sinks because they attract the orbits of the points around them while +/-1 are called repelling fixed points or sources for the opposite reason.

Investigating the system for values of c around c=1/4,

Y




X c >1/4 c = 1/4 -3/4 < c < 1/4 For c=1/4, for all seeds, for which the solution doesn’ t approach zero, it converges to x=0.5. For values of c (-0.75<c<0.25), the values of x around 1.5 repel away towards x=0.5. Hence x=1.5 is a source and x=0.5 is a sink in the system. For values of c<-0.75, the sink bifurcates into two sinks and hence the values of x, instead of converging to a particular point, start oscillating between two specific points and as the value of “c” decreases further, the period of oscillation increases until for values of “c” below -1.4 where in the system enters a chaotic regime and no particular behavior can be predicted. The value of “x” instead of oscillating between some values, starts moving in the entire solution space, say [-2, 2] in this case. The BIFURCATION diagram, as shown below, depicts the same.

Figure 6: The bifurcation diagram for the solution set of simultaneous equations discussed in Example 6.




3. Trying our hand at “Chaos”

However, if we do discover a complete theory, it should in time be understandable in broad principle by everyone, not just a few scientists. Then we shall all, philosophers, scientists, and just ordinary people, be able to take part in the discussion of the question of why it is that we and the universe exist.

-Stephen Hawking As rightly mentioned by great Stephen Hawking, ordinary graduate students like us got into getting a taste of this theory ourselves and this chapter happens to be an important pillar of our attempts on unraveling the mystery of “Chaos theory” .

Our Motivation

We are four graduate students (three from Aerospace and one from Mechanical Engineering), who are taking a course in Industrial Mathematics under the revered guidance of Dr. McCuan of the Mathematics department at Georgia Tech. This course caught our attention from its description since it was supposed to capture the mathematics of our everyday research problems, something meant for engineers. A project on a relevant topic was supposed to be the culmination of the course. Interestingly, our group had been caught up with some idea about doing experiments in Vibration Mechanics when suddenly one day Dr. McCuan mentioned to us about something that we could look for a prospective project topic in the realm of Chaos theory. It all started then and believe us, it has been an amazing experience. We won’ t say that we have unraveled all the mystery yet since this is a universal phenomenon with scores of examples still unnoticed. We have just tried to take a look at the surface, maybe even scratched it a bit. �




The following sections shall discuss about our attempts at modeling some systems, which researchers have proclaimed in the past, exhibit chaotic behavior.

a.) Double Pendulum This experiment lured us from the very beginning in its very simplistic setup but having a profound explanation to how this system would behave under different initial conditions. Figure 7 and Figure 8 show the schematic and the experimental setup for this experiment that we performed at the Aerospace Combustion Lab at Georgia Tech.

Figure 7: Schematic diagram of the Double Pendulum Experiment setup with all the concerned variables.

Numerical simulation of the problem behavior

x-axis

y-axis




The governing equations for the double pendulum can be derived using the “Principle of Variation” . When dynamical systems tend to get more complicated, as in this case, in terms of the action and reaction of forces, it makes much more sense to shift away from Newtonian mechanics and concentrate on Energy methods to describe the behavior of such a system. From Figure 7, it can be seen that the pendulum consists of two straight rods O1-O2 (mass m1 and length L1) and O2-O3 (mass m3 and length L3). Additional masses m2 and m3 are attached to the system at O2 and O3. The center of gravity of the rods is assumed at their middle point. Let the following variables be defined the way, as mentioned below: (x1, y1) : Co-Ordinates of the center of the rod (O1-O2) (x2, y2) : Co-Ordinates of the first additional mass, m2 (x3, y3) : Co-Ordinates of the center of the rod (O2-O3) (x4, y4) : Co-Ordinates of the second additional mass, m4 The dependence of the Cartesian co-ordinates on the angular co-ordinates of the system are as follows:

11

1 sin*2

θ��

��

�=l

x 11

1 cos*2

θ��

��

�−=l

y

112 sin* θlx = 112 cos* θly −=

1122

3 sin*sin*2

θθ ll

x +��

��

�= 1122

3 cos*cos*2

θθ ll

y −��

��

�−=

11224 sin*sin* θθ llx += 11223 cos*cos* θθ lly −−=

The PE and KE of the bobs and the rods are as follows:

