Nonlinear dynamics bifurcation

1. Jul 21, 2011

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

2. Jul 21, 2011

pmsrw3

Have you read Strogatz, Nonlinear Dynamics and Chaos?

3. Jul 22, 2011

Eynstone

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

4. Jul 23, 2011

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?

5. Jul 23, 2011

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.

6. Aug 3, 2011

gato_

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