# Hopf bifurcation for systems where the dynamics is homogeneous of degree 1

Consider dynamical system in dimension 3 $$x'(t)=f(x(t),d)$$ where the dynamics f is homogeneous of degree 1 and there is exactly one line of equilibrium points. This line is independent of the parameter d The linearized system ha one zero eigenvalue and we assume that, the other two eigenvalues are $$\lambda (d)= \mu (d) \pm i\nu (d).$$ Assume that, at d=d_0, two purely imaginary eigenvalues appear with. $$\mu (d_0)=0, \quad and \quad\mu' (d_0)\not = 0.$$

In the books on bifurcations (e .g. the book of Kuznetsov) this case is studied as the so called Fold-Hopf bifurcation which may give rise to quite complex dynamics.

However my nonlinear system is very special. Indeed the dynamics is homogeneous of degree 1. Is this case already known in the literature or easy to study? I guess that the homogeneity assumption should allow to simplify the analysis but I do not know how and I do not know any reference in the literature.