CDS 140a Winter 2014 Homework 8

 R. Murray Issued: 25 Feb 2014 (Tue) ACM 101b/AM 125b/CDS 140a, Winter 2014 Due: 5 Mar 2014 (Wed) @ noon

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1. Perko, Section 4.1, Problem 1:
(a) Consider the two vector fields
<amsmath>
 f(x) = \begin{bmatrix} -x_2 \\ x_1 \end{bmatrix}, \qquad
g(x) = \begin{bmatrix} -x_2 + \mu x_1 \\ x_1 + \mu x_2 \end{bmatrix}.

</amsmath>

Show that $\|f-g\|_1 = |\mu| (\max_{x \in K} \|x\| + 1)$, where $K \subset {\mathbb R}^2$ is a compact set containing the origin in its interior.

(b) Show that for $\mu \neq 0$ the systems

<amsmath>
 \aligned \dot x_1 &= -x_2 \\ \dot x_2 &= x_1 \endaligned \quad\text{and}\quad
\aligned \dot x_1 &= -x_2 + \mu x_1 \\ \dot x_2 &= x_1 + \mu x_2 \endaligned

</amsmath>

are not topologically equivalent.

Hint: Let $\phi_t$ and $\psi_t$ be the flows defined by these two systems and assume that there is a homeomorphism $H:{\mathbb R}^2 \to {\mathbb R}^2$ and a strictly increasing, continuous function $t(\tau)$ mapping $\mathbb R$ onto $\mathbb R$ such that $\phi_{t(\tau)} = H^{-1} \circ \psi_\tau \circ H$. Use the fact that $\lim_{t\to\infty} \phi_t(1,0) \neq 0$ and that for $\mu < 0$, $\lim_{t \to \infty} \psi_t(x) = 0$ for all $x \in {\mathbb R}^2$ to arrive at a contradiction.

2. Consider the dynamical system
<amsmath>
 m \ddot q + b \dot q + k q = u(t), \qquad
u(t) = \begin{cases} 0 & t = 0, \\ 1 & t > 0, \end{cases} \qquad
q(0) = \dot q(0) = 0,

</amsmath>

which describes the "step response" of a mass-spring-damper system.

(a) Derive the differential equations for the sensitivities of <amsmath>q(t) \in {\mathbb R}</amsmath> to the parameters <amsmath>b</amsmath> and <amsmath>k</amsmath>. Write out explicit systems of ODEs for computing these, including any initial conditions. (You don't have to actually solve the differential equations explicitly, though it is not so hard to do so.)

(b) Compute the sensitivities and the relative (normlized) sensitivies of the equilibrium value of <amsmath>q_e</amsmath> to the parameters <amsmath>b</amsmath> and <amsmath>k</amsmath>. You should give explicit formulas in terms of the relevant parameters and initial conditions.

(c) Sketch the plots of the relative sensitivities <amsmath>S_{q,b}</amsmath> and <amsmath>S_{q,k}</amsmath> as a function of time for the nominal parameter values <amsmath>m = 1</amsmath>, <amsmath>b = 2</amsmath>, <amsmath>k = 1</amsmath>.

3. Perko, Section 4.2, Problem 4: Consider the planar system
<amsmath>
 \aligned
\dfot x &= \mu x - x^2 \\
\dot y &= -y.
\endaligned

</amsmath>

Verify that the system satisfies the conditions for a transcritical bifurcation (equation (3) in Section 4.2) and determine the dimensions of the various stable, unstable and center manifolds that occur.

4. Perko, Section 4.2, Problem 7: Consider the two dimensional system
<amsmath>
 \aligned
\dot x &= -x^4 + 5 \mu x^2 - 4 \mu^2 \\
\dot y &= -y.
\endaligned

</amsmath>

Determine the critical points and the bifurcation diagram for this system. Draw phase portraits for the various values of $\mu$ and draw the bifurcation diagram.