ISSUES RELATED TO THE PROBLEM

THIS PROBLEM IS INTENDED TO INTRODUCE MODELING OF DYNAMIC SYSTEMS, AND IN PARTICULAR TO DEVELOP AN UNDERSTANDING FOR THEIR QUALITATIVE BEHAVIOR, WITHOUT REQUIRING THE SOLUTION OF DIFFERENTIAL EQUATIONS. DIFFERENCE EQUATIONS ARE USED INSTEAD.

This problem may look rather dry, but actually it is not. Linear difference equations such as the ones cited are commonly used in modeling linear systems of many kinds. Often analysts want to investigate the limiting behavior of the system: will it predict a decay to zero, unbounded growth, or what? Deeply imbedded in the discussion is the size of the solution,
so there must be an agreed on "norm" or way of measuring the size of the solution vector.

This problem is really intended to introduce the student to some of the key concepts of linear computation, without getting deep into jargon.

You have probably realized that this problem is closely related to the analysis of the matrix A,
| .989899 .10101 |
A = | |
| -0.10101 1.0101 |

and in particular, to its eigenvalues and eigenvectors. It is possible to learn something without ever mentioning this, however. Note that the above matrix has one eigenvalues that are complex. This is because the system exhibits oscillatory behavior. Note that the real parts have positive signs. This is the indicator that the system will be "blowing up".

Output =

{{xn1 -> 0.10101 vn0 + 0.989899 xn0, 
 
   vn1 -> 1.0101 vn0 - 0.10101 xn0}}

Input := 

A = { {.989899, .10101},{ -.10101, 1.0101}}

Output =

{{0.989899, 0.10101}, {-0.10101, 1.0101}}

Input := 

Eigenvalues[A]

Output =

{0.999999 + 0.100504 I, 
 
  0.999999 - 0.100504 I}

If the student approaches the problem with Lagrange multipliers, the Lagrange multipliers will turn out to be the square roots of the eigenvalues of ATA.

(1) The solution to this part can be carried out as a minimax problem in a function of two variables, with a constraint. See the solution below.
(2) & (3). If we bound the energy with the largest multiplier to the power n, where n is the step number we get a "sure-fire" bound which is considerably greater than the actual value of energy that we would get if we took the solution out to n steps. The reason is explained below. The student might try the average of the two multipliers that he or she finds. That actually gives a fairly good estimate of the energy as a function of step number.