Sketch the phase portrait for the following systems: (a) u̇1=u1-3 u2, u̇2=-3 u1+u2 .

Step 1: Analyze the system (a) $u005Cbegin{aligned} u0026 u005Cdot{u}_1u003Du_1u002D3 u_2, u005Cu005C u0026 u005Cdot{u}_2u003Du002D3 u_1+u_2

Question

Sketch the phase portrait for the following systems: (a) $\begin{aligned} & \dot{u}_1=u_1-3 u_2, \\ & \dot{u}_2=-3 u_1+u_2 .\end{aligned}$ (b) $$ \begin{aligned} & \dot{u}_1=u_1-4 u_2, \\ & \dot{u}_2=u_1-u_2 . \end{aligned} $$ (c) $\dot{u}_1=u_1+u_2$, (d) $$ \begin{aligned} & \dot{u}_1=u_1+u_2, \\ & \dot{u}_2=u_2 . \end{aligned} $$ (e) $\dot{u}_1=\frac{3}{2} u_1+\frac{5}{2} u_2$. $$ \dot{u}_2=-\frac{5}{2} u_1+\frac{3}{2} u_2 \text {. } $$

    Sketch the phase portrait for the following systems:
(a) $\begin{aligned} & \dot{u}_1=u_1-3 u_2, \\ & \dot{u}_2=-3 u_1+u_2 .\end{aligned}$
(b)
$$
\begin{aligned}
& \dot{u}_1=u_1-4 u_2, \\
& \dot{u}_2=u_1-u_2 .
\end{aligned}
$$
(c) $\dot{u}_1=u_1+u_2$,
(d)
$$
\begin{aligned}
& \dot{u}_1=u_1+u_2, \\
& \dot{u}_2=u_2 .
\end{aligned}
$$
(e) $\dot{u}_1=\frac{3}{2} u_1+\frac{5}{2} u_2$.
$$
\dot{u}_2=-\frac{5}{2} u_1+\frac{3}{2} u_2 \text {. }
$$

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Peter J. Olver,… 2nd Edition

Chapter 10, Problem 5

Instant Answer

Step 1

\end{aligned}$ - Write the system in matrix form: $\dot{\mathbf{u}} = A\mathbf{u}$ where $\mathbf{u} = \begin{bmatrix} u_1 \\ u_2 \end{bmatrix}$ and $A = \begin{bmatrix} 1 & -3 \\ -3 & 1 \end{bmatrix}$. - Find the eigenvalues of $A$ by solving $\det(A - \lambda I) Show more…

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Sketch the phase portrait for the following systems: (a) $\begin{aligned} & \dot{u}_1=u_1-3 u_2, \\ & \dot{u}_2=-3 u_1+u_2 .\end{aligned}$ (b) $$ \begin{aligned} & \dot{u}_1=u_1-4 u_2, \\ & \dot{u}_2=u_1-u_2 . \end{aligned} $$ (c) $\dot{u}_1=u_1+u_2$, (d) $$ \begin{aligned} & \dot{u}_1=u_1+u_2, \\ & \dot{u}_2=u_2 . \end{aligned} $$ (e) $\dot{u}_1=\frac{3}{2} u_1+\frac{5}{2} u_2$. $$ \dot{u}_2=-\frac{5}{2} u_1+\frac{3}{2} u_2 \text {. } $$

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