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noun.What scientists and mathematicians mean by chaos is very much related to the spirit of the definitions given above. We state that systems are chaotic if they:
1. A condition or place of total disorder or confusion: ``emotions in complete chaos.'' 2. Often [Chaos]. The disordered state of unformed matter and infinite space supposed by some religious cosmological views to have existed prior to the ordered universe. 3. (Obsolete). A vast abyss or chasm.
###################################################### # Begin by loading plot library with(plots): # # Animate the curve animate([sin(u*t*2)+cos(u*t*3),sin(u*t)-cos((u*t)^2), t=0..1], u=0..2*Pi, frames=60,numpoints=100); ######################################################The path traced out by end of the line as the animation proceeds certainly looks chaotic (in the general meaning of that word) as shown below.
However, the general meaning of chaos implies unpredictability of the path. Certainly as you trace the animation out frame-by-frame you would be hard put to guess the next move of the end of the line, but, since the curve is generated from a mathematical formula, you can in fact predict exactly what the curve will do next in the animation and what its final shape will be. So what might appear to the naked eye as infathomably complex is, in reality, governed by a relatively simple mathematical expression. In essence, there is no difference between the complex path shown above and, say, a path along a straight line or sinusoidal curve. Its simply that a straight line path leads the eye in an obvious way from the start point to the end point.
#################################################### # The animation for a line is particularly simple # if we use a parametric description. If the slope # is 1, then animate([u*t, u*t, t=0..1], u=0..2, frames=20); # # produces a boring path traced out by the end of # the line as the animation moves along ######################################################
The trick, in practice, of predicting the path of something or the future course of an evolving system described by a complex algorithm comes in the sensitivity of the algorithm to small changes in the initial point. Let's take the case of a straight line and the complex curve we looked at previously. If the line is described by y = m*x + b, then making a change in the starting x point amounts to shifting the origin, i.e. we make x -> x + a where a is the small shift in the starting point. We know precisely what effect this has on the line mathematically. We also can see "by eye" what effect this would have in predicting where we would end up. There's no need to use Maple to know that the line ends up a distance a from the previous end point predicted before the shift. For the case of our complex curve, things get even more complicated than they were before. For example, let's consider what a small shift in the initial position does.
##################################################################### # Just add in the shift, in this case 0.1 animate([sin(0.1+u*t*2)+cos(0.1+u*t*3),sin(0.1+u*t)-cos((0.1+u*t)^2), t=0..1], u=0..2*Pi, frames=60,numpoints=100); #####################################################################
Now we note that our animated curve generally follows the same loopy pattern we saw before, BUT it ends up considerably further from the previous endpoint than the 0.1 shift might have led us to believe (compare the figure below with our previous one if you didn't do the Maple animation)
The essence of chaos in science is just that: a relatively complex behavior which is strictly governed by a mathematical algorithm, but, is nonetheless unpredictable due to sensitivity to initial conditions. So, although in principle we can predict how a system will behave to an arbitrary level of precision, in practice we can't find the initial starting point of the system accurately enough to be able to predict in detail what will happen beyond a short period of time. Small mismeasurements eventually add up to a big discrepancy between calculated and observed behavior.
The surefire way to have a system described by an algorithm that exhibits chaotic behavior is to have it be nonlinear. The importance of studying chaotic behavior lies in the fact that most systems encountered in the real world are nonlinear to some extent and either exhibit chaotic behavior or can be made to exhibit it. Prime examples are weather prediction, population kinetics (i.e. fluctuations in populations from generation to generation), fluid flow, mechanical and electrical oscillatory phenomena (e.g. heart beats or the electrical activity of your brain), the tumbling motion of the moon Hyperion in its orbit around the planet Saturn, economic systems, and many, many other phenomena. In fact, chaos is observed in so many systems in the real world that some scientists rank the understanding of chaos as being as important as the theories of relativity and quantum mechanics in that its ramifications stretch into every aspect of scientific study.
Nonetheless, how do you know nonlinear behavior when you see it? Let's start by looking at an example. A relatively simple one is shown in the diagram below.
