We investigate noise-induced transitions in non-gradient systems when complex
invariant sets emerge. Our example is the Lorenz system in three representative Rayleigh number
regimes. It is found that before the homoclinic explosion bifurcation, the only transition state is
the saddle point, and the transition is similar to that in gradient systems. However, when the
chaotic invariant set emerges, an unstable limit cycle continues from the homoclinic trajectory. This
orbit, which is embedded in a local tube-like manifold around the initial stable stationary point as a
relative attractor, plays the role of the most probable exit set in the transition process. This example
demonstrates how limit cycles, the next simplest invariant set beyond fixed points, can be involved
in the transition process in smooth dynamical systems.