60 lines
2.5 KiB
Markdown
60 lines
2.5 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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Example:
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$\dot x = y$
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$\dot y = -x -y + x^2 + y^2$
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Assume $\zeta(x,y) = 1$
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$\partial / \partial x (y) + \partial / \partial y (-x - y +x^2 +y^2) /rightarrow 0 + (-1+2y)$
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Assume $\zeta(x,y) = e^{\alpha x}$
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$\partial / \partial x (e^{\alpha x} y) + \partial / \partial y (e^{\alpha x} (-x - y +x^2 +y^2))$
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$\alpha e^{\alpha x} y + 2 y e^{\alpha x} - e^{\alpha x}$
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$e^{\alpha x}((\alpha+2) y -1)$
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Now let $\alpha = -2$
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$\nabla \cdot (\zeta f) = e^{-2 x}$
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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.
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### Lyapunov Function
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Aleksander Lyapunov (Liapunov)
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$V(\vec x) = V(x,y) \leftarrow$ a scalar function
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$V(\vec x) > 0 \forall \vec x\neq \vec x^*$
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$V(\vec x^*) = 0$
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$\dot V = \frac{dV}{dt} <0$
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$V(\vec x)$ is a positive definite function.
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Then the system is stable ISL (in the sense of Lyapunov).
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The system will always asymptotically approach the equilibrium point.
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$\frac{dV}{dt} = \frac{dV}{dx} \frac{dx}{dt} + \frac{dV}{dy} \frac {dy}{dt} = \dot x \frac{dV}{dx} + \dot y \frac{dV}{dy}$
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Example:
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$\dot x = y - x^3$
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$\dot y = -x-y^3$
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$V(x,y) = c_1 x^2 + c_2 y^2$
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$\frac{dV}{dt} = 2 c_1 x \dot x + 2 c_2 y \dot y$
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$= 2 c_1 x(y-x^3) + 2c_2 y(-x-y^3)$
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Assume $c_1 = c_2$
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... $\therefore \frac{dV}{dt} = -2c(x^4+y^4) < 0$
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Therefore limit cycles are not possible.
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