Differential equations equilibrium points

[sana2476]

Homework Statement

x'=x(x^2+y^2)
y'=y(x^2+y^2)

i) Find all the equilibrium points and describe the behavior of the associated linearized system
ii) Describe the phase portrait for the nonlinear system
iii) Does the linearized system accurately describe the local behavior near the equilibrium points?

Homework Equations

Solving for equilibrium, you get: x=0 and x^2+y^2=0. Likewise with second function, you get: y=0 and x^2+y^2=0. But I don't know what the phase portrait will behave.

 

[sana2476]

I know that getting the Jacobian matrix and evaluating it at (0,0). You get your eigenvalues as 0 as well.

 

[sana2476]

I really need help...can someone try and help me with this please!

 

[squidsoft]

I really need help...can someone try and help me with this please!

Hello sana. I can't help you directly here as I have limited access to the web. But I can give you really good advice in the interim: The book "Differential Equations" by Blanchard, Devaney and Hall does a really good job of working with these problems: First linearize it by calculating the partials and form the Jacobian matrix, find the eigenvalues and determine the types of fixed points based on the kinds of eigenvalues. Draw the phase portrait. Mathematica ver 7 has an excellent function for drawing in an instant what use to take hours: It's called "StreamPlot" (I think, or something like this) and draws the entire phase portrait in a single operation.

Note linearization fails for some systems. Find out which types of systems these are.