REPLY First order oscillation (SD6462)

[SDMAIL yaman barlas]

Posted by  yaman barlas <ybarlas at boun.edu.tr>

Dear Cory;

It IS possible for a 1st order time-discrete model (i.e difference) equation to
oscillate. It does NOT even have to be non-linear. Here is a short proof: take the
simplest linear, constant-coefficient, 1st order equation x(k+1) = ax(k). 
Now try an a value between -1 and 0.  You will get damped oscillations. For
a<-1, you will get growing oscillations. You may want to play with different
a's. 

As for the interpretation: there is NO numerical stability/error problem here
at all. When you iterate this first order model, what you obtain is actually its
EXACT mathematical behavior. So, the true behavior of a first order
time-discrete system can be oscillatory. 

What we are used to hear in SD community ('first order systems can not
oscillate') assumes the model is continuous. 

It is great that you brought this up. The implication of this important
difference between continuous and discrete systems is that we must be  careful
and EXPLICIT in our assumption in this respect. When we build a continuous
model, we must mean it and must never use arbitrary/careless dt values. (Such
dt values would imply some 'arbitrary' discrete-time models, with potentially
very different dynamics). Similarly, when we build time-discrete models and use
dt=1, we must again mean it and know the fact that would be delaing with quite
different dynamics. Time continuous and discrete models are quite different
creatures. 

About your final question on chaos in the first order discrete logistic model,
yes, it is well known with this property. (The articles you refer to are very
important classics). Since this particular model is non-linear, it CAN exhibit
strange attractors and chaos, even though it is first order. Again, here is an
example: The simplest discrete logistic equation is: x(k+1) = ax(k)(1-x(k)). 
It is known to start exhibiting near chaotic behavior,  after a=3.8 and becoming
wilder and wilder as it approaches 4.   Perhaps you may want to play with it?
best wishes,
Yaman Barlas
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Professor,  Industrial Engineering Dept.            
Bogazici University,                  
34342 Bebek, Istanbul, TURKEY              
Posted by  yaman barlas <ybarlas at boun.edu.tr>
posting date  Fri, 1 Jun 2007 19:44:18 +0300