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

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:
4. Stable - Trajectories pull onto the limit cycle
5. 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. Finding \zeta is tricky.

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)