111

1 cos*2

θglmPE −=

1122 cosθglmPE −=

��

��

� +−= 22

1133 cos2

cos θθ llgmPE

��

��

� +−= 22

1144 cos2

cos θθ llgmPE




21

2

111 22

1 θ��

��

�=l

mKE

21

2122 2

1 θ�lmKE =

( ) )]**cos*(2

[2

1212121

21

2

221

2133 θθθθθθ �� −+�

�

��

�+= lll

lmKE

( ) )]**cos*(*2[2

1212121

22

22

21

2144 θθθθθθ �� −++= llllmKE

L (Lagrangian) = ΣKE - ΣPE The Euler’s equations are as follow:

011

=��

��

�

∂∂−

∂∂

θθ �

L

dt

dL

022

=��

��

�

∂∂−

∂∂

θθ �

L

dt

dL

Combining all these, we can obtain a governing set of equations of the form,

( ) ( )

( ) ( ) 02

sinsin*sin*2

cos*cos*22

14131211

121214212132

2

2121421213

22

12

2

11

214

2131

=��

��

� ++++��

�

� −+−−

+

��

�

� −+−−

+��

�

�

��

�

�−�

�

��

�−+−

glmglmglmglm

llmllm

llmllm

lml

mlmlm

θθθθθθ

θθθθθθ

�

��

( ) ( )( )

( ) ( ) 02

sinsin*sin*2

2cos*cos

2423

221214212132

1

224

2

2322121212131

=��

��

� ++��

�

� −+−+

��

�

�

��

�

�−�

�

��

�−+−+−−

glmglm

llmllm

lml

mllllm

θθθθθθ

θθθθθθ

�

��

The coupled differential equations above were then solved, using the Runge-Kutta method, with varying initial conditions in a MATLAB. For the present work the standard




ODE solver (ode45) of MATLAB was used. “ode45” uses a 4th order Runge-Kutta scheme for solving a system of linear ODE’s. The MATLAB codes (* .m files) for numerical solution to this problem are attached herewith.

Experimental setup for the problem

Figure 8: The experimental set up of for the Double Pendulum problem

Figure 9: Zoom in look at the double pendulum Figure 8 shows the experimental setup for this system. It consists of two aluminum rods with additional masses attached to its ends and free to swing like a double pendulum. For different initial conditions, the double pendulum shows different sets of motion. The chaotic nature of the double pendulum is demonstrated by capturing the motion of the pendulum for two slightly different sets of initial conditions. To measure the position of the pendulum, a perforated inch board was used as a background. A particular coordinate system (as shown in Figure 7) was chosen about the point of suspension of the double pendulum. A high-speed camera having capability of taking pictures in the range of 60-5000 frames per second was used. As the double pendulum has a low frequency of

Computer Camera is connected to

Redlake High-Speed Camera

Double Pendulum

Graduated Board

Double Pendulum

Graduated Board




oscillation, the camera was used at its lowest mode. The data acquisition was done by coupling the camera to a desktop and storing the frames as JPEG files, which were then processed thoroughly to track the motion of the swinging pendulum.

Results and Discussion Our attempts at modeling the numerical and the experimental facets of the double pendulum problem can be studies under two different branches.

a.) We studied the nature of the solution as obtained by numerical simulation and by experimentation

b.) We also studied the varying nature of the solution when the system has almost

similar initial conditions. a.) We performed the experiment under varying initial conditions and the snapshots of all

