CDS 140a Winter 2013 Homework 2 - Revision history

Murray at 18:47, 15 January 2013

2013-01-15T18:47:22Z

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	{{CDS homework		{{CDS homework
	\| instructor = R. Murray, D. MacMartin		\| instructor = R. Murray, D. MacMartin

Murray at 18:38, 15 January 2013

2013-01-15T18:38:03Z

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	</amsmath></center>		</amsmath></center>
	(a) Compute the solutions to the differential equation. You should provide the matrices used to transform the system to Jordan form along with the appropriate matrix exponential of the relevant Jordan form matrix (you don't need to multiply everything out to get the solution in the original basis).		(a) Compute the solutions to the differential equation. You should provide the matrices used to transform the system to Jordan form along with the appropriate matrix exponential of the relevant Jordan form matrix (you don't need to multiply everything out to get the solution in the original basis).
			:* Note: you should show the various (regular and generalized) eigenvectors associated with each eigenvalue. OK to check your answer with MATLAB, but be sure to show that you know how to solve it by hand.

	(b) Find the stable, unstable and center subspaces for this system.		(b) Find the stable, unstable and center subspaces for this system.

Murray at 18:48, 13 January 2013

2013-01-13T18:48:30Z

← Older revision		Revision as of 18:48, 13 January 2013
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			{{warning\|This homework set is still being written. Do not start working on these problems until this banner is removed.}}

	{{CDS homework		{{CDS homework
	\| instructor = R. Murray, D. MacMartin		\| instructor = R. Murray, D. MacMartin

Murray at 18:47, 13 January 2013

2013-01-13T18:47:17Z

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	\dot x = \begin{bmatrix} 0 & 2 & 0 \\ -2 & 0 & 0 \\ 2 & 0 & 6 \end{bmatrix} x.		\dot x = \begin{bmatrix} 0 & 2 & 0 \\ -2 & 0 & 0 \\ 2 & 0 & 6 \end{bmatrix} x.
	</amsmath></center>		</amsmath></center>
	(a) Compute the solutions to the differential equation. You ~~can~~ provide the matrices used to transform the system to Jordan form along with the appropriate matrix exponential of the relevant Jordan form matrix (you don't need to multiply everything out to get the solution in the original basis).		(a) Compute the solutions to the differential equation. You should provide the matrices used to transform the system to Jordan form along with the appropriate matrix exponential of the relevant Jordan form matrix (you don't need to multiply everything out to get the solution in the original basis).

	(b) Find the stable, unstable and center subspaces for this system.		(b) Find the stable, unstable and center subspaces for this system.

Murray at 18:46, 13 January 2013

2013-01-13T18:46:22Z

@@ Line 66: / Line 66: @@
 </li>
-<li> '''Perko, Section 1.10, problem 2''': Use Theorem 1 to solve the nonhomogeneous linear system
+<li> '''Perko, Section 1.10, problem 2''': Use Theorem 1 in Section 1.10 to solve the nonhomogeneous linear system
 <center><amsmath>
 \dot x = \begin{bmatrix} 1 & 1 \\ 0 & -1 \end{bmatrix} x + \begin{bmatrix} t \\ 1 \end{bmatrix}
@@ Line 74: / Line 74: @@
 x(0) = \begin{bmatrix} 1 \\ 0 \end{bmatrix}.
 </amsmath></center>
 </li>
 </ol>

Murray at 18:44, 13 January 2013

2013-01-13T18:44:43Z

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	</li>		</li>

	<li> '''Perko, Section 1.9, problem 3''': ~~Solve~~ the system		<li> '''Perko, Section 1.9, problem 3''' (modified): Consider the linear system
	<center><amsmath>		<center><amsmath>
	\dot x = \begin{bmatrix} 0 & 2 & 0 \\ -2 & 0 & 0 \\ 2 & 0 & 6 \end{bmatrix} x.		\dot x = \begin{bmatrix} 0 & 2 & 0 \\ -2 & 0 & 0 \\ 2 & 0 & 6 \end{bmatrix} x.
	</amsmath></center>		</amsmath></center>
	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~~.		(a) Compute the solutions to the differential equation. You can provide the matrices used to transform the system to Jordan form along with the appropriate matrix exponential of the relevant Jordan form matrix (you don't need to multiply everything out to get the solution in the original basis).

