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Question: What is the exact difference between a linear and a non-linear 
differential equation, and how does this difference transfer over to the 
physical world?  For example, what is the difference between a linear and non-
linear system.  I suppose my exact question is what exactly is being said when 
a system is described as having "non-linear" behavior- perhaps a driven, 
damped oscillator might be an example (such as a mass on a spring that is 
somehow driven and damped).  I apologize if the question is not very clear but 
I am having trouble with this concept.
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This is hard, but I will try to explain as much as I can in the 
simplest manner possible.  This is a "new" area and was a great surprise when 
it became clear.  Linear systems are systems in which the forces are either 
constants or are linear functions of space variables.  A spring and a mass 
(ideal spring) are exactly a linear system.  The motion for such a system is 
very simple and represents simple oscillatory motion (often easily represented 
by sines and cosines of time).  Non-linear systems have forces that are not 
linear functions of the space variables and the surprise is that the non-
linear systems have very little long time predictability.  In other words, two 
such systems that started out with very close to the same initial conditions 
(position, velocity) can end up after a long enough time very far apart.  This
means that we cannot make accurate predictions over long times.  The weather 
is an excellent example. It appears that we cannot get accurate predictions 
from our computer models for longer than about 10 to 14 days.  Running them 
over this time will not be dependable.  It is not a problem of inadequate 
starting data, but has to do with the nature of the forces between air masses.  
Let me know if this was an understandable answer.
Samuel B. Bowen
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