the initial conditions along with the comparative plot of the numerical and experimental results is included herewith, as dp_plot_1, dp_plot_2, dp_plot_3, dp_plot_4, dp_plot_5, dp_plot_6, dp_plot_7, dp_plot_8 and dp_plot_9. As observed from these plots, the numerical plots for θθθθ1 and θθθθ2 have much higher amplitude and differ widely from the experimental trends. An obvious explanation into this difference is the non-inclusion of damping in our numerical formulation. There is a tangible amount of friction at the hinges O1 and O2 and the system is always subjected to damping, predominantly due to the fixtures. As a result, even though the numerical simulation may predict that one of the rods will make numerous complete rotations about its corresponding hinge (depicted by the angular distance value rising to as high as 15 to 20 radians) in the numerical plots, the experimental plot show that a complete rotation would happen only once or maybe twice (as depicted by the small amplitude values of θθθθ1 and θθθθ2. These factors result in a tangible difference in the numerical and experimental results for some of the runs. dp_plot_1 and dp_plot_2 show very similar trends towed by the numerical and the experimental results. However, as it can be inferred from these plots that the behavior of the system would definitely vary from one set of initial conditions and the next subsection will concentrate on how sensitive would this be to the change in initial conditions, hinting towards the very chaotic nature of this system. Damping, as mentioned before plays an important role in the difference of the numerical and experimental plots for some of the runs and damping not only influences the amplitude of the motion but the complete behavior of the system. The plots for the experimental values are only for short duration, less than ten seconds. This is because the system damps really fast and towards the end of the run, the oscillations become more or less periodic in nature. The loss in energy of the




system can only result in small variations in the angular distances traveled by the two rods and hence the motion more or less becomes predictable and eventually dies off. All result plots (dp_plot_1 through dp_plot_9) show the above.

b.) The second strain of evaluations that can be drawn from the numerical and the experimental results would point to the true chaotic nature of the double pendulum problem. We conducted these experiments for different set of initial conditions and we performed some of them under almost similar initial conditions and studied the behavior of the system. Please refer to the following sets of results:

(i) First set of similar initial conditions The plots dp_plot_7 and dp_plot_8 show the results of the first test run we did for similar initial conditions. The position from which the system is left free (initial condition) doesn’ t provide the system with enough potential energy to sustain motion. Damping, in addition, damps down the motion to an equilibrium state very quickly. As it can be seen from the plots, its only for the first few seconds that the motion of the pendulum shows some a non periodic nature and thereafter the motion dampens to a more or less periodic nature (since the amplitude drops down drastically). However, the most important thing to observe in comparing these runs is first few seconds. In spite of having almost indistinguishable initial conditions, the trajectory followed by the two bobs and the rods (indicated by the plots of θθθθ1 and θθθθ2 with respect to time) are different. This was the main aim for performing this test run. The next couple of runs that we performed with similar initial conditions are ones in which the system has a higher level of potential energy and hence the chaotic behavior can be observed for a longer time interval. The test run for this set has been able to show the inability to predict the behavior of the system since the behavior of the system is very sensitive to initial conditions and measuring the initial conditions with infinite accuracy is impossible. This is chaos. (ii) Second set of similar initial conditions The plots dp_plot_4 and dp_plot_5 show the results for the 2nd experimental run. The results shown are for two different cases with slightly different initial conditions. The plots show a comparison of numerical and experimental values of the variation of θθθθ1 and θθθθ2 with time. As the amplitude of oscillations for the upper arm of the pendulum is relatively smaller when compared to that for the lower arm, it can be noticed that the experimental prediction of θθθθ1 shows rapid damping with time as opposed to that for θθθθ2 for both the cases. The damping effect is an obvious consequence of air resistance. It is observed that for slightly different initial conditions, the trajectory traversed by both the arms is completely different whether the prediction is based on numerical computation or experimental observation. The reason for this behavior lies in the precision with which




the initial conditions can be applied for the system. Theoretically, an infinite precision in the initial conditions would amount to replication of the pendulum motion but for any real system the uncertainty always prevails. The large difference observed in numerical and the experimental observation is due to the inherent damping offered by any real system, which for the present study reflects in the form of large reduction in the amplitudes of motion. This behavior is in fact typical of any system that is chaotic, and thus forms the basis of present study. (iii) Third set of similar initial conditions The plots dp_plot_6 and dp_plot_9 show the results for the 3rd experimental run set carried out in this study. The four curves in each of the plots correspond to the Numerical and Experimental values of θθθθ1 and θθθθ2 with time. These runs (dp_plot_6 and dp_plot_9) correspond to almost similar initial conditions but it can be seen that the trajectory traversed by pendulum varies a lot. The graphs with smaller amplitude correspond to variation of θθθθ1 with time where as those with the higher ones refer to θθθθ2

variation. The plots indicate that for slight variation in initial condition the trajectory followed by the upper arm changes completely. It should be noticed that the differences in amplitudes of variation remain relatively high for the initial part of the run, which eventually reduces with time. A similar trend is followed by the lower arm as well. These three special test runs show that a small perturbation in the initial condition changes the dynamical behavior of the system that would eventually damps down to an equilibrium state. This signifies that the system does exhibit a chaotic phenomenon for the considered set of initial conditions.




