ODE system. Limit cycle; Hopf bifurcation.

[spitz]

Homework Statement

The following two-dimensional system of ODEs possesses a limit-cycle solution for certain values of the parameter $a$. What is the nature of the Hopf bifurcation that occurs at the critical value of $a$ and state what the critical value is.

Homework Equations

$\dot{x}=-y+x(a+x^2+(3/2)y^2)$

$\dot{y}=x+y(a+x^2+(3/2)y^2)$

The Attempt at a Solution

By setting each equation to zero, i found the only equilibrium point to be $(0,0)$.

For the Jacobian matrix at $(0,0)$, I have:

$J(0,0)=\left( \begin{array}{cc} a & -1\\ 1 & a\end{array} \right)$

So:

$\tau=2a$

$\delta=a^2+1>0$

$\bigtriangleup=4a^2-4a^2-4=-4<0$

which gives:

$a<0$: $(0,0)$ is an attractor spiral.

$a>0$: $(0,0)$ is a repellor spiral.

$a=0$: $(0,0)$ is a center.

Does this mean that paths spiral into $(0,0)$ for negative $a$, and then spiral out towards a stable limit cycle. And, for positive $a$, beyond the limit cycle, paths spiral in towards it.

How do I find the critical value?

 

[Adrmay]

Is it just 0? The nature of the Hopf bifurcation that occurs at the critical value of a is that a stable limit cycle appears when a passes through 0. The critical value is a=0.