vault backup: 2024-09-30 13:04:30

1 Daily Notes/2024/9 September/2024-09-30.md

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 # Calendar Tasks
-- [[300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-09-30]] [startTime:: 12:30] [endTime:: 15:00] 
+- [[2024-09-30 Limit Cycles]] [startTime:: 12:30] [endTime:: 15:00] 
 - 

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

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+# What is a limit cycle?
+Isolated, closed trajectories.
+1. Not like a center.
+2. Centers are closed, but not isolated.
+3. Neighboring trajectories are NOT closed. 
+Different forms:
+1. **Stable** - Trajectories pull onto the limit cycle
+2. **Unstable** - Trajectories are repelled by the limit cycle.
+
+**A imit cycle is a explicitly nonlinear phenomenon.**
+
+You can't identify if there is a limit cycle by using linearizing methods.
+# How do we find limit cycles?
+## How do we rule out a closed loop?
+### Dulac's Criterion:
+If we have some flow field:
+$$ \dot{\vec{x}}= f(\vec x)$$
+- If we can find a function $\zeta(x,y)$ such that $\nabla \cdot (\zeta f))$ does not change sign in some region of $R$, then there's no limit cycle in that region. 
+- If in some region $R$, $\zeta(x,y)$ s.t :
+  $$ \frac{\partial}{\partial x} (\zeta(x,y) f_1(x,y)) + \frac{\partial}{\partial y}(\zeta(x,y) f_2(x,y))$$
+  is of constant sign, then there are no closed orbits in R. 
+<mark style="background: #FFF3A3A6;">Finding $\zeta$ is tricky.</mark>