vault backup: 2024-09-30 13:15:33

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. 
 <mark style="background: #FFF3A3A6;">Finding $\zeta$ is tricky.</mark>
+
+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 $\alpha = -2$
+$\nabla \cdot (\zeta f) = e^{-2 x}$