The climate is a complex nonequilibrium dynamical system that relaxes toward a steady state under the continuous input of solar radiation and dissipative mechanisms. The steady state is not necessarily unique. A useful tool to describe the possible steady states under different forcing is the bifurcation diagram, which reveals the regions of multistability, the position of tipping points, and the range of stability of each steady state. However, its construction is highly time consuming in climate models with a dynamical deep ocean, whose relaxation time is of the order of thousand years, or other feedback mechanisms that act on even longer time scales, like continental ice or carbon cycle. Using a coupled setup of the MIT general circulation model, we test two techniques for the construction of bifurcation diagrams with complementary advantages and reduced execution time. The first is based on the introduction of random fluctuations in the forcing and permits to explore a wide part of phase space. The second reconstructs the stable branches using estimates of the internal variability and of the surface energy imbalance on each attractor, and is more precise in finding the position of tipping points.