Difference between revisions of "CDS 140a Winter 2015 Homework 2"

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  | pdf = cds140-wi15_hw2.pdf
 
  | pdf = cds140-wi15_hw2.pdf
 
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'''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
 
'''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
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A lot of students also had questions about problem
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* A lot of students also had questions about problem 4 I believe it was, kind of wondering if the solution was really as easy as it looked, and the students were told that yes, it was that easy.  
4 I believe it was, kind of wondering if the solution was really as easy
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as it looked, and the students were told that yes, it was that easy.
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* On problem 4, many students got the prinicipal axes wrong, and they were also frequently confused about the ... [rest of text missing]
 
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<li> '''Perko, Section 2.5, problem 4''':  
 
<li> '''Perko, Section 2.5, problem 4''':  
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Students didn't seem to understand what they were supposed to do.  
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* Students didn't seem to understand what they were supposed to do. They were showed an example of a 1-d system with a pitchfork bifurcation, and also a bifurcation plot of a 2-d system with a pitchfork bifurcation to help get intuition about what they were actually looking for, so they could apply the concepts to the 3-d system.  I didn't see that they had to classify the equilibrium points because that was in the Note at the end of the problem, and I didn't realize that Notes contained further instruction, so I told a few students they only needed to find mu where the single equilibrium split into several equilibriums.  So we decided to grade lightly for this problem.  The students were also showed how to do the stability analysis for the example of 1-d system (and qualitatively for the 2-d system with the phase portrait), in hopes it would give them more intuition about why this problem was important,  (not realizing that the Note at the end of the problem asked the students to actually classify the equilibrium points).
They were showed an example of a 1-d system with a pitchfork bifurcation,
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and also a bifurcation plot of a 2-d system with a pitchfork bifurcation
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* On problem 6, students not performing stability analysis on the equilibrium points for mu > 1 as required.
to help get intuition about what they were actually looking for, so they
 
could apply the concepts to the 3-d system.  I didn't see that they had to
 
classify the equilibrium points because that was in the Note at the end of
 
the problem, and I didn't realize that Notes contained further
 
instruction, so I told a few students they only needed to find mu where
 
the single equilibrium split into several equilibriums.  So we decided to
 
grade lightly for this problem.  The students were also showed how to do
 
the stability analysis for the example of 1-d system (and qualitatively
 
for the 2-d system with the phase portrait), in hopes it would give them
 
more intuition about why this problem was important,  (not realizing that
 
the Note at the end of the problem asked the students to actually classify
 
the equilibrium points).  
 
 
-->
 
-->
 
<!-- Dropped for 2015
 
<!-- Dropped for 2015

Latest revision as of 23:40, 19 January 2015

R. Murray Issued: 12 Jan 2015
CDS 140, Winter 2015

(PDF)

Due: 21 Jan 2015 at 12:30 pm
In class or to box across 107 STL

__MATHJAX__

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 2.2, problem 5: Let $V$ be a normed linear space. If $T:V \to V$ satisfies
    <amsmath>

    \|T(u) - T(v) \| \leq c\|u - v \|

    </amsmath>

    for all $u, v \in V$ with $0 < c < 1$ then $T$ is called a contraction mapping. It can be shown that contraction mappings give rise to unique solutions of the equation $T(u) = v$:

    Theorem (Contraction Mapping Principle) Let $V$ be a complete normed linear space and $T:V \to V$ a contraction mapping. Then there exists a unique $u \in V$ such that $T(u) = v$.

    Let $f \in C^1(E)$ and $x_0 \in E$. For $I = [-a, a]$ and $u \in C(I)$, let

    <amsmath>
     T(u)(t) = x_0 + \int_0^t f(u(s)) ds.
    
    </amsmath>

    Define a closed subset $V$ of $C(I)$ and apply the Contraction Mapping Principle to show that the integration equation (7) in Perko, Section 2.2 has a unique solution $u(t)$ for all $t \in [-a, a]$ provided the constant $a > 0$ is sufficiently small.

  2. Perko, Section 2.3, problem 1: Use the fundamental theorem for linear systems in Chapter 1 of Perko to solve the initial value problem
    <amsmath>
     \dot x = A x, \qquad x(0) = y.
    
    </amsmath>

    Let $u(t, y)$ denote the solution and compute

    <amsmath>
     \Phi(t) = \frac{\partial u}{\partial y}(t, y).
    
    </amsmath>

    Show that $\Phi(t)$ is the fundamental matrix solution of

    <amsmath>
     \dot \Phi = A \Phi, \qquad \Phi(0) = I.
    