b.) Magnetic Pendulum This experiment as inspired by one of the show pieces that we had and which happened to be our experimental setup. The behavior of a pendulum (a magnetic bob suspended from a fixed point by a rigid bar) under the action of underlying magnets of opposite polarity to the magnetic bob, a Magnetic Pendulum as it is called (Figure 9), also happens to be one of the many good examples of chaotic behavior of non linear dynamical systems. The experimental setup for this experiment is as shown in Figure 10.

Figure 10: Magnetic Pendulum

Figure 11: Experimental setup for the Magnetic Pendulum problem

Magnet 1

Magnet 2

Magnet 3

Magnetic bob

Pendulum

Redlake High-Speed Camera

Computer camera is connected to




Numerical simulation of the problem behavior The magnetic pendulum consists of three magnets kept at different spatial locations. The mass hanging vertically exhibits a chaotic motion (divergent behavior under seemingly same initial conditions). The force acting on the moving mass is due to its weight, the frictional resistance due to air, and the magnetic force exerted by the underlying magnets. If the magnetic force constant and the frictional resistance constant can be expressed using single parameters C and R, then the equation governing the motion of the pendulum can be written as,

( )

( )

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

=

=

−′′ ′+ − + =− + − +

−′′ ′+ − + =− + − +

where ( ix , iy ) refers to the spatial coordinates of the three magnets with respect to the origin (directly underneath the hanging magnetic bob). The above set of simultaneous differential equations was solved using the ODE45 solver in MATLAB, which utilize the 4th order Runge-Kutta method.

Experimental setup for the Magnetic Pendulum Figure 10 and 11 shows the magnetic pendulum set up that was used for the experiment. The magnetic pendulum consists of a rigid plastic pipe suspended from a fixed support, with a small magnet attached to its free end. The pendulum stand consists of a rigid base perpendicular to the pendulum arm, on which three magnets of nearly same magnetic strength were placed. A high-speed camera was used to capture the motion of the pendulum as it swings past the fixed magnets. The camera output was processed to obtain JPEG files for tracking the pendulum trajectory.




Results and discussion The behavior of magnetic pendulum was studied on the basis of numerical computations and experimental observations. The major objective was to find out how the system responds to changes in prescribed initial conditions. The study was broadly divided into following two categories.

a. Numerical prediction of the path traced by the magnetic pendulum for specific set of parameters.

b. Experimental observation of the behavior of magnetic pendulum for small

changes in initial conditions. a) The efforts pertaining to numerical computations was organized to determine the chaotic nature of the system. It was observed that the dynamical behavior of magnetic pendulum is closely associated with the initial conditions. The numerical computations involved solving the system of Ordinary Differential Equations (as described in previous section), which was accomplished using ode45 solver in MATLAB. The solution of the governing set of equations requires an initial condition to be specified, which turns out to be an important factor for this system. It is a fact that the precision with which an initial condition can be specified in numerical computations is relatively much higher than that for an experimental run. So, an attempt was made to assess the behavior of the system by specifying the initial conditions much more precisely (cor rect up to 2nd decimal place) for numer ical calculation, without paying much attention to replicating the system as shown in the exper imental setup. Computations were performed by using representative values of the coefficients C and R (which combine the effects of magnetic forces and frictional resistance) from available literature. For the present study C and R was fixed as 4.0 and 0.05. The plots mp_num_run1, mp_num_run2, mp_num_run3 and mp_num_run4 show the computational results obtained by solving the governing equations in MATLAB. It can be observed that the change in initial conditions (which corresponds to the spatial point from which the pendulum is released) for all the runs is almost negligible for all practical purposes. The computation was performed for time duration of 100 seconds and the plots in fact refer to the projected trajectory of the pendulum in the XY plane. Thus, it can be observed that for four different sets of initial conditions with small perturbations, the pendulum shows completely different behavior. To be precise, the equilibrium position of the pendulum after 100 seconds turns out to be concentrated around different magnets. This behavior is typical of any chaotic system, which proves the point that the numerical computations do convey the chaotic nature of the system under consideration. b) In order to validate our conclusions from numerical computations, experimental observations were made. In all, four experimental runs were performed with slight




