From 8bd60e4b2bdaec49cdb7008623fc32210f8baa18 Mon Sep 17 00:00:00 2001 From: Dane Sabo Date: Mon, 30 Sep 2024 13:30:32 -0400 Subject: [PATCH] vault backup: 2024-09-30 13:30:32 --- .../2024-09-30 Limit Cycles.md | 20 ++++++++++++++++++- 1 file changed, 19 insertions(+), 1 deletion(-) 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 5c85cc31..b18ff032 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 @@ -38,4 +38,22 @@ $\nabla \cdot (\zeta f) = e^{-2 x}$ Now a special note: These functions can define where limit cycles can't be. If the function doesn't change sign for a subset of R, there can't be a limit cycle contained in that subset. There CAN be a limit cycle that crosses the point the function changes sign. ### Lyapunov Function -Aleksander Lyapunov (Liapunov) \ No newline at end of file +Aleksander Lyapunov (Liapunov) +$V(\vec x) = V(x,y) \leftarrow$ a scalar function +$V(\vec x) > 0 \forall \vec x\neq \vec x^*$ +$V(\vec x^*) = 0$ +$\dot V = \frac{dV}{dt} <0$ +$V(\vec x)$ is a positive definite function. +Then the system is stable ISL (in the sense of Lyapunov). +The system will always asymptotically approach the equilibrium point. +$\frac{dV}{dt} = \frac{dV}{dx} \frac{dx}{dt} + \frac{dV}{dy} \frac {dy}{dt} = \dot x \frac{dV}{dx} + \dot y \frac{dV}{dy}$ + +Example: +$\dot x = y - x^3$ +$\dot y = -x-y^3$ +$V(x,y) = c_1 x^2 + c_2 y^2$ +$\frac{dV}{dt} = 2 c_1 x \dot x + 2 c_2 y \dot y$ +$= 2 c_1 x(y-x^3) + 2c_2 y(-x-y^3)$ +Assume $c_1 = c_2$ +... $\therefore \frac{dV}{dt} = -2c(x^4+y^4) < 0$ +Therefore limit cycles are not possible.