1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Phase portrait of nonlinear system of differential equations

  1. Nov 7, 2009 #1
    1. The problem statement, all variables and given/known data

    Describe the phase portrait of the nonlinear system x' = x^2, y' = y^2
    Also, find the equilibrium points and describe the behaviour of the associated linearized system.

    3. The attempt at a solution

    We have an equilibrium point at (0,0).
    The associated linearized system is x' = 0, y' = 0. The phase portrait for this consists of lines of equilibria along x = 0, and y = 0.

    For the nonlinear system, I have found solutions x(t) = -1/t and y(t) = -1/t. I don't know what these solutions mean in terms of a phase portrait. Nor can I express the solutions in terms of constants x_0 and y_0.
  2. jcsd
  3. Nov 8, 2009 #2


    Staff: Mentor

    It looks like you have left off the constant of integration when you found solutions x(t) and y(t). Both of your differential equations are separable.

    dx/dt = x2 ==> dx/x2 = dt ==> [itex]\int dx/x^2 = \int dt[/itex]
    ==> -1/x = t + C1 ==> x = -1/(t + C1)

    Similarly, y = -1/(t + C2)
    You should be able to determine the constant from your initial conditions.

    In the special case where C1 = C2 = 0, the trajectories follow the line y = x. If t > 0, the solution points approach the origin along the part of the line in the third quadrant. If t < 0, the solution points approach the origin along the part of the line in the first quadrant. Different initial conditions will generate different trajectories, but I believe all of them will be straight lines pointing into the origin.

    Does that make sense? It has been a lot of years since I studied dynamical systems, so I might be a little off base on some of this.
  4. Nov 8, 2009 #3
    Thanks :) It makes sense.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook