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300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-09-30 Limit Cycles.md
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@ -12,7 +12,7 @@ Different forms:
You can't identify if there is a limit cycle by using linearizing methods. You can't identify if there is a limit cycle by using linearizing methods.
# How do we find limit cycles? # How do we find limit cycles?
## How do we rule out a closed loop? ## How do we rule out a closed loop?
### Dulac's Criterion: ### Bendixon's Criterion:
If we have some flow field: If we have some flow field:
$$ \dot{\vec{x}}= f(\vec x)$$ $$ \dot{\vec{x}}= f(\vec x)$$
- 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 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.