changes in initial conditions. The path traced by the magnetic pendulum was recorded using a digital camera and the data was processed manually to get the trajectory of the pendulum motion. The plots mp_exp_run1, mp_exp_run2, mp_exp_run3 and mp_exp_run4 show the experimental results obtained. The experimental runs were recorded for about 15 to 17 seconds and hence the plots refer to the trajectory traced by the pendulum for duration of about 17 seconds. An interesting aspect of this experiment was the specified initial conditions. For all practical purposes, the initial conditions were kept identical but since the precision with which any initial condition can be specified has practical limitations, the response of the system was different for different runs. This brought forth the interesting fact that the nature of this system highly depends on initial conditions. The practical limitations in specifying initial conditions with infinite accuracy, reflects as four different sets of initial conditions with small variations, for the different runs considered. It is thus observed that, for all the four runs, the solution regime varies a lot giving perfect evidence of the chaotic nature of the system, a fact that was already confirmed using numerical computations.




c.) Vibrating Spr ing This was another experiment that we tried to conduct. Some probing into literature had hinted that a ferromagnetic string fixed at two ends and under the effect of an electromagnet shows some oscillatory motion and for some characteristic of this system, it starts exhibiting a chaotic motion. We were unfortunate with this experiment and were not able to consummate it. With the available resources, we were not able to locate the chaotic regime for this problem and all the runs that we performed had the string vibrating in complete predictable periodic motion with a fixed frequency (10 Hz). Figure 11 shows the schematic setup that we had for this experiment.

Figure 12: Experimental setup for the Vibrating String




Figure 13: Vibrating String Experimental Setup The setup for this experiment included an electromagnet, a steel string, fixed supports, a function generator, an oscilloscope, a video camera, a point light source and a graduated board. As per literature, a simple string attached between two fixed support, when excited electro-magnetically, shows chaotic behavior (behavior highly sensitive to initial conditions) for certain range of frequency and amplitude of the electrical input signal. A function generator was used to obtain a sinusoidal waveform, which served as a power source for magnetizing the electromagnet. The change in polarity of the input signal causes a forced oscillation of the string, which ultimately starts vibrating about its mean position. The shadow of the vibrating string was magnified using a light source and was measured against a graduated background. A video camera was used to track the motion of the string. It was observed that the amplitude of the oscillation of the steel string was very small and the frequency of vibration was very large. This made it difficult to track the motion of the midpoint of the vibrating string with the available camera. We would need a much stronger electromagnet in the future.

Ruler

White Board

Wire

Electromagnet

Function Generator

Oscilloscope




d.) Swinging Spr ing This experiment was performed in ACE lab at the Mathematic Department, making use of the available high-speed digital camera. Figure 12 shows the experimental setup used for this system.

Figure 14: Experimental setup for the Swinging Spring The setup consists of a simple spring suspended from a fixed support. An additional mass of known value was fastened to the free end of the spring. The spring constant was found out experimentally by measuring the spring extension for different sets of know masses. A graduated background was used to accurately track the motion of the swinging spring. The movie obtained form the digital camera was processed using Window Movie Maker in Windows XP, to get an assessment of the Cartesian X and Y positions of the spring were made with time.

