From 4f7b6ebe73226445214381c5fd6d94f49a56e34e Mon Sep 17 00:00:00 2001 From: Dane Sabo Date: Mon, 28 Oct 2024 15:30:42 -0400 Subject: [PATCH] vault backup: 2024-10-28 15:30:42 --- .../2024-10-28 Stability.md | 19 +++++++++++++++++- Pasted image 20241028152704.png | Bin 0 -> 65376 bytes 2 files changed, 18 insertions(+), 1 deletion(-) create mode 100644 Pasted image 20241028152704.png diff --git a/300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-10-28 Stability.md b/300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-10-28 Stability.md index 03d9f414f..3bb46570b 100644 --- a/300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-10-28 Stability.md +++ b/300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-10-28 Stability.md @@ -22,5 +22,22 @@ $$\text{dist}[x, c] = \min_{y \in C}|x-y|$$ Where c is a curve. Where in the plane we're using the minimum of the 2 norm. +## Summary Stable half-paths can be generally stable, approaching an equilibrium, or periodic. -Unstable half-paths exceed the bound $\epsilon$ somewhere. \ No newline at end of file +Unstable half-paths exceed the bound $\epsilon$ somewhere. + +Poincare cannot handle the time dependency of systems. As a result, we can't really use Poincare to handle real systems. That leads us to..... + +# Lyapunov Stability +Basically extend the 2D distance formula we talked about last time to include n dimensions. (May need to analyze complex solutions as well). + +Let's define Lyapunov Stability: +>[!note] Lyapunov Stability Definition +>Let $x^*$ be a real or complex solution of $x = X(x,t)$. Then, +>1. $x^*$ is lyapunov stable iff for each value of $\epsilon>0$ however small there is a corresponding value of $\delta>0$ such that +>![[Pasted image 20241028152704.png]] +>2. If the system is autonomous, then we can disregard the idea of $t_0$ in 1. +>3. Otherwise, we call the system unstable in the sense of Lyapunov. + +This stability definition defines that for an autonomous system, Lyapunov stability is sufficient for Poincare stability. + diff --git a/Pasted image 20241028152704.png b/Pasted image 20241028152704.png new file mode 100644 index 0000000000000000000000000000000000000000..ef616b1351ba1bf669dbd6df62f8871553988852 GIT binary patch literal 65376 zcmeFYWmg?R(>981+}&a0uECv+ySoPn?(XgyG{H4EY#f5SOK^90cgV@}KKFgr`{n$D zGaqKUx@)G_s;a4}zWSPI6(wmDL_$Oe2nZBe83{EA2S6(5>ELK@!R%`0VqxLnYUSv51>G(DPl@?|l*C;u zOx>&<9e^6v_7)I|{~8>;KxH#GAO{-0009Ankd+YA z@X9{#^vuqY_AmQ8(Q(iy_=i=x`kI)uI)t%~#HH*MQ zPELjHM;`VKei@s{&Y&*7-N1Z2d7-`8=50lNi=$iQP{?w+#=~M%XLc-8IF*7$lGHhtC2^&6g1YfM+A_}iv9LVaQ^q4iKSG=;BFcYsVMBKSLv-Q)uhi=VWtIy1--rKCCvPyjT*3dD>i?)@_u!Yz{LjSy zdxMfm-2XihEt?cQ=YPkLBe3xQ|7rhYy8mw@)1vO`@boyXQ;Pm{HunlO{jc#I{ZcX`*bAw@mVm;M29!#*4xZ)!Se!EW%g z`#}WX3OQI(3+Vv9F4y}~&j*>ERa;-glnZh}_|m%09W&w))K&Qs{uD#>NN3O6T=WhG 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