# Chapter 1: Differential Equations and their Solutions

Click here to see the first 35 Pages

## Section 1: First Order ODE: Existence and Uniqueness

This movie shows the metaphorical “smoke particle blowing in
the wind” mentioned on page 5 of the text. The path of the smoke
particle is given by a dotted curve. The dots are laid down at a
constant rate, so the distance between them gives an indication of the
wind speed along the path. Notice how the wind keeps changing
direction both in space and in time, and that the smoke particle keeps
changing its direction to that of the local wind direction at that
time.

NOTE: This movie was produced by a freely available mathematical visualization program called 3D-XplorMath. We encourage you to use this program to visualize differential equations. It is available in both Mac and cross-platform Java versions at 3d-xplormath.org.

Here is a whole family of one- , two-, and three-dimensional ODE applets produced using the Java version.
(See: 3D-XplorMath ODE Gallery)
We highly recommend these applets
as a default playground for experimenting with all types of ODE.

## Section 5: Chaos — Or a Butterfly Spoils Laplace's Dream

### Lorenz ODE

This movie shows the evolution of a solution of the Lorenz ODE. This is the equation studied by Edward Lorenz that led him to
make his famous remark about the flapping of a butterfly's wing in
Brazil setting off a tornado in Texas.

The important thing to notice about these movies is the sudden jumps
of the orbit between the upper and lower loops. Two orbits with very
close initial conditions will stay close for a reasonably long time,
but then one of them will make the jump considerably earlier than the
other, and after that the solutions are far apart and
un-correlated. This is the essence of chaotic behavior.

### Rössler System

Here is a movie of another famous chaotic equation, the Rössler
System. Here the point to notice is the way an orbit slows down
greatly and then speeds up again as it moves around the phase
space. Points that start out close together in the phase space will at
first remain close together, but one will experience one of these
speed changes well before the other and after that their paths will be
far apart and uncorrelated.