The picture depicts a pendulum with a magnet (blue) as the bob on the end of a rigid rod which is free to swing on a supporting bar. Two other magnets (red) are fixed in position on either side of the equilibrium position (in the absence of the red magnets) of the blue magnet. The red magnets must be placed so that pendulum can rest in equilibrium with the blue magnet directly above either of the fixed magnets. Hence we have a system with two possible equilibrium positions. If the pendulum is pulled aside along the y axis and released, it quickly begins to execute an extremely complicated motion. Since the magnetic force is a strong function of the relative distance between the magnets and the magnetic forces can provide acceleration, deceleration, and damping (the damping force comes about because of induced currents, a topic you will learn more about in Physics 2 or 151). There is also damping due to friction of the pendulum support on the supporting rod.
The damping eventually causes the blue magnet to move to one of the equilibrium points, BUT which one? That depends very much on the initial condition. If the pendulum starts its motion ever so slightly closer to one magnet than the other, then its motion will eventually become highly perturbed. The complexity of the motion is high enough so as to make it nearly impossible to determine which equilibrium position the pendulum will choose given certain starting positions. Of course the motion of the pendulum is strictly governed by Newton's Laws so we have a mathematical description of the motion. Certainly the motion is complex and sensitive to initial conditions. Therefore, this is a perfect example of chaotic motion!
The figures below show the path of the pendulum magnet as projected on the plane with the fixed magnets. The white dot represents the position of one fixed magnet while the blue dot marks the position of the other fixed magnet. The colored path shows that the pendulum follows a complicated path to its eventual equilibrium position around one of the fixed magnets.
The next figure displays which equilibrium position the magnet winds up on according to its initial position as projected on the plane containing the fixed magnets. The blue and white regions show initial positions which correspond to the magnet coming to equilibrium around either the blue or white fixed magnet.
If we could blow up the region around the boundaries between blue and white areas, we would find that they are not infinitely sharp. Instead, we would see a complex structure which is termed a fractal. Fractals have fractional dimensions and the unique property of self-similarity to all levels of magnification. If you magnify any part of a fractal, you see a minature recreation of the overall fractal structure repeated on the small scale. Magnify a small piece of any part of the small structure and you see the overall structure repeated again and again, ad infinitum. The coastlines of continents and the structure of snowflakes are just two of the many examples of fractals found in nature. The most famous shape among the fractals is the Mandelbrot set shown below. Click on the image and you will see a movie that zooms in on the figure. As you go to higher and higher magnification, you see the same shape repeated over and over, albeit with surprising variations on combinations of shapes.
Fractals, as you can well imagine, are well-represented on the Net. You can find nifty pictures like the ones below in a repository in France.
Click on the above and wait if you want to
see more pictures.
The differential equation describing the motion of a pendulum that is damped and driven (i.e. some outside force provides the energy to keep the pendulum in motion) can be described most conveniently in terms of angles rather than position. This convention is explained in next week's readings but we've already mentioned it in the discussion of trigonometric functions. We simply say for now that we define angles in terms of radians, where the connection between displacement and angle is
where s is the arc-length or distance the pendulum swings along its arc and R is simply the length of the pendulum arm (i.e. the radius of its swinging motion). Given the angular displacement, we can, by analogy with straight line motion, define the angular velocity and angular acceleration
Keeping these terms in mind, you will learn in Physics 150 that the equation of motion we want, in angular terms, can be written down as follows:
with b and C being constants which characterize the amount of damping and the strength of the driving force, respectively. The cosine term multiplying C specifies the frequency of the driving force. This differential equation is clearly second order in the angular displacement since the angular acceleration on the left-hand side of the equation is the second derivative of the angular displacement. Since the angular velocity in the damping term (1st term on the right-hand side (which we will abbreviate as rhs) of the equation) is just the first derivative of the angular displacement and the driving force term is independent of the pendulum displacement, the differential equation is linear except for the sine in the middle term on the rhs of the equation. The sine term is what makes the differential equation nonlinear. Physically, this term simply relates the effect of the gravitational force on a pendulum bob. Mathematically, it provides the coupling with the other terms on the rhs of the equation to make the pendulum capable of chaotic motion for particular values of the damping and driving constants.
One of the most intellectually appealing aspects of chaos is its intimate mathematical connection to other curious entities such as fractals, Penrose tilings, and quasi-crystals. You can interactively explore these on the Web. Try the U. of Minnesota Geometry Center site for an interactive quasi-tiler for example in generating interestingly complex and colorful pictures. Another game showing the complexity of seemingly simple equations is orbifold pinball, which shows how hard it would be to play pinball on a curved surface. We will come back to this interesting notion of motion in a curved space when we discuss Einstein's theory of gravity later.
