41 lines
1.8 KiB
Markdown
41 lines
1.8 KiB
Markdown
# 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>
|
|
|
|
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 $\alpha = -2$
|
|
$\nabla \cdot (\zeta f) = e^{-2 x}$
|
|
|
|
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) |