Where Does Homoclinic Bifurcation Occur in a 2-D Nonlinear Dynamics System?

[phyalan]

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

 

[pmsrw3]

Have you read Strogatz, Nonlinear Dynamics and Chaos?

 

[Eynstone]

Could you describe how the system depends on the parameters ( atleast qualitatively) ?

 

[phyalan]

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
$\dot{x}=-x+\frac{yx^{2}}{1+ax^{2}}C$
$\dot{y}=\frac{1}{r}(1-y-\frac{byx^{2}}{1+ax^{2}})$
where r and C are constants and a,b are the parameters that cause bifurcation
Can anyone give me some ideas?

 

[Eynstone]

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.

 

[gato_]

Look for the stationary points $$\dot{x}=\dot{y}=0$$ You have two algebraic equations depending on the parameters