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?