Stability of nonlinear system

i need show that at the following system the zero solution is nominally stable, using some change of variable that transforme in a linear system

$$\frac{dx}{dt}=-x + \beta (x^2+ y^2)$$

$$\frac{dy}{dt}=-2y + \gamma x y$$

i tried with the eigenvalues of the Jacobian matrix at (0,0), but one of them is positive , then the system is unstable...

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What does the linearized matrix look like?

HallsofIvy
Homework Helper
No, it is pretty obvious just looking at this system what the eigenvalues are and they are both negative.

epenguin
Homework Helper
Gold Member
What point are you linearising about?

OK now I see you say only (0, 0) is required. Doesn't seem to me you need any transformation for that point.

If I am not mistaken there is the possibility of three 'equilibrium' points - the other two may be more interesting than (0, 0).

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the problem says:
"show that the zero solution is nonlinear stable. For this, find the change of variable that transforms this system in a linear system"....

i dont understand

epenguin
Homework Helper
Gold Member
i need show that at the following system the zero solution is nominally stable, using some change of variable that transforme in a linear system

the problem says:
"show that the zero solution is nonlinear stable. For this, find the change of variable that transforms this system in a linear system"....

i dont understand

Can anyone tell me what 'nominally stable' means? I know what 'locally stable' is, which would usually be the question.

To transform the whole system in which there are in general three different stationary points into a linear one would seem on the face of it impossible, isn't it?:uhh:

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