Direction fields, fractals, and damped and forced vibrations
The direction field of a first order differential equation
is obtained by drawing a small line segment centered at the point (x,y) and with slope f(x,y). For example, the equation
has this AAAdir-e.gifBBB direction field. CCC
A solution to the differential equation should have tangents
in the direction field. Here is the same direction field
with some of the AAAdir-e2.gifBBB solutions CCC
laid upon it.
Some other examples are:
AAAdir-p.gifBBB direction field CCC
AAAdir-p2.gifBBB solutions CCC
AAAdir-s.gifBBB direction field CCC
AAAdir-s2.gifBBB solutions CCC
AAAdir-d.gifBBB direction field CCC
AAAdir-d2.gifBBB solutions CCC
The Mandelbrot set is obtained by iterating the complex function
Given any c in the complex numbers let 
, 
,
and so forth, so that 
. If 
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:
If we color the set according to how fast the 
's escape to
infinity, we get the following picture:
AAAmandel.gifBBB mandelbrot set CCC
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 
and then iterate:
In the case, 
, this would be:
Notice that this makes sense even when 
is a complex number.
In the complex numbers, there are three roots to the equation
namely 1, , and 
, all on the unit circle
and exactly 120 degrees apart. The
AAAnewt.gifBBB Newton's method fractal CCC
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 AAAfractal.txtBBB Fractals. CCC
The equation:
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
has no friction and its solution set 
or equivalently, , consists of various
AAAdamp1.gifBBBsine curves. CCC
If we raise the constant term b but keep it small, say
we get oscillating AAAdamp2.gifBBBsolutions CCC which quickly die off, e.g., functions of the form
for negative 
.
If we raise it very high, say
we get
AAAdamp4.gifBBBsolutions CCC
which do not oscillate, in fact, cross the time axis at most
once and have the form 
for negative and . The critial value
for b is 2:
where we get
AAAdamp3.gifBBBsolutions CCC
of the form 
, 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:
For AAAdamp5.gifBBB example, CCC
or another AAAdamp6.gifBBB exampleCCC
The equation
has solutions of the form:
Except in the interesting case when . In this case the AAAdamp7.gifBBB solution set CCC is:
The function 
is an oscillating blow up.
A famous example of an oscillating system which blew up, is
the AAAtacoma.gifBBB Tacoma bridge disaster. CCC
For pictures (and video) visit the site:
AAAhttp://www.civeng.carleton.ca/Exhibits/"
Tacoma bridge. CCC