What is a limit cycle?

Isolated, closed trajectories.

Not like a center.
Centers are closed, but not isolated.
Neighboring trajectories are NOT closed. Different forms:
Stable - Trajectories pull onto the limit cycle
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) V(\vec x) = V(x,y) \leftarrow a scalar function V(\vec x) > 0 \forall \vec x\neq \vec x^* V(\vec x^*) = 0 \dot V = \frac{dV}{dt} <0 V(\vec x) is a positive definite function. Then the system is stable ISL (in the sense of Lyapunov). The system will always asymptotically approach the equilibrium point. \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}

Example: \dot x = y - x^3 \dot y = -x-y^3 V(x,y) = c_1 x^2 + c_2 y^2 \frac{dV}{dt} = 2 c_1 x \dot x + 2 c_2 y \dot y = 2 c_1 x(y-x^3) + 2c_2 y(-x-y^3) Assume c_1 = c_2 ... \therefore \frac{dV}{dt} = -2c(x^4+y^4) < 0 Therefore limit cycles are not possible.

Index Method

This is a method covered in the book. Sometimes is used to rule out limit cycles.

Poincare - Bendixon Theorem.

Book!

2.6 KiB Raw Blame History