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I've been checking out numerically an ODE model of a gene circuit. Just from simulations, it appears that once a parameter passes some critical value a stable fixed point splits into three other fixed points (two stable, and one unstable) and an unstable limit cycle.

Initially, I thought it was just a supercritical pitchfork bifurcation followed by a Hopf bifurcation (around the unstable fixed point). However, in some piece-wise linear approximation of the model (for which, given some parameter values, it is possible to compute the trajectories analytically and get a definite answer on how unstable/stable fixed points/limit cycles there are) the emergence of the two stable fixed points and the limit cycles seems to happen at the same parameter value.

Does this type of bifurcation (one stable fixed point $\rightarrow$ one unstable limit cycle, one unstable and two stable fixed points) exists? If so what is its name?

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  • $\begingroup$ Is this a system in the plane? I suppose there is a stable fixed point inside the limit cycle, but where are the other fixed points? It would be interesting to see what the equations are. $\endgroup$ – Oberdada Jun 21 '13 at 15:30
  • $\begingroup$ No, it is not, in the simplest case it is eight dimensional. The equations are $$\dot{x}_i = \frac{c}{1+z_{i-1}^h}-x_i,$$$$\dot{z}_i = x_i-z_i,$$ where $i=1,\dots,4$, $c>0$, $h$ is an integer larger than zero (say $10$) and I'm using the convention $z_0\equiv z_4$. The bifurcation occurs when $c$ is slightly greater than $1$, however it is hard to see the limit cycle in simulations without increasing $c$ further. $\endgroup$ – jkn Jun 21 '13 at 15:45
  • $\begingroup$ what is exactly the stable fixed point that you have calculated prior to transition? $\endgroup$ – al-Hwarizmi Jun 23 '13 at 17:28
  • $\begingroup$ While there is no complete theory of bifurcation for 8 dimensional systems, you might be able to at least check for consistency by using some kind of a degree argument. If you search for "Browuer degree and bifurcation theory", you may get something on those lines. $\endgroup$ – nonlinearism Jun 24 '13 at 1:23

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