Periodic orbits: Bendixson-Dulac

Consider the system

$\displaystyle \begin{aligned} &\dot{x}=f(x, y)\\ &\dot{y}=g(x, y) \end{aligned}$

How do we get information about the number of periodic orbits of this system? Can we show that there are no periodic orbit given some system? What are some sufficient conditions?

Bendixson: On a simply connected domain ${D \subset \mathbb{R}^2}$ , if the quantity ${ \frac{\partial f}{\partial x}+\frac{\partial g}{\partial y} }$ doesn’t change sign, then the above system has not periodic orbits.

Proof: If ${\gamma}$ is a periodic orbit and ${S}$ is the region enclosed by ${\gamma}$ , we see that (Stoke’s theorem)

$\displaystyle \int_{S}\left(\frac{\partial f}{\partial x}+\frac{\partial g}{\partial y}\right) d x d y =\int_{\gamma} f d y-g d x = 0$

where the last equality follows from the fact that the vector field ${(f,g)}$ is tangential to the curve.

Example:

$\displaystyle \begin{aligned} &\dot{x}=y\\ &\dot{y}=x-x^{3}-\delta y \end{aligned}$

$\displaystyle \frac{\partial f}{\partial x}+\frac{\partial g}{\partial y}=-\delta$

Hence we see that for ${\delta \neq 0}$ , the above system has no closed orbits.

What if $\displaystyle \frac{\partial f}{\partial x}+\frac{\partial g}{\partial y}$ doesn’t have the property we want and changes sign? The following generalized statement might help.

Bendixson-Dulac: We are using that the vector field is tangential to the a closed curve and hence by Stokes theorem the integral over the region enclosed is required to be zero. We can generalize by noting that by multiplying by a scalar field $V(x,y)$ , the vector field ${V(x,y) (f, g)}$ obtained will also be tangential to a periodic orbit. On a simply connected domain ${D \subset \mathbb{R}^2}$ , if the quantity
${ \frac{\partial Vf}{\partial x}+\frac{\partial Vg}{\partial y} }$ doesn’t change sign, then the system has not periodic orbits.

We can also cosnider ${l}$ -connected domains and in this case stokes theorem says that if the quantity ${ \frac{\partial Vf}{\partial x}+\frac{\partial Vg}{\partial y} }$ doesn’t change sign, maximum number of periodic orbits is ${l.}$

Example:
$\displaystyle \dot{x}=x(a x+b y+c)$
$\displaystyle \dot{y}=y(d x+e y+f)$

If a trajectory intersects the axes , it’s easy to see that it cannot be periodic. So we assume that our domain is a quadrant-let’s just say the first quadrant.

We consider Dulac function ${V(x, y)=x^{A} y^{B}}$

$\displaystyle \frac{\partial Vf}{\partial x}+\frac{\partial Vg}{\partial y} =x^{A} y^{B}((a A+d B+2 a+d) x+(b A+e B+2 e+b) y+(c A+f B+c+f))$

If ${ae-bd\neq 0}$ , we can choose ${A,B}$ such that the above quantity is ${\frac{a b f+c e d-a e f-a c e}{a e-b d} x^{A} y^{B}}$ by making the coefficients of ${x,y}$ zero.
If ${a b f+c e d-a e f-a c e \neq 0}$ , by applying Bendixson-Dulac, we see that this system cannot have periodic orbits.
If ${a b f+c e d-a e f-a c e = 0}$ , the system can actually have periodic orbit – take a Lotka-Volterra system for instance.
If ${ae-bd=0,}$ the differential equation can be simplified to ${\dot{x}=p x, \dot{y}=q y,}$ which clearly doesn’t have periodic orbits.

One can generalize this result in many ways to get bounds on the number of limit cycles of a system in a region
https://core.ac.uk/download/pdf/78536536.pdf

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