Graph Paper

Hanging Rod

Spring and Mass




4. Conclusion

It has been a really enriching experience trying to model experiments and do a complete theoretical evaluation simultaneously in an attempt to observe a new concept. “Chaos” isn’ t something new, considering Lorenz’s findings in 1960, but it certainly is one of the topics that our group decided to work on since it would be something we hadn’ t done before. This was something different from the everyday research that we conduct at our labs. To begin with, we have been able to conduct a good literature survey in the given possible time frame and have been able to understand the important theoretical concepts. Chaos is irregularity, but an unstable dynamical system approaches it in a very regular manner. The basic nature of these chaotic dynamical systems is their sensitivity to the initial conditions. A probe into the behavior of a system of two simultaneous equations (Example 6 in the Introduction (Chapter 1)) has been really helpful in understanding the above and the way the concepts behind the bifurcation diagram. In the light of the above, we have attempted to model some examples, both their experimental and numerical setup. We have successfully been able to demonstrate the chaotic behavior of systems like the Double Pendulum. We have also done a rigorous theoretical model for this example by deriving the equations of motion using the Hamilton’s principle and the corresponding Lagrangian equations and then solved the coupled differential equations using numerical solvers in MATLAB. In fact, there are similar systems that can be extended from the same concept which have “n” (n-pendulums) such simple pendulums instead of 2 (double pendulum). We have also been able to model the Magnetic Pendulum and show that for almost similar initial conditions (leaving the magnetic bob free from almost the same position), the underlying magnet on which the bob rests differs and hence due to unpredictability of this behavior, this system behaves chaotically. As before, we have been able to perform a rigorous numerical evaluation as well as an experimental evaluation of the behavior of the system. The Vibrating String experiment has been one of the examples where we haven’ t been able to catch the system parameters for which this system enters the chaotic regime. This is definitely something that we intend to follow up on in the future. The Swing Spring experiment also shows varied behavior under the same set of initial conditions but we haven’ t been able to numerically model this chaotic behavior. This is also something that we intend to follow up on in the future.




5. The Road Ahead…

Although we have been able to model some examples successfully and demonstrated their chaotic behavior, we still have the following potential areas of improvement:

a.) It certainly is to be seen as to how we can develop a better experimental fixture for the Double Pendulum model. As of now, the experimental set up induces a lot of damping from the point “go” and we have been able to show the unpredictable behavior for this system under different sets of initial conditions only for few seconds. The motion thereafter damps off and the system approaches more or less a periodic equilibrium position. It is also to be seen if we can have a way to induce a forcing function into this system and then study the behavior of the same both numerically and experimentally.

b.) We have been able to model the Magnetic Pendulum example too but it is still to

be seen as to for what values of the height of the pendulum from the plane of the underlying magnets would the behavior be unpredictable. There are some unanswered questions regarding this model, such as How would this system behave if some of the magnets were attracting and some of them repelling. We should be able to numerically model this situation.

ANALYTICAL EVALUATION FOR (a.) and (b.) For the cases above, where we performed a numerical evaluation of the dynamical equations derived, we would like to perform an analytical analysis. It still needs to be seen as to how would the above set of equations behave when the points in the neighborhood one initial condition approach that initial condition. This would help us to appreciate the sensitivity of the system to the initial conditions better.

c.) With the Vibrating String experiment, our biggest drawback has been the use of a

weak electromagnet. Within the available resources and the time frame, we guess this is one of the major reasons as to why our attempts have failed to catch the chaotic regime. We have the freedom to vary the current amplitude and frequency only to within a small interval. We would definitely like to improve on this experiment by using a suitable electromagnet for the purpose and also a function generator that would output higher voltage amplitude.

d.) With the Swinging Spring, most of our literature search has directed us to

mathematical modeling of some modes of periodic behavior of the system and within the available time frame we haven’ t been able to model the system numerically. Apart from this, we would also like to have a better experimental fixture for this set up. The bob hanging from the spring should have the degree of




freedom to bounce back above the plane of support and this is something we would like to improve on in the future.




6. References

Since this was our first attempt into understanding “Chaos” , we found the references at the following websites more useful. We were working at the grass roots level and needed to implement some experiments and hence these website proved to mode much more helpful that actual textbooks.

[1]. http://www-chaos.umd.edu/ [2]. http://math.bu.edu/DYSYS/ [3]. http://order.ph.utexas.edu/chaos/ [4]. http://www.math.sunysb.edu/dynamics/ [5]. http://hypertextbook.com/chaos/http://hypertextbook.com/chaos/ [6]. http://www.wfu.edu/~petrejh4/chaosind.htm [7]. http://www.apmaths.uwo.ca/~bfraser/nonlinearlab.html [8]. http://www.around.com/chaos.html [9]. http://www.yiin.ca/chaos/ [10]. http://www.nbi.dk/CATS/research/ar96/node31.html [11]. http://www.geocities.com/CapeCanaveral/Lab/4430/volterra.htm [12]. http://www.maths.tcd.ie/~plynch/SwingingSpring/SS_Home_Page.html [13]. http://scienceworld.wolfram.com/physics/DoublePendulum.html

Is it all about Chaos? - ACE Mathematics Laboratory · Is it all about Chaos? Final Project Report...

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