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Nonlinear dynamics bifurcation

  1. Jul 21, 2011 #1
    Hi everyone,
    I am studying a 2-D nonlinear dynamics system, with two key parameters. But I have trouble when I want to locate where the homoclinic bifurcation occurs in the parameter space. Can anyone give me some ideas or reference readings? Thx
     
  2. jcsd
  3. Jul 21, 2011 #2
    Have you read Strogatz, Nonlinear Dynamics and Chaos?
     
  4. Jul 22, 2011 #3
    Could you describe how the system depends on the parameters ( atleast qualitatively) ?
     
  5. Jul 23, 2011 #4
    I have had a quick look at the book Nonlinear dynamics and chaos but still can't get the answer numerically. Below is my system
    [itex]\dot{x}=-x+\frac{yx^{2}}{1+ax^{2}}C[/itex]
    [itex]\dot{y}=\frac{1}{r}(1-y-\frac{byx^{2}}{1+ax^{2}})[/itex]
    where r and C are constants and a,b are the parameters that cause bifurcation
    Can anyone give me some ideas?
     
  6. Jul 23, 2011 #5
    We could try the following special cases in which the equation becomes tractable (numerically) :
    1. b=0 or a=0. The equation can be solved for y & for x in turn. Check how the solution depends on a,b.
    2.When a is small, approximate 1/{1+ax^{2}} by 1- ax^{2} .
    3. When a is large, the system behaves as dx/dt=-x & dy/dt=frac{1}{r}(1-y)
    So, find a suitably large 'a' & observe the behaviour as a is decreased.
     
  7. Aug 3, 2011 #6
    Look for the stationary points [tex]\dot{x}=\dot{y}=0[/tex] You have two algebraic equations depending on the parameters
     
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