University of Notre Dame
Aerospace and Mechanical Engineering

ME 698: Geometric Nonlinear Control
Homework 1

B. Goodwine
Fall, 1999
Issued: August 26, 1999
Due: September 2, 1999
  1. (From Sastry's text). A friend tells you that a good way to compute the square root of two is to iterate the map:


    1. Determine whether this is correct by
      1. showing that the square root of 2 is a fixed point for the map and
      2. numerically iterating the map to determine whether the choice of initial condition matters.

        You may want simply modify the code for the logistic map used in class; however, you can use any programming language, package or environment that you want.

    2. By thinking about what is a fixed point of the map, what would be the formula for
      1. the square root of 3?
      2. the square root of 5?
      3. the square root of a?
  2. Is the set of real numbers with the operation of real multiplication a group? If it is not, is there an easy way to make it one?
  3. Is the set of n by m matrices with the operation of matrix addition a vector space?
  4. Is the set of n by m matrices with the operation of matrix multiplication a vector space?
  5. Is the set of 3 vectors (R3) with the dot product an algebra?
  6. Is the set of real numbers with the usual addition as the vector space addition and the usual multiplication an algebra?
  7. (Updated: typo corrected). Verify by numerical experimentation that the phase plot shown in class for the Van der Pol equation

    was, at least qualitatively, accurate.

  8. Verify by numerical experimentation that the phase plot shown in class for Duffing's equation

    was, at least qualitatively, accurate.


As with the tone-deaf, they don't know what they miss. They give a pitying chuckle at the news of scientists who have never read a major work of English literature. They dismiss them as ignorant specialists. Yet their own ignorance and their own specialization is just as startling. A good many times I have been present at gatherings of people who, by the standards of the traditional culture, are thought highly educated and who have with considerable gusto been expressing their incredulity at the illiteracy of scientists. Once or twice I have been provoked and have asked the company how many of them could describe the Second Law of Thermodynamics. The response was cold: it was also negative. Yet I was asking something which is about the scientific equivalent of: Have you read a work of Shakespeare's?

I now believe that if I had asked an even simpler question -- such as, What do you mean by mass, or acceleration, which is the scientific equivalent of saying, Can you read? -- not more than one in ten of the highly educated would have felt that I was speaking the same language. So the great edifice of modern physics goes up, and the majority of the cleverest people in the western world have about as much insight into it as their neolithic ancestors would have had.

--- C. P. Snow, The Two Cultures.


Last updated: August 26, 1999.
B. Goodwine (jgoodwin@nd.edu)