# Possible to find a gradient system for this?

1. Feb 21, 2013

### Jamin2112

1. The problem statement, all variables and given/known data

Does

x' = xex2tanh(x+y) + (1/2)ex2sech2(x+y)
y' = (1/2)ex2sech2(x+y)

contain a limit cycle? Possibly-relevant theorem below.

2. Relevant equations

Theorem. Closed orbits are impossible in gradient systems.

Definition. If x' = -ΔV for some cont. diff'ble, single-valued scalar func. V(x), then such a system is called a gradient system with potential function V.

3. The attempt at a solution

So the problem looks like a setup to apply this theorem .... but I can't figure out how. I've started at it for like 1 hr.

2. Feb 22, 2013

### haruspex

You mean -∇V, right?

3. Feb 22, 2013

### haruspex

Since you want that to look like -∂V/∂y, what does it suggest for V?

4. Feb 22, 2013

### Jamin2112

That V(x,y) should contain ∫-∂V/∂y dy somewhere

5. Feb 22, 2013

### Jamin2112

But I don't know how to integrate that. I can't even find it on an integral chart.

6. Feb 22, 2013

### haruspex

Remember, you're only trying to integrate that wrt y. You should find sech2 on the chart. If not, what's the integral of sec2?

7. Feb 22, 2013

### Jamin2112

Ah, okay. So I know, for one thing, that V(x,y) somehow involves (1/2)xex2tanh(x+y). And ....... I see it now. If V(x,y) = (1/2)xex2tanh(x+y), then ∂V/∂x = xex2tanh(x+y) + (1/2)xex2sech2(x+y). But, I want <-∂V/∂x, -∂V/∂y> = < xex2tanh(x+y) + (1/2)ex2sech2(x+y), (1/2)ex2sech2(x+y)>.

Last edited: Feb 22, 2013
8. Feb 22, 2013

### Jamin2112

Ah, wait. I could do

V(x,y) = (1/2)ex2tanh(-(x+y)). Then

-∂V/∂x = -[xex2 * tanh(-(x+y)) + (1/2)ex2 * -sech2(-(x+y))]
= xex2 * tanh(x+y) + (1/2)ex2 * sech2(x+y)

since sech is even and tanh is odd.

9. Feb 22, 2013

That's it.