# Difference between revisions of "CDS 140a Winter 2013 Homework 4"

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{{CDS homework | {{CDS homework | ||

| instructor = R. Murray, D. MacMartin | | instructor = R. Murray, D. MacMartin | ||

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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 | ||

this homework set (including reading). | this homework set (including reading). | ||

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<ol> | <ol> | ||

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</li> | </li> | ||

− | <li> | + | <li> Prove that if |

− | + | <center><amsmath> | |

+ | \aligned | ||

+ | \dot{x}&=f(x,y),\qquad x\in\mathbb{R}^k\\ | ||

+ | \dot{y}&=g(x,y),\qquad g\in\mathbb{R}^m | ||

+ | \endaligned | ||

+ | </amsmath></center> | ||

+ | then the manifold $S=\{(x,y)\in\mathbb R^k\times\mathbb R^m|y=h(x)\}$ is an invariant manifold of the system if | ||

+ | <center><amsmath> | ||

+ | g(x,h(x))=Dh(x)f(x,h(x)) | ||

+ | </amsmath></center> | ||

+ | Use this result to compute the stable manifold for the system of problem 2 and 3 above using the Taylor series for $h(x)$ to define $S=\{(x_1,x_2)|x_2=h(x)\}$ and matching coefficients to solve for $h(x)$. | ||

+ | '''Hint:''' One way to show $S$ is an invariant manifold in $\mathbb R^2$ is to show that the normal vector (orthogonal to the tangent to $S$ at $(x,h(x))$) is orthogonal to the vector field $(f,g)$ at that point. (It is sufficient to prove the result for $\mathbb R^2$.) | ||

+ | </li> | ||

+ | <li> '''Perko, Section 2.7, Problem 6''' | ||

+ | Let $E$ be an open subset of $\mathbb{R}^n$ containing the origin. Use the fact that if $F\in C^1(E)$ then for all $x, y\in N_\delta(0)\subset E$ there exists a $\xi\in N_\delta(0)$ such that | ||

+ | <center><amsmath> | ||

+ | |F(x)-F(y)|\leq\|DF(\xi)\|\,|x-y| | ||

+ | </amsmath></center> | ||

+ | (cf. Theorem 4.19 and the proof of Theorem 9.19 in [R]) to prove that if $F\in C^1(E)$ and $F(0)=DF(0)=0$ then given any $\epsilon>0$ there exists a $\delta>0$ such that for all $x, y\in N_\delta(0)$ we have | ||

+ | <center><amsmath> | ||

+ | |F(x)-F(y)|<\epsilon |x-y| | ||

+ | </amsmath></center> | ||

+ | </li> | ||

<li>'''Perko, Section 2.9, problem 2(a)(b)''' | <li>'''Perko, Section 2.9, problem 2(a)(b)''' | ||

Determine the stability of the equilibrium points of the system (1) with $f(x)$ given by | Determine the stability of the equilibrium points of the system (1) with $f(x)$ given by | ||

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</li> | </li> | ||

+ | |||

+ | <!-- | ||

<font color=blue>And any two of the following:</font> | <font color=blue>And any two of the following:</font> | ||

<li> | <li> | ||

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Note that these formulae actually determine the global stable and unstable manifolds $W^s(0)$ and $W^u(0)$ respectively. | Note that these formulae actually determine the global stable and unstable manifolds $W^s(0)$ and $W^u(0)$ respectively. | ||

</li> | </li> | ||

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<li> | <li> | ||

'''Perko, Section 2.8, Problem 1''' | '''Perko, Section 2.8, Problem 1''' | ||

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W^u(0)=\{x\in{\mathbb{R}}|y_1=0,y_2=z^2/3\}. | W^u(0)=\{x\in{\mathbb{R}}|y_1=0,y_2=z^2/3\}. | ||

