Reconstructing bifurcation behavior of a nonlinear dynamical system by introducing weak noise

•For a model nonlinear dynamical system, it is shown how to obtain its bifurcation behavior by introducing noise into the dynamics and then studying the resulting noisy (Langevin) dynamics in the weak-noise limit.•We employ two complementary approaches of analysis of stochastic processes, the Fokker–Planck and the path-integral approach, to study the noisy dynamics, and derive a number of analytical results.•Our work primarily serves as a proposal of a theoretical framework to systematically obtain the stability properties of the noiseless dynamics from a suitable analysis of the noisy one, and as an illustration of how one may derive analytical expressions of the quantities involved in the latter analysis. For a model nonlinear dynamical system, we show how one may obtain its bifurcation behavior by introducing noise into the dynamics and then studying the resulting Langevin dynamics in the weak-noise limit. A suitable quantity to capture the bifurcation behavior in the noisy dynamics is the conditional probability to observe a microscopic configuration at one time, conditioned on the observation of a given configuration at an earlier time. For our model system, this conditional probability is studied by using two complementary approaches, the Fokker–Planck and the path-integral approach. The latter has the advantage of yielding exact closed-form expressions for the conditional probability. All our predictions are in excellent agreement with direct numerical integration of the dynamical equations of motion.