UBC Theses and Dissertations
Asymptotic behaviour of tipping points in non-autonomous systems with random and periodic forcing Zhu, Jielin
We consider the effect on tipping from an additive periodic forcing and an additive white noise in a canonical model with a saddle-node bifurcation and a slowly varying bifurcation parameter. Using a multiple scales analysis, we consider the effect of amplitude and frequency of the periodic forcing relative to the drifting rate of the varying bifurcation parameter. We show that a high frequency oscillation drives an earlier tipping when the bifurcation parameter varies more slowly, with the advance of the tipping point proportional to the square of the ratio of amplitude to frequency. In the low frequency case the position of the tipping point is affected by the frequency, amplitude and phase of the low frequency oscillation. The results are based on an analysis of the local concavity of the trajectory. The tipping point location increases with the amplitude of the periodic forcing, with critical amplitudes where there are jumps in the location. Using a WKB method and a multi-scale analysis, we consider the effect of the amplitude of the noise relative to the drifting rate of the bifurcation parameter. We show that the early tipping is likely to happen if the drifting rate of the bifurcation parameter is smaller than the square of the amplitude of the noise. The WKB approximation shows that the quasi-stationary PDF is related to the potential well of the system. The probability of early tipping, which is related to the PDF near the unstable equilibrium, should be considered locally through a convection-diffusion process. The analysis of the Morris-Lecar model with oscillations shows that the method of multiple scales we used is applicable to the higher dimensional system. For the sea ice model, the method of multiple scales is adapted for the