</amsmath></center> | </amsmath></center> | ||

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</li> | </li> | ||

+ | </!-- | ||

</ol> | </ol> | ||

## Latest revision as of 03:09, 4 February 2013

R. Murray, D. MacMartin | Issued: 29 Jan 2013 (Tue) |

ACM 101/AM 125b/CDS 140a, Winter 2013 | Due: 5 Feb 2013 (Tue) |

**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).

**Perko, Section 2.7, problem 1**Write the system<amsmath> \aligned \dot{x}_1&=x_1+6x_2+x_1x_2,\\ \dot{x}_2&=4x_1+3x_2-x_1^2 \endaligned

</amsmath>in the form

<amsmath> \dot{y}=By+G(y)

</amsmath>where

<amsmath> B=\begin{bmatrix}\lambda_1&0\\0&\lambda_2\end{bmatrix}

</amsmath>with $\lambda_1<0$, $\lambda_2>0$ and $G(y)$ is quadratic in $y_1$ and $y_2$.

**Perko, Section 2.7, problem 2**Find the first three successive approximations $u^{(1)}(t,a)$, $u^{(2)}(t,a)$, and $u^{(3)}(t,a)$ for<amsmath> \aligned \dot{x}_1&=-x_1,\\ \dot{x}_2&=x_2+x_1^2 \endaligned

</amsmath>and use $u^{(3)}(t,a)$ to approximate $S$ near the origin. Also approximate the unstable manifold $U$ near the origin for this system. Note that $u^{(2)}(t,a)=u^{(3)}(t,a)$ and therefore $u^{(j+1)}(t,a)=u^{(j)}(t,a)$ for $j\geq 2$. Thus $u(t,a)=u^{(2)}(t,a)$ which gives the exact function defining $S$.

**Perko, Section 2.7, problem 3**Solve the system in Problem 2 and show that $S$ and $U$ are given by<amsmath> S:\,x_2=-\frac{x_1^2}{3}

</amsmath><amsmath> U:\,x_1=0

</amsmath>Sketch $S$, $U$, $E^s$ and $E^u$.

- Prove that if
<amsmath> \aligned \dot{x}&=f(x,y),\qquad x\in\mathbb{R}^k\\ \dot{y}&=g(x,y),\qquad g\in\mathbb{R}^m \endaligned

</amsmath>then the manifold $S=\{(x,y)\in\mathbb R^k\times\mathbb R^m|y=h(x)\}$ is an invariant manifold of the system if

<amsmath> g(x,h(x))=Dh(x)f(x,h(x))

</amsmath>Use this result to compute the stable manifold for the system of problem 2 and 3 above using the Taylor series for $h(x)$ to define $S=\{(x_1,x_2)|x_2=h(x)\}$ and matching coefficients to solve for $h(x)$.

**Hint:**One way to show $S$ is an invariant manifold in $\mathbb R^2$ is to show that the normal vector (orthogonal to the tangent to $S$ at $(x,h(x))$) is orthogonal to the vector field $(f,g)$ at that point. (It is sufficient to prove the result for $\mathbb R^2$.) -
**Perko, Section 2.7, Problem 6**Let $E$ be an open subset of $\mathbb{R}^n$ containing the origin. Use the fact that if $F\in C^1(E)$ then for all $x, y\in N_\delta(0)\subset E$ there exists a $\xi\in N_\delta(0)$ such that<amsmath> |F(x)-F(y)|\leq\|DF(\xi)\|\,|x-y|

</amsmath>(cf. Theorem 4.19 and the proof of Theorem 9.19 in [R]) to prove that if $F\in C^1(E)$ and $F(0)=DF(0)=0$ then given any $\epsilon>0$ there exists a $\delta>0$ such that for all $x, y\in N_\delta(0)$ we have

<amsmath> |F(x)-F(y)|<\epsilon |x-y|

</amsmath> **Perko, Section 2.9, problem 2(a)(b)**Determine the stability of the equilibrium points of the system (1) with $f(x)$ given by<amsmath> (a)\quad\begin{bmatrix}x_1^2-x_2^2-1\\2x_2\end{bmatrix}

</amsmath><amsmath> (b)\quad\begin{bmatrix}x_2-x_1^2+2\\2x_2^2-2x_1x_2\end{bmatrix}

</amsmath>