Van der Pol oscillator + Hopf bifurcation

[nicksauce]

Homework Statement

Consider the biased van der Pol oscillator: $$\frac{d^2x}{dt^2}=u (x^2-1)\frac{dx}{dt} + x = a$$. Find the curves in (u,a) space at which Hopf bifurcations occur. (Strogatz 8.21)

Homework Equations

The Attempt at a Solution

Not even sure where to start with this question, other than turning it into a 2-d system. Any advice on how to approach this?

 

[quantum boy]

I'm attempting a very similar question. I've managed to turn it into a 2D system but haven't got much further.

dx/dt = y

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

A Hopf bifurcation is one where, as you vary u, the eigenvalues of the Jacobian change from having a negative real component to having a positive real component.

I'll let you know if I make more progress.

 

[quantum boy]

I've just looked in Strogatz at question 8.21 and I think you've copied out the question incorrectly:
you wrote an equals sign between d^2x/dt^2 and u(x^2 -1)
but Strogratz wrote a plus.

Luckily, this means we're working on exactly the same question.
The equations for the 2D system are not as written above; insteat they are:

dx/dt = y

dy/dt = a - x - u(x^2 -1)y

 

[quantum boy]

I think I've answered the question. My method was as follows:

I wrote down the Jacobian and got an expression for its eigenvalues. I then found the fixed points. I found what the eigenvalues are at the fixed points. It's then just a matter of inspecting your solution, looking at the definition of a Hopf bifurcation and seeing what values u and a need to take.

Best of luck,
quantum boy

 

[damian001]

i took did the jacobian and got eigenvalues of 0 and u(a^2)-u...cause i found a fixed point of (a,0) from nullclines...now what do I do...