Cusp Bifurcation Subject to Additive Noise

This study investigates the cusp bifurcation in a one-dimensional normal form system subject to additive white noise. First, we demonstrate that the corresponding random attractor of the system is a singleton across a range of bifurcation parameters. Further, analysis reveals a qualitative transition in the system’s random dynamics at the underlying deterministic cusp bifurcation point. This transition is manifested in two key aspects: the loss of uniform attractivity of the random attractor, and the emergence of positive Finite-Time Lyapunov Exponents (FTLEs). Moreover, we derive large deviation estimates and implement a computer-assisted proof methodology to quantify the nonvanishing probabilities associated with positive FTLEs. Numerical computations based on this methodology yield explicit bounds for the exponential decay rate, which characterizes the transient instabilities at representative parameter values.