    </amsmath>
    • Note: this problem works through the more general result for nonlinear systems (Corallary on page 83) for the special case of a linear system.
  3. Perko, Section 2.5, problem 4: Sketch the flow of the linear system
    <amsmath>
     \dot x = A x \quad\text{with}\quad A = \begin{bmatrix} -1 & -3 \\ 0 & 2 \end{bmatrix}
    
    </amsmath>

    and describe $\phi_t(N_\epsilon(x_0))$ for $x_0 = (-3, 0)$, $\epsilon = 0.2$.

  4. Perko, Section 2.5, problem 5: Determine the flow $\phi_t:{\mathbb R}^2 \to {\mathbb R}^2$ for the nonlinear system
    <amsmath>
     \dot x = f(x) \quad\text{with}\quad f(x) = \begin{bmatrix} -x_1 \\ 2 x_2 + x_1^2 \end{bmatrix}
    
    </amsmath>

    and show that the set $S = \{x \in {\mathbb R}^2| x_2 = -x_1^2/4\}$ is invariant with respect to the flow $\phi_t$.

  5. Choose one of the following systems and determine all of the equilibrium points for the system, indicating whether each is a sync, source, or saddle.

    (a) Moore-Greitzer model: The Moore-Greitzer equations model rotating stall and surge in gas turbine engines are given by
    <amsmath>
     \aligned 
       \frac{d\psi}{dt} &= \frac{1}{4 B^2 l_c}\left(\phi - \Phi_T(\psi) \right), \\
       \frac{d\phi}{dt} &= \frac{1}{l_c}\left( \Psi_c(\phi) - \psi + \frac{J}{8} 
         \frac{\partial^2 \Psi_c}{\partial \phi^2} \right), \\
       \frac{dJ}{dt} &= \frac{2}{\mu + m} \left(
          \frac{\partial \Psi_c}{\partial \phi} + \frac{J}{8}
             \frac{\partial^3 \Psi_c}{\partial \phi^3} \right) J,
     \endaligned
    
    </amsmath>

    where

    <amsmath>
     \aligned 
         B &= 0.2, & \Phi_T(\psi) &= \sqrt{\psi},\\
         l_c &= 6, & \Psi_c(\phi) &= 1 + 1.5 \phi - 0.5 \phi^3, \\
         \mu &= 1.256, &\qquad\qquad m &= 2.
     \endaligned
    
    </amsmath>

    This is a model for the dynamics of the compression system (first part of a jet engine) with $\psi$ representing the pressure rise across the compressor, $\phi$ representing the mass flow through the compressor and $J$ representing the amplitude squared of the first modal flow perturbation (corresponding to a rotating stall disturbance).


    (b) Genetic toggle switch: Consider the dynamics of two transcriptional repressors connected together in a cycle. It can be shown that the normalized dynamics of the system can be written as

    <amsmath>
     \frac{dz_1}{d\tau} = \frac{\mu}{1 + z_2^n} - z_1 - v_1,\qquad
     \frac{dz_2}{d\tau} = \frac{\mu}{1 + z_1^n} - z_2 - v_2.
    
    </amsmath>

    where $z_1$ and $z_2$ represent scaled versions of the protein concentrations, $v_1$ and $v_2$ represent external inputs and the time scale has been changed. Let $\mu = 2.16$, $n = 2$ and $v_1 = v_2 = 0$.


    (c) Congestion control: A simplified model for congestion control between $N$ computers connected by a router is given by the differential equation

    <amsmath>
     \aligned 
       \frac{dx_i}{dt} &= -b \frac{x_i^2}{2} + (b_{\text{max}} - b), \qquad
       \frac{db}{dt} &= \Bigl( \sum_{i=1}^N x_i \Bigr) - c,
    \endaligned 
    
    </amsmath>

    where $x_i \in {\mathbb R}$, $i = 1, \ldots, N$ are the transmission rates for the sources of data, $b \in {\mathbb R}$ is the current buffer size of the router, $b_{\text{max}} > 0$ is the maximum buffer size and $c > 0$ is the capacity of the link connecting the router to the computers. The $\dot x_i$ equation represents the control law that the individual computers use to determine how fast to send data across the network and the $\dot b$ equation represents the rate at which the buffer on the router fills up. Consider the case where $N = 2$ (so that we have three states, $x_1$, $x_2$ and $b$) and take $b_{\text{max}} = 1$ Mb and $c = 2$ Mb/s.


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.