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