From e29821bdc581e8f8473117c56e6c0d5b984bc800 Mon Sep 17 00:00:00 2001 From: mmm0226 Date: Sun, 4 May 2025 14:24:15 -0500 Subject: [PATCH 01/12] Create critical point for 2d systems --- critical point for 2d systems | 1 + 1 file changed, 1 insertion(+) create mode 100644 critical point for 2d systems diff --git a/critical point for 2d systems b/critical point for 2d systems new file mode 100644 index 000000000..8b1378917 --- /dev/null +++ b/critical point for 2d systems @@ -0,0 +1 @@ + From 5f1518f82e02306af80a9da24ef037b8433f2e78 Mon Sep 17 00:00:00 2001 From: mmm0226 Date: Wed, 7 May 2025 17:35:22 -0500 Subject: [PATCH 02/12] Update critical point for 2d systems --- critical point for 2d systems | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/critical point for 2d systems b/critical point for 2d systems index 8b1378917..f0b187201 100644 --- a/critical point for 2d systems +++ b/critical point for 2d systems @@ -1 +1 @@ - +Types of Critical Points for 2d Systems From d3995f0b19662e2ca421d8a858d25ee31099baab Mon Sep 17 00:00:00 2001 From: mmm0226 Date: Wed, 7 May 2025 17:51:53 -0500 Subject: [PATCH 03/12] Update critical point for 2d systems --- critical point for 2d systems | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/critical point for 2d systems b/critical point for 2d systems index f0b187201..d2fb9c620 100644 --- a/critical point for 2d systems +++ b/critical point for 2d systems @@ -1 +1,3 @@ -Types of Critical Points for 2d Systems +### 3. Types of Critical Points for 2d Systems + +We We can use the eigenvalues and eigenvectors to anticipate what solutions around $\underline{critical points}$ (y' = 0, steady states) look like. This will be really useful for controls! From c2b3c0299a55fd5102e50032c83a17698dc8c9b6 Mon Sep 17 00:00:00 2001 From: mmm0226 Date: Wed, 7 May 2025 17:53:11 -0500 Subject: [PATCH 04/12] Update critical point for 2d systems --- critical point for 2d systems | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/critical point for 2d systems b/critical point for 2d systems index d2fb9c620..d7f928a4f 100644 --- a/critical point for 2d systems +++ b/critical point for 2d systems @@ -1,3 +1,5 @@ ### 3. Types of Critical Points for 2d Systems -We We can use the eigenvalues and eigenvectors to anticipate what solutions around $\underline{critical points}$ (y' = 0, steady states) look like. This will be really useful for controls! +We can use the eigenvalues and eigenvectors to anticipate what solutions around $\underline{critical points}$ (y' = 0, steady states) look like. This will be really useful for controls! + +If the eigenvalues are $\underline{real}$ and distinct: From deeb7da3439ab70d2e1c8939d9c3d680bf14f1dd Mon Sep 17 00:00:00 2001 From: mmm0226 Date: Wed, 7 May 2025 17:54:27 -0500 Subject: [PATCH 05/12] Update critical point for 2d systems From 785f258ad7ef0a10f86d898a14e02f6d811f3c7e Mon Sep 17 00:00:00 2001 From: mmm0226 Date: Thu, 8 May 2025 16:57:23 -0500 Subject: [PATCH 06/12] Update critical point for 2d systems --- critical point for 2d systems | 20 ++++++++++++++++++++ 1 file changed, 20 insertions(+) diff --git a/critical point for 2d systems b/critical point for 2d systems index d7f928a4f..ee133d615 100644 --- a/critical point for 2d systems +++ b/critical point for 2d systems @@ -3,3 +3,23 @@ We can use the eigenvalues and eigenvectors to anticipate what solutions around $\underline{critical points}$ (y' = 0, steady states) look like. This will be really useful for controls! If the eigenvalues are $\underline{real}$ and distinct: + +\begin{align} +\begin{equation} +\lambda_1 > \lambda_2 > 0 +\end{equation} + +improper node (unstable) + +\begin{equation} +\lambda_1 < \lambda_2 < 0 +\end{equation} + +improper node (stable) + +\begin{equation} +\lambda_1 > 0, & \lambda_2 < 0 +\end{equation} + +saddle +\end{align} From 74a9c26dc93d1c3f5072d960f6d5b9d4acc9c9d1 Mon Sep 17 00:00:00 2001 From: mmm0226 Date: Thu, 8 May 2025 17:26:05 -0500 Subject: [PATCH 07/12] Update critical point for 2d systems --- critical point for 2d systems | 24 ++++++++++++++++++++++++ 1 file changed, 24 insertions(+) diff --git a/critical point for 2d systems b/critical point for 2d systems index ee133d615..c62736631 100644 --- a/critical point for 2d systems +++ b/critical point for 2d systems @@ -23,3 +23,27 @@ improper node (stable) saddle \end{align} + +If the eigenvalues are $\underline{complex}$ & \lambda = \alpha \pm \omega i: + +\begin{align} +\begin{equation} +\alpha > 0 +\end{equation} + +spiral (unstable) + +\begin{equation} +\alpha < 0 +\end{equation} + +spiral (stable) + +\begin{equation} +\alpha = 0 +\end{equation} + +center +\end{align} + +The direction depends on the matrix and can be From d320e36a7437696b54b29fac97cad3aa9cbcd398 Mon Sep 17 00:00:00 2001 From: mmm0226 Date: Thu, 8 May 2025 17:32:22 -0500 Subject: [PATCH 08/12] Update critical point for 2d systems --- critical point for 2d systems | 13 ++++++++++++- 1 file changed, 12 insertions(+), 1 deletion(-) diff --git a/critical point for 2d systems b/critical point for 2d systems index c62736631..b243efa88 100644 --- a/critical point for 2d systems +++ b/critical point for 2d systems @@ -46,4 +46,15 @@ spiral (stable) center \end{align} -The direction depends on the matrix and can be +The direction depends on the matrix and can be checked for some point (e.g. (1,0)). +\begin{align} +If \vv{A} = +& +\begin{bmatrix} +-7 & 2 & -3 \\ +2 & -4 & -6 \\ +-1 & -2 & -3 +\end{bmatrix} & , +\end{align} + + From 64ca0d95b78b707c1cd5535af1e4a34233e4ad25 Mon Sep 17 00:00:00 2001 From: mmm0226 Date: Thu, 8 May 2025 17:37:15 -0500 Subject: [PATCH 09/12] Update critical point for 2d systems --- critical point for 2d systems | 6 ++++++ 1 file changed, 6 insertions(+) diff --git a/critical point for 2d systems b/critical point for 2d systems index b243efa88..63a717ec3 100644 --- a/critical point for 2d systems +++ b/critical point for 2d systems @@ -57,4 +57,10 @@ If \vv{A} = \end{bmatrix} & , \end{align} +b-c>0 & \to & clockwise + +b-c<0 & \to & counterclockwise + +If the eigenvalues are $\underline{real}$ and $\underline{repeated}$ + From 3458b8a0e6ab11d364df45c9726f31b59ae4ed11 Mon Sep 17 00:00:00 2001 From: mmm0226 Date: Thu, 8 May 2025 17:47:24 -0500 Subject: [PATCH 10/12] Update critical point for 2d systems --- critical point for 2d systems | 18 +++++++++++++++++- 1 file changed, 17 insertions(+), 1 deletion(-) diff --git a/critical point for 2d systems b/critical point for 2d systems index 63a717ec3..5572caa20 100644 --- a/critical point for 2d systems +++ b/critical point for 2d systems @@ -61,6 +61,22 @@ b-c>0 & \to & clockwise b-c<0 & \to & counterclockwise -If the eigenvalues are $\underline{real}$ and $\underline{repeated}$ +If the eigenvalues are $\underline{real}$ and $\underline{repeated}$ & (\lambda_1 = \lambda_2): + +\begin{align} +\vv{A} is a multiple of \vv{I} + +proper node/star + +direction depends on \vv{A} as for complex + +degenerate node + +unstable vs. stable + +\begin{equation} +\lambda > 0 & \lambda < 0 +\end{equation} +\end{align} From 65b0332f5458725a30b1cd45ce4f4a9b3c3a6a95 Mon Sep 17 00:00:00 2001 From: Michael Howard Date: Fri, 9 May 2025 14:03:56 -0500 Subject: [PATCH 11/12] Add images --- .../homogeneous-constant-coefficient.md | 2 +- systems-first-order-odes/_images/center.png | Bin 0 -> 33353 bytes .../_images/degenerate_node.png | Bin 0 -> 71652 bytes .../_images/proper_node.png | Bin 0 -> 65079 bytes systems-first-order-odes/_images/saddle.png | Bin 0 -> 50993 bytes .../_images/stable_improper_node.png | Bin 0 -> 45625 bytes .../_images/stable_spiral.png | Bin 0 -> 36293 bytes .../_images/unstable_improper_node.png | Bin 0 -> 45097 bytes .../_images/unstable_spiral.png | Bin 0 -> 36234 bytes .../homogeneous-constant-coefficient.md | 128 ++++++++++++++++++ 10 files changed, 129 insertions(+), 1 deletion(-) create mode 100644 systems-first-order-odes/_images/center.png create mode 100644 systems-first-order-odes/_images/degenerate_node.png create mode 100644 systems-first-order-odes/_images/proper_node.png create mode 100644 systems-first-order-odes/_images/saddle.png create mode 100644 systems-first-order-odes/_images/stable_improper_node.png create mode 100644 systems-first-order-odes/_images/stable_spiral.png create mode 100644 systems-first-order-odes/_images/unstable_improper_node.png create mode 100644 systems-first-order-odes/_images/unstable_spiral.png 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 0000000000000000000000000000000000000000..e6f702a35d1a61b814d0f832241b7da5d0f53f9d GIT binary patch literal 33353 zcmZ^~1yoeg`vp3~P(w?1C?QA;At5a&4N}rEAR^Ksok|TMN)D2WN|&TCz(^?#k_wUn zqDXhXGyeXy-dm5g#u@KDw@!TD-uv4Zbx&7~oP?PK0)dcgsH+-4Ab7kG2y~X15NxSj 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+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 From 0f8c2bbc526fdbf1277beef339902a2767e6a252 Mon Sep 17 00:00:00 2001 From: Michael Howard Date: Fri, 9 May 2025 14:04:27 -0500 Subject: [PATCH 12/12] Remove unused file --- critical point for 2d systems | 82 ----------------------------------- 1 file changed, 82 deletions(-) delete mode 100644 critical point for 2d systems diff --git a/critical point for 2d systems b/critical point for 2d systems deleted file mode 100644 index 5572caa20..000000000 --- a/critical point for 2d systems +++ /dev/null @@ -1,82 +0,0 @@ -### 3. Types of Critical Points for 2d Systems - -We can use the eigenvalues and eigenvectors to anticipate what solutions around $\underline{critical points}$ (y' = 0, steady states) look like. This will be really useful for controls! - -If the eigenvalues are $\underline{real}$ and distinct: - -\begin{align} -\begin{equation} -\lambda_1 > \lambda_2 > 0 -\end{equation} - -improper node (unstable) - -\begin{equation} -\lambda_1 < \lambda_2 < 0 -\end{equation} - -improper node (stable) - -\begin{equation} -\lambda_1 > 0, & \lambda_2 < 0 -\end{equation} - -saddle -\end{align} - -If the eigenvalues are $\underline{complex}$ & \lambda = \alpha \pm \omega i: - -\begin{align} -\begin{equation} -\alpha > 0 -\end{equation} - -spiral (unstable) - -\begin{equation} -\alpha < 0 -\end{equation} - -spiral (stable) - -\begin{equation} -\alpha = 0 -\end{equation} - -center -\end{align} - -The direction depends on the matrix and can be checked for some point (e.g. (1,0)). -\begin{align} -If \vv{A} = -& -\begin{bmatrix} --7 & 2 & -3 \\ -2 & -4 & -6 \\ --1 & -2 & -3 -\end{bmatrix} & , -\end{align} - -b-c>0 & \to & clockwise - -b-c<0 & \to & counterclockwise - -If the eigenvalues are $\underline{real}$ and $\underline{repeated}$ & (\lambda_1 = \lambda_2): - -\begin{align} -\vv{A} is a multiple of \vv{I} - -proper node/star - -direction depends on \vv{A} as for complex - -degenerate node - -unstable vs. stable - -\begin{equation} -\lambda > 0 & \lambda < 0 -\end{equation} -\end{align} - -