Homework 3 - Due Oct 28th

  • Due Oct 28, 2019 by 2p.m.
  • Points 20
  1. Problems in Lynch, Section 3.7 -- 4 (b), (f)
  2. Problems in Lynch, Section 8.6 -- 2 and 3.

  3. Consider the nonlinear system

    \begin{pmatrix} \dot{x} \\ \dot{y} \end{pmatrix} =\begin{pmatrix} x \\ -y+x^3 \end{pmatrix}.

    (a) Show that the origin is a fixed point of the system and linearize the system at the origin. What is the expected behaviour of solutions near the origin?

    (b) Given the initial condition (x_0,y_0), what is the solution to the system?

    (c) Hence, write down an explicit formula for the flow of the system \phi_t : \mathbb{R}^2 \to \mathbb{R}^2.

        (d) Use part (c) to find the stable and unstable manifolds passing through the origin. Show that they are tangent to the stable and unstable spaces of the linearization.

        (This is an example of stable manifold theorem in action.)

        (e) Use part (c) (and (d)) to show that the conclusion you obtained using the linearization is, in fact, true. 

        (This is an example of linearized stability theorem in action.)

        (f) Sketch phase portrait of the system in a small neighbourhood of the origin.  

4.  (a) Show that the origin is a (globally) asymptotically stable fixed point of \dot{x}=-x. 

      (b)  Find a differentiable function f : \mathbb{R}\to \mathbb{R} so that there are solutions of \dot{x}=-x+f(t) that do not converge to the origin as  t \to \infty.

      (This is an example showing limitations of the stability results we discussed in class.)

5.  (a) Show that the eigenvalues of the matrix

A(t)=\begin{pmatrix} -1+\frac{3}{2}\cos^2 t & 1-\frac{3}{2}\cos t \sin t \\ -1-\frac{3}{2}\cos t \sin t & -1+\frac{3}{2}\sin^2 t \end{pmatrix}

             are independent of t, and that they have negative real parts.

        (b) Show that

\begin{pmatrix} -e^{t/2}\cos t & e^{-t}\sin t \\ e^{t/2}\sin t & e^{-t}\cos t \end{pmatrix}

               is a fundamental matrix solution of \dot{\underline{x}}=A(t)\underline{x}

         (c) Use part (b) to show that the origin is an unstable fixed point.

         (d) Observe that the matrix has eigenvalues with negative real parts but the origin is unstable. Why doesn’t this contradict the stability theorems we discussed in the lectures?

        (This is an example showing limitations of the stability results we discussed in class.)

SOLUTIONS

1572285600 10/28/2019 02:00pm