diff --git a/second-order-odes/homogeneous-constant-coefficient.md b/second-order-odes/homogeneous-constant-coefficient.md index 97f479b08..ddb0b8274 100644 --- a/second-order-odes/homogeneous-constant-coefficient.md +++ b/second-order-odes/homogeneous-constant-coefficient.md @@ -45,7 +45,7 @@ give the solution type: 2. $\lambda_{1,2} = \alpha \pm i \omega$ are complex: \begin{equation} - y = e^{\omega x} \left[c_1 \cos(\omega x) + c_2 \sin(\omega x)\right] + y = e^{\alpha x} \left[c_1 \cos(\omega x) + c_2 \sin(\omega x)\right] \end{equation} 3. $\lambda_1 = \lambda_2 = \lambda $ is real and repeated: diff --git a/systems-first-order-odes/_images/center.png b/systems-first-order-odes/_images/center.png new file mode 100644 index 000000000..e6f702a35 Binary files /dev/null and b/systems-first-order-odes/_images/center.png differ diff --git a/systems-first-order-odes/_images/degenerate_node.png b/systems-first-order-odes/_images/degenerate_node.png new file mode 100644 index 000000000..507515500 Binary files /dev/null and b/systems-first-order-odes/_images/degenerate_node.png differ diff --git a/systems-first-order-odes/_images/proper_node.png b/systems-first-order-odes/_images/proper_node.png new file mode 100644 index 000000000..79e5d5e44 Binary files /dev/null and b/systems-first-order-odes/_images/proper_node.png differ diff --git a/systems-first-order-odes/_images/saddle.png b/systems-first-order-odes/_images/saddle.png new file mode 100644 index 000000000..dc9f1856c Binary files /dev/null and b/systems-first-order-odes/_images/saddle.png differ diff --git a/systems-first-order-odes/_images/stable_improper_node.png b/systems-first-order-odes/_images/stable_improper_node.png new file mode 100644 index 000000000..aa02f295f Binary files /dev/null and b/systems-first-order-odes/_images/stable_improper_node.png differ diff --git a/systems-first-order-odes/_images/stable_spiral.png b/systems-first-order-odes/_images/stable_spiral.png new file mode 100644 index 000000000..dee29f548 Binary files /dev/null and b/systems-first-order-odes/_images/stable_spiral.png differ diff --git a/systems-first-order-odes/_images/unstable_improper_node.png b/systems-first-order-odes/_images/unstable_improper_node.png new file mode 100644 index 000000000..9bf96264d Binary files /dev/null and b/systems-first-order-odes/_images/unstable_improper_node.png differ diff --git a/systems-first-order-odes/_images/unstable_spiral.png b/systems-first-order-odes/_images/unstable_spiral.png new file mode 100644 index 000000000..ff0ef0b6e Binary files /dev/null and b/systems-first-order-odes/_images/unstable_spiral.png differ diff --git a/systems-first-order-odes/homogeneous-constant-coefficient.md b/systems-first-order-odes/homogeneous-constant-coefficient.md index 3a8f9a4f7..6d1c2468b 100644 --- a/systems-first-order-odes/homogeneous-constant-coefficient.md +++ b/systems-first-order-odes/homogeneous-constant-coefficient.md @@ -263,6 +263,134 @@ c_2(t) &= \frac{1}{2} \left(1 - e^{-0.2 t}\right) ```` +## Types of critical points + +We can use the eigenvalues and eigenvectors to anticipate what solutions +around *critical points* ($\vv{y}' = \vv{0}$, or steady states) look like. We +will focus our discussion on only 2 x 2 systems. + +- If the eigenvalues are real and distinct, $\lambda_1 \ne \lambda_2$: + + ::::{grid} + :::{grid-item-card} Unstable improper node + + $$ + \lambda_1 > \lambda_2 > 0 + $$ + + ```{image} ./_images/unstable_improper_node.png + :alt: Unstable improper node + :width: 150px + :align: center + ``` + + ::: + :::{grid-item-card} Stable improper node + + $$ + \lambda_1 < \lambda_2 < 0 + $$ + + ```{image} ./_images/stable_improper_node.png + :alt: Unstable improper node + :width: 150px + :align: center + ``` + + ::: + :::{grid-item-card} Saddle + + $$ + \lambda_1 > 0, \quad \lambda_2 < 0 + $$ + + ```{image} ./_images/saddle.png + :alt: Saddle + :width: 150px + :align: center + ``` + + ::: + :::: + +- If the eigenvalues are complex, $\lambda_{1,2} = \alpha \pm i\omega$: + + ::::{grid} + :::{grid-item-card} Unstable spiral + + $$ + \alpha > 0 + $$ + + ```{image} ./_images/unstable_spiral.png + :alt: Unstable spiral + :width: 150px + :align: center + ``` + + ::: + :::{grid-item-card} Stable spiral + + $$ + \alpha < 0 + $$ + + ```{image} ./_images/stable_spiral.png + :alt: Stable spiral + :width: 150px + :align: center + ``` + + ::: + :::{grid-item-card} Center + + $$ + \alpha = 0 + $$ + + ```{image} ./_images/center.png + :alt: Center + :width: 150px + :align: center + ``` + + ::: + :::: + + The direction of the orbit depends on the matrix and can be checked for + some point. If + + \begin{equation} + \vv{A} = \begin{bmatrix}a & b \\ c & d\end{bmatrix} + \end{equation} + + The orbit is clockwise if $b > c$ and counterclockwise if $c < b$. + +- If the eigenvalues are real and repeated, $\lambda_1 = \lambda_2$: + + ::::{grid} + :::{grid-item-card} Proper node / star + + **A** is a multiple of **I**. + + ```{image} ./_images/proper_node.png + :alt: Proper node + :width: 150px + :align: center + ``` + + ::: + :::{grid-item-card} Degenerate node + + ```{image} ./_images/degenerate_node.png + :alt: Degenerate node + :width: 150px + :align: center + ``` + + ::: + :::: + ## Skill builder problems Solve the inital value problem $ \vv{y}' = \vv{A}\vv{y}$ with