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300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-09-30 Limit Cycles.md

@@ -39,3 +39,21 @@ $\nabla \cdot (\zeta f) = e^{-2 x}$
 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)
+$V(\vec x) = V(x,y) \leftarrow$ 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.