From c5a6542e147e5ede489ba485cc278d443b27c7b1 Mon Sep 17 00:00:00 2001 From: Dane Sabo Date: Mon, 30 Sep 2024 13:15:33 -0400 Subject: [PATCH] vault backup: 2024-09-30 13:15:33 --- .../2024-09-30 Limit Cycles.md | 15 +++++++++++++++ 1 file changed, 15 insertions(+) diff --git a/300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-09-30 Limit Cycles.md b/300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-09-30 Limit Cycles.md index 0ac2a6d5..cacaa97b 100644 --- a/300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-09-30 Limit Cycles.md +++ b/300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-09-30 Limit Cycles.md @@ -20,3 +20,18 @@ $$ \dot{\vec{x}}= f(\vec x)$$ $$ \frac{\partial}{\partial x} (\zeta(x,y) f_1(x,y)) + \frac{\partial}{\partial y}(\zeta(x,y) f_2(x,y))$$ is of constant sign, then there are no closed orbits in R. Finding $\zeta$ is tricky. + +Example: +$\dot x = y$ +$\dot y = -x -y + x^2 + y^2$ + +Assume $\zeta(x,y) = 1$ +$\partial / \partial x (y) + \partial / \partial y (-x - y +x^2 +y^2) /rightarrow 0 + (-1+2y)$ + +Assume $\zeta(x,y) = e^{\alpha x}$ + +$\partial / \partial x (e^{\alpha x} y) + \partial / \partial y (e^{\alpha x} (-x - y +x^2 +y^2))$ +$\alpha e^{\alpha x} y + 2 y e^{\alpha x} - e^{\alpha x}$ +$e^{\alpha x}((\alpha+2) y -1)$ +Now let $\alpha = -2$ +$\nabla \cdot (\zeta f) = e^{-2 x}$