• Support PF! Buy your school textbooks, materials and every day products Here!

Stability of nonlinear system

  • #1
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

[tex] \frac{dx}{dt}=-x + \beta (x^2+ y^2) [/tex]

[tex] \frac{dy}{dt}=-2y + \gamma x y [/tex]

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

Answers and Replies

  • #2
6,054
390
What does the linearized matrix look like?
 
  • #3
HallsofIvy
Science Advisor
Homework Helper
41,833
955
No, it is pretty obvious just looking at this system what the eigenvalues are and they are both negative.
 
  • #4
epenguin
Homework Helper
Gold Member
3,724
765
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).
 
Last edited:
  • #5
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
 
  • #6
epenguin
Homework Helper
Gold Member
3,724
765
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:
 
Last edited:

Related Threads on Stability of nonlinear system

  • Last Post
Replies
3
Views
500
  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
2
Views
948
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
5
Views
1K
Replies
7
Views
960
  • Last Post
Replies
13
Views
2K
Replies
1
Views
1K
Replies
1
Views
1K
Top