## Stability of the equilibrium at (0,0)

x'=y-x^3 and y'=-x^5

I've worked the jacobian which is
[-3x^2 1;-5x^4 0] and the equilibrium is at (0,0)
so jac = [0 1;0 0]

and eigenvalues are both 0

so is the stability non isolated point??? and what i can say about the basin of attraction of the origin?

Could anyone help me?

thanks a lot
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 Recognitions: Homework Help Science Advisor Not an area I know much about, but here goes... It's fairly clear that the trajectory spirals clockwise around the origin. Question is, does it spiral in or out? Maybe it spirals in within some annulus and out in another. For |y| >> |x5|, it approximates the ellipse-like curves 3y2+x6 = c. So set f(x,y) = 3y2+x6 and calculate f'. This pretty much settles the overall behavior, I think.

 Quote by haruspex Not an area I know much about, but here goes... It's fairly clear that the trajectory spirals clockwise around the origin. Question is, does it spiral in or out? Maybe it spirals in within some annulus and out in another. For |y| >> |x5|, it approximates the ellipse-like curves 3y2+x6 = c. So set f(x,y) = 3y2+x6 and calculate f'. This pretty much settles the overall behavior, I think.
but using the λ^2-τ*λ+Δ=0

if Δ=0 at least one of the eigenvalues is zero. then the origin is not an isolated fixed point. there is either a whole line of fixed points or a plane of fixed points if A=0

Spirals only satisfy τ^2-4*Δ<0.

I don't know how to draw it and pplane and XPP just confuse me.

Thanka a lot

## Stability of the equilibrium at (0,0)

Google for the terms "stable" or "asymptotically stable" statioary point. First you have to
understand this terms and then you will find a characterization based on the eigenvalues.

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