CDS 140a Winter 2011 Homework 2

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R. Murray, G. Buzi Issued: 11 Jan 2011 (Tue)
ACM 101/AM 125a/CDS 140a, Winter 2011 Due: 20 Jan 2011 (Thu)

Note: In the upper left hand corner of the second page of your homework set, please put the number of hours that you spent on this homework set (including reading).

  1. Perko, Section 1.7, problem 3, parts (a) and (d): Solve the initial value problem (1) with the matrix
    <amsmath>

    \text{(a)}\quad A = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & 1 \\ 0 & -1 & 1 & 0 \end{bmatrix}, \qquad \text{(d)}\quad A = \begin{bmatrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 1 & 1 \end{bmatrix}, \qquad

    </amsmath>
  2. Perko, Section 1.8, problem 10: Suppose that the elementary blocks <amsmath>B</amsmath> in the Jordan form of the matrix <amsmath>A</amsmath>, given by (2) or (3), have no ones or <amsmath>I_2</amsmath> blocks off the diagonal. Show that if all of the eigenvalues of <amsmath>A</amsmath> have nonpositive real parts, then for each <amsmath>x_0 \in {\mathbb R}^n</amsmath> there is a positive constant <amsmath>M</amsmath> such that <amsmath>|x(t)| \leq M</amsmath> for all <amsmath>t \geq 0></amsmath> where <amsmath>x(t)</amsmath> is the solution of the initial value problem (4).
  3. Perko, Section 1.9, problem 3: Solve the system
    <amsmath>

    \dot x = \begin{bmatrix} 0 & 2 & 0 \\ -2 & 0 & 0 \\ 2 & 0 & 6 \end{bmatrix} x.

    </amsmath>

    Find the stable, unstable and center subspaces for this system and sketch the phase portrait. Show that the flow of initial points in each of the subspaces remains in that subspace.

  4. Perko, Section 1.9, problem 5, parts (c), (d1) and (d2): Let <amsmath>A</amsmath> be an <amsmath>n \times n</amsmath> nonsingular matrix and let <amsmath>x(t)</amsmath> be the solution of the initial value problem (1) with <amsmath>x(0) = x_0</amsmath>. Show that
    • If <amsmath>x_0 \in E^c \sim \{0\}</amsmath> and <amsmath>A</amsmath> is semisimple, then there are postive constants <amsmath>m</amsmath> and <amsmath>M</amsmath> such that for all <amsmath>t \in R</amsmath>, <amsmath>m \leq |x(t)| \leq M</amsmath>;
      • Note: Perko defines <amsmath>\sim</amsmath> to mean "set subtraction". So <amsmath>E \sim \{0\}</amsmath>is the set <amsmath>E</amsmath> minus the point 0.
    • If <amsmath>x_0 \in E^c \sim \{0\}</amsmath> and <amsmath>A</amsmath> is not semisimple, then there is an <amsmath>x_0 \in {\mathbb R}^n</amsmath> such that <amsmath>\lim_{t \to \pm \infty} |x(t)| = \infty</amsmath>;
      • Note: this problem should be modified to state that the projection of A onto the center subspace is not semisimple. (Hint: put the system into Jordan form; the center subspace is now the set of coordinate directions corresponding to eigenvalues with zero real part.)
    • If <amsmath>E^s \neq \{0\}</amsmath>, <amsmath>E^u \neq \{0\}</amsmath>, and <amsmath>x_0 \in E^s \oplus E^u \sim (E^s \cup E^u)</amsmath>, then <amsmath>\lim_{t \to \pm \infty} |x(t)| = \infty</amsmath>;
  5. Perko, Section 1.10, problem 2: Use Theorem 1 to solve the nonhomogeneous linear system
    <amsmath>

    \dot x = \begin{bmatrix} 1 & 1 \\ 0 & -1 \end{bmatrix} x + \begin{bmatrix} t \\ 1 \end{bmatrix}

    </amsmath>

    with the initial condition

    <amsmath>

    x(0) = \begin{bmatrix} 1 \\ 0 \end{bmatrix}.

    </amsmath>
  6. Perko, Section 1.10, problem 3: Show that
    <amsmath>

    \Phi(t) = \begin{bmatrix} e^{-2t} \cos t & -\sin t \\ e^{-2t} \sin t & \cos t \end{bmatrix}

    </amsmath>

    is a fundamental matrix of the nonautonomous linear system

    <amsmath>

    \dot x = A(t) x

    </amsmath>

    with

    <amsmath>

    A(t) = \begin{bmatrix} -2 \cos^2 t & -1 - \sin 2t \\ 1 - \sin 2t & -2 \sin^2 t \end{bmatrix}

    </amsmath>

    Find the inverse of <amsmath>\Phi(t)</amsmath> and use Theorem 1 and Remark 1 to solve the nonhomogeneous linear system

    <amsmath>

    \dot x = A(t) x + b(t)

    </amsmath>

    with <amsmath>A(t)</amsmath> given above and <amsmath>b(t) = (1, e^{-2t})^T</amsmath>. Note that, in general, if <amsmath>A(t)</amsmath> is a periodic matrix of period <amsmath>T</amsmath>, then corresponding to any fundamental matrix <amsmath>\Phi(t)</amsmath> there exists a periodic matrix <amsmath>P(t)</amsmath> of period <amsmath>T</amsmath> and a constant matrix <amsmath>B</amsmath> such that

    <amsmath>

    \Phi(t) = P(t) e^{B t}

    </amsmath>

    Show that <amsmath>P(t)</amsmath> is a rotation matrix and <amsmath>B = \text{diag}[-2, 0]</amsmath> in this problem.

    • Note: A rotation matrix is defined as a matrix <amsmath>R</amsmath> that satisfies <amsmath>R R^T = I</amsmath> and <amsmath>\det R = 1</amsmath>.

Notes:

  • The problems are transcribed above in case you don't have access to Perko. However, in the case of discrepancy, you should use Perko as the definitive source of the problem statement.
  • There are a number of problems that can be solved using MATLAB. If you just give the answer with no explanation (or say "via MATLAB"), the TAs will take off points. Instead, you should show how the solutions can be worked out by hand, along the lines of what is done in the textbook. It is fine to check everything with MATLAB.