  Complex dynamical systems may exhibit multiple steady states, including
time-periodic limit cycles, where the final trajectory depends on initial
conditions. With tuning of parameters, limit cycles can proliferate or merge at
an exceptional point. Here we ask how dynamics in the vicinity of such a
bifurcation are influenced by noise. A pitchfork bifurcation can be used to
induce bifurcation behavior. We model a limit cycle with the normal form of the
Hopf oscillator, couple it to the pitchfork, and investigate the resulting
dynamical system in the presence of noise. We show that the generating
functional for the averages of the dynamical variables factorizes between the
pitchfork and the oscillator. The statistical properties of the pitchfork in
the presence of noise in its various regimes are investigated and a scaling
theory is developed for the correlation and response functions. The analysis is
done by perturbative calculations as well as numerical means. Finally,
observables illustrating the coupling of a system with a limit cycle to a
pitchfork are discussed and the phase-phase correlations are shown to exhibit
non-diffusive behavior with universal scaling.