			(b) Find the stable, unstable and center subspaces for this system.
	</li>		</li>

Murray at 18:36, 13 January 2013

2013-01-13T18:36:38Z

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	(c) 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>;		(c) 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>;
	* ~~<font color=blue>~~Note:~~</font>~~ 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.		:* 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.

	(d1) If <amsmath>x_0 \in E^c \sim \{0\}</amsmath> and the projection of <amsmath>A</amsmath> onto the ~~cetner~~ subspace 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>		(d1) If <amsmath>x_0 \in E^c \sim \{0\}</amsmath> and the projection of <amsmath>A</amsmath> onto the center subspace 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>
	* Hint: put the system into Jordan form; the center subspace is now the set of coordinate directions corresponding to eigenvalues with zero real part.		:* Note: there is a bug in the statement in Perko; use the modified version here
			:* Hint: put the system into Jordan form; the center subspace is now the set of coordinate directions corresponding to eigenvalues with zero real part.

	(d2) 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>;		(d2) 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>;

Murray at 18:35, 13 January 2013

2013-01-13T18:35:27Z

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	* <font color=blue>Note:</font> 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.		* <font color=blue>Note:</font> 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.

	(d1) If <amsmath>x_0 \in E^c \sim \{0\}</amsmath> and the projection of <amsmath>A</amsmath> onto the cetner subspace 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>;		(d1) If <amsmath>x_0 \in E^c \sim \{0\}</amsmath> and the projection of <amsmath>A</amsmath> onto the cetner subspace 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>
	* Hint: put the system into Jordan form; the center subspace is now the set of coordinate directions corresponding to eigenvalues with zero real part.		* Hint: put the system into Jordan form; the center subspace is now the set of coordinate directions corresponding to eigenvalues with zero real part.

Murray at 18:34, 13 January 2013

2013-01-13T18:34:52Z

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	</li>		</li>

	<li> '''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		<li> '''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<br>

	(c) 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>;		(c) 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>;
	* <font color=blue>Note:</font> 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.		* <font color=blue>Note:</font> 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.

	(d1) 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>;		(d1) If <amsmath>x_0 \in E^c \sim \{0\}</amsmath> and the projection of <amsmath>A</amsmath> onto the cetner subspace 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>;
	* ~~<font color=blue>Note: this problem should be modified to state that the projection of ''A'' onto the center subspace is not semisimple</font>. (~~Hint: put the system into Jordan form; the center subspace is now the set of coordinate directions corresponding to eigenvalues with zero real part.)		* Hint: put the system into Jordan form; the center subspace is now the set of coordinate directions corresponding to eigenvalues with zero real part.

	(d2) 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>;		(d2) 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>;

Murray at 18:33, 13 January 2013

2013-01-13T18:33:10Z

← Older revision		Revision as of 18:33, 13 January 2013
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	<li> '''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		<li> '''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>;
	** <font color=blue>Note:</font> 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.		(c) 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>;
	* 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>;		* <font color=blue>Note:</font> 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.
	** <font color=blue>Note: this problem should be modified to state that the projection of ''A'' onto the center subspace is not semisimple</font>. (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>;		(d1) 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>;
			* <font color=blue>Note: this problem should be modified to state that the projection of ''A'' onto the center subspace is not semisimple</font>. (Hint: put the system into Jordan form; the center subspace is now the set of coordinate directions corresponding to eigenvalues with zero real part.)

			(d2) 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>;
	</li>		</li>