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Direction fields, fractals, and damped and forced vibrations

The direction field of a first order differential equation

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is obtained by drawing a small line segment centered at the point (x,y) and with slope f(x,y). For example, the equation

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has this direction field. A solution to the differential equation should have tangents in the direction field. Here is the same direction field tex2html_wrap_inline63 with some of the solutions laid upon it. Some other examples are:

tex2html_wrap_inline65 direction field solutions

tex2html_wrap_inline67 direction field solutions

tex2html_wrap_inline69 direction field solutions


The Mandelbrot set is obtained by iterating the complex function

Given any c in the complex numbers let tex2html_wrap_inline75 , tex2html_wrap_inline77 , and so forth, so that tex2html_wrap_inline79 . If tex2html_wrap_inline81 goes off to infinity, then c is not in the Mandelbrot set, otherwise it is. Hence, the subset M of the complex plane is defined by:

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If we color the set according to how fast the tex2html_wrap_inline89 's escape to infinity, we get the following picture: mandelbrot set

Another popular fractal, is based on Newton's method for find roots. To find r such that f(r)=0 we start with an arbitrary guess tex2html_wrap_inline95 and then iterate:

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In the case, tex2html_wrap_inline99 , this would be:

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Notice that this makes sense even when tex2html_wrap_inline103 is a complex number. In the complex numbers, there are three roots to the equation

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namely 1, , and tex2html_wrap_inline111 , all on the unit circle and exactly 120 degrees apart. The Newton's method fractal is obtained by plotting each point in the complex plane one of three colors depending on which of the three roots Newton's method converges to when started with that value. The interesting thing about this fractal is that every boundary point of any two of the colors is also a boundary point of the third color.

Many more fractal programs are out there, see Fractals.


The equation:

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where a, b, and c are positive constants is known as the damped oscillator. The constant b is determined by the amount of friction in the system. The equation

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has no friction and its solution set tex2html_wrap_inline125 or equivalently, , consists of various sine curves.

If we raise the constant term b but keep it small, say

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we get oscillating solutions which quickly die off, e.g., functions of the form

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for negative tex2html_wrap_inline135 . If we raise it very high, say

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we get solutions which do not oscillate, in fact, cross the time axis at most once and have the form tex2html_wrap_inline139 for negative and . The critial value for b is 2:

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where we get solutions of the form tex2html_wrap_inline151 , where it looks damped but changing the equation by just a little would bring us over into the oscillating range.

If we add a outside imposed force onto our system we get an equation of the form:

displaymath153

For example,

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or another example

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The equation

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has solutions of the form:

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Except in the interesting case when . In this case the solution set is:

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The function tex2html_wrap_inline167 is an oscillating blow up. A famous example of an oscillating system which blew up, is the Tacoma bridge disaster. For pictures (and video) visit the site: Tacoma bridge.


Executable and source (part 3): progs4.zip. These programs are written in TrueBasic for MSDos computers and zipped into an archive using PKZIP (use PKUNZIP) to unzip.




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Next: About this document

Arnold Miller
Fri Dec 13 15:52:05 CST 1996