Results 1 to 3 of 3

Math Help - non-linear differential equation system

  1. #1
    Newbie
    Joined
    Sep 2008
    Posts
    17

    non-linear differential equation system

    Hi,

    I am having trouble with the following question..

    dx/dt = y
    dy/dt = -x + (1-x^2)y

    Find the critical points and determine their stability.


    Now, I found the critical point to be (0,0).
    The matrix representing the linearized system is
    A = [ 0, 1; -1, 0 ] and eigenvalues of this matrix are -i and i, which are complex with the real parts equal to zero, thereofore Linearization theorem can't be used here.

    so.. using coordinate shift I get X=x and Y=y
    thus dY/dX = -X/Y + (1-X^2).. to solve this I get an integrating factor
    e^(0.5*x^2)... which can't be integrated. This is where I get stuck (assuming I have done everything else above correctly)

    Does this mean the critical point is unstable and that the orbits of the non-linear system cannot be approximated by the linear approximation?

    Or have I done something wrong along the way?

    Thanks in advance
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Aug 2008
    Posts
    903
    I get 1/2\pm \sqrt{3}/2 for the eigenvalues: The origin is a source but that's true only for solutions near the origin. Further qualitative analysis would show that solutions close to the origin spiral outward to the equilibrium solution which is periodic and solutions outside the equilibrium solution spiral towards it. Here's some Mathematica code to illustrate these two scenarios. Me personally, I'd always do it numerically just to check things.

    Code:
    tmax = 50; 
    xinit = 0.2; 
    yinit = 0.2; 
    sol = NDSolve[{Derivative[1][x][t] == y[t], 
        Derivative[1][y][t] == 
         -x[t] + (1 - x[t]^2)*y[t], 
        x[0] == xinit, y[0] == yinit}, 
       {x[t], y[t]}, {t, 0, tmax}]
    p1 = ParametricPlot[Evaluate[{x[t], y[t]} /. 
         sol], {t, 0, tmax}, PlotStyle -> Red]
    
    tmax = 50; 
    xinit = 2.5; 
    yinit = 2.5; 
    sol = NDSolve[{Derivative[1][x][t] == y[t], 
        Derivative[1][y][t] == 
         -x[t] + (1 - x[t]^2)*y[t], 
        x[0] == xinit, y[0] == yinit}, 
       {x[t], y[t]}, {t, 0, tmax}]
    p2 = ParametricPlot[Evaluate[{x[t], y[t]} /. 
         sol], {t, 0, tmax}, PlotStyle -> Blue]
    Show[{p1, p2}, PlotRange -> 
       {{-3, 3}, {-3, 3}}]
    Attached Thumbnails Attached Thumbnails non-linear differential equation system-eqsol.jpg  
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Sep 2008
    Posts
    17
    Hey,

    thanks shawsend. I figured out my mistake yday.. so it's all good
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Difficult non-linear system of two differential equations
    Posted in the Differential Equations Forum
    Replies: 9
    Last Post: March 29th 2011, 07:36 AM
  2. System of Linear Differential Equations with Constant Coefficients
    Posted in the Differential Equations Forum
    Replies: 8
    Last Post: February 28th 2011, 05:51 AM
  3. Runge Kutta, linear differential system
    Posted in the Math Software Forum
    Replies: 5
    Last Post: December 4th 2010, 11:38 PM
  4. Replies: 0
    Last Post: September 12th 2010, 01:51 PM
  5. Replies: 4
    Last Post: April 23rd 2010, 08:11 AM

Search Tags


/mathhelpforum @mathhelpforum