09.18.2016

Dear Student,

Before I begin comment on the lectures and homework for Week 5, I would appreciate it if you could let me know whether you have completed the lectures and homework for Week 4 (Lectures 11 and 12).  Just a brief email note would be adequate.

Our class schedule for Week 5 prescribes that you should complete Lectures 12, 13, and 14 and the associated homework by Sunday, September 25.  I hope that the comments and suggestions below will help you as you work through this material.
 Week 5 -- Content Summary for Lectures 12, 13, and 14

Lectures 12, 13, and 14 comprise a short unit on non-linear systems of differential equations. The analysis of such systems is important in many engineering applications.

Recall that in our discussion of linear systems of differential equations y'(t) = A y(t) + g(t) where A is a constant matrix, we showed that we can find all solutions of the system by using the Method of Undetermined Coefficients for some choices of g(t), and more generally by the Method of Variation of Parameters and also by using the exponential matrix y.  We showed that systems of linear differential equations of order 2 or higher can also be solved completely by increasing the size of the matrix A. 

By way of contrast, complete algebraic solutions of systems of non-linear differential equations are almost impossible to find!  However, we will show in this unit that even though general solutions of systems of non-linear differential equations are not available, we can learn much about the behavior of the solutions of such systems by approximating them with linear systems.  This process is called linearization of the non-linear system near equilibrium states of such systems.

Physical systems often have equilibrium positions or states. (In a mechanical system, the equilibrium position is often a rest position; in a chemical system, the equilibrium state might be a steady concentration of a compound in the system.)  An important question about such systems is:  Are these equilibrium positions or states stable or unstable?  That is, if the system is disturbed from an equilibrium state, will it return to that equilibrium state in time, or will it move further and further away from equilibrium? (i.e., "fly off" or "blow up" or whatever other term you want to use for something bad happening) as time goes on) We refer to this sort of question about the behavior of time-dependent systems as stability behavior or asymptotic behavior of the system.

Lecture 12 introduces the mathematical terminology associated with the analysis of the stability behavior of non-linear systems of two differential equations of first order. Many important physical systems can be modeled with two variables, and the analysis of the two-variable case can be pictured in the plane very nicely.  Higher dimensional systems can be analyzed in a similar way from an analytic point of view, but the rich graphical interpretations are more difficult, so we will stick to the two dimensional case in this course, except for a brief foray into n by n systems at the beginning of Lecture 12

Lecture 13 gives a complete stability analysis of all possible systems of linear differential equations in two variables in terms of eigenvalues and eigenvectors of the matrix of the system. These cases are summarized nicely at the beginning of Lecture 14.  You really need to know exactly how the eigenvalues and eigenvectors of any linear system completely describe the asymptotic behavior of the system. 

In Lectures 13 and 14 the method of linearization for non-linear systems is developed and applied.  This is a very slick method that produces a lot of useful information about the stability and asymptotic behavior of the solutions of such systems even though there are no methods to actually compute exact solutions.  The stability analysis of linear systems in Lecture 13 is quite detailed, with plenty of examples.  You should find the quick summary of all of the cases at the beginning of Lecture 14 to be especially helpful when you do the homework.

I hope that you find these explanatory comments helpful as you study the lectures and do the homework this week.  Please contact me by e-mail or cell phone (217-840-2871) if you have any questions, and I will do my best to clear them up.

Have a great week!

Tony Peressini

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