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We study the impact of noise on the rate dependent transitions in a noisy bistable oscillator using a thermoacoustic system as an example. As the parameter—the heater power—is increased in a quasi-steady manner, beyond a critical value, the thermoacoustic system undergoes a subcritical Hopf bifurcation and exhibits periodic oscillations. We observe that the transition to this oscillatory state is often delayed when the control parameter is varied as a function of time. However, the presence of inherent noise in the system introduces high variability in the characteristics of this critical transition. As a result, if the value of the system variable—the acoustic pressure—approaches the noise floor before the system crosses the unstable manifold, the effect of rate on the critical transition becomes irrelevant in determining the transition characteristics, and the system undergoes a noise-induced tipping to limit-cycle oscillations. The presence of noise-induced tipping makes it difficult to identify the stability regimes in such systems by using stability maps for the corresponding deterministic system.
Noise is an inherent part of practical systems. When a system is nonlinear, noise can have nontrivial effects on its dynamics. We study the effect of inherent noise on dynamic bifurcations in a nonlinear system as a parameter of the system is varied in time. We show that noise can have varied effects on the dynamic bifurcation in a system, depending on the initial conditions of the system and the rate at which the parameter of the system is varied. We use a Rijke tube to experimentally demonstrate our findings from the theoretical analysis.
