  We consider the dynamics of a two-dimensional ordinary differential equation
exhibiting a Hopf bifurcation subject to additive white noise and identify
three dynamical phases: (I) a random attractor with uniform synchronisation of
trajectories, (II) a random attractor with non-uniform synchronisation of
trajectories and (III) a random attractor without synchronisation of
trajectories. The random attractors in phases (I) and (II) are random
equilibrium points with negative Lyapunov exponents while in phase (III) there
is a so-called random strange attractor with positive Lyapunov exponent.
  We analyse the occurrence of the different dynamical phases as a function of
the linear stability of the origin (deterministic Hopf bifurcation parameter)
and shear (ampitude-phase coupling parameter). We show that small shear implies
synchronisation and obtain that synchronisation cannot be uniform in the
absence of linear stability at the origin or in the presence of sufficiently
strong shear. We provide numerical results in support of a conjecture that
irrespective of the linear stability of the origin, there is a critical
strength of the shear at which the system dynamics loses synchronisation and
enters phase (III).
