# Difference between revisions of "CDS 140a Winter 2013 Homework 5"

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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). | ||

+ | |||

+ | <ol> | ||

+ | <li>'''Perko, Section 2.9, problem 3''' | ||

+ | Use the Lyapunov function $V(x)=x_1^2+x_2^2+x_3^2$ to show that the origin is an asymptotically stable equilibrium point of the system | ||

+ | <center><amsmath> | ||

+ | \dot{x}=\begin{bmatrix}-x_2-x_1x_2^2+x_3^2-x_1^3\\ | ||

+ | x_1+x_3^2-x_2^2\\ | ||

+ | -x_1x_3-x_3x_1^2-x_2x_3^2-x_3^5\end{bmatrix} | ||

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

+ | Show that the trajectories of the linearized system $\dot{x}=Df(0)x$ for this problem lie on circles in planes parallel to the $x_1,\,x_2$ plane; hence, the origin is stable, but not asymptotically stable for the linearized system. | ||

+ | </li> | ||

+ | <li>'''Perko, Section 2.12, problem 2''' | ||

+ | Use Theorem 1 [Centre Manifold Theorem] to determine the qualitative behaviour near the non-hyperbolic critical point at the origin for the system | ||

+ | <center><amsmath> | ||

+ | \aligned | ||

+ | \dot{x}&=y\\ | ||

+ | \dot{y}&=-y+\alpha x^2+xy | ||

+ | \endaligned | ||

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

+ | for $\alpha\neq 0$ and for $\alpha=0$; i.e., follow the procedure in Example 1 after diagonalizing the system as in Example 3. | ||

+ | </li> | ||

+ | </ol> |

## Revision as of 04:43, 3 February 2013

WARNING: This homework set is still being written. Do not start working on these problems until this banner is removed. |

R. Murray, D. MacMartin | Issued: 5 Feb 2013 (Tue) |

ACM 101/AM 125b/CDS 140a, Winter 2013 | Due: 12 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 2.9, problem 3**Use the Lyapunov function $V(x)=x_1^2+x_2^2+x_3^2$ to show that the origin is an asymptotically stable equilibrium point of the system<amsmath> \dot{x}=\begin{bmatrix}-x_2-x_1x_2^2+x_3^2-x_1^3\\ x_1+x_3^2-x_2^2\\ -x_1x_3-x_3x_1^2-x_2x_3^2-x_3^5\end{bmatrix}

</amsmath>Show that the trajectories of the linearized system $\dot{x}=Df(0)x$ for this problem lie on circles in planes parallel to the $x_1,\,x_2$ plane; hence, the origin is stable, but not asymptotically stable for the linearized system.

**Perko, Section 2.12, problem 2**Use Theorem 1 [Centre Manifold Theorem] to determine the qualitative behaviour near the non-hyperbolic critical point at the origin for the system<amsmath> \aligned \dot{x}&=y\\ \dot{y}&=-y+\alpha x^2+xy \endaligned

</amsmath>for $\alpha\neq 0$ and for $\alpha=0$; i.e., follow the procedure in Example 1 after diagonalizing the system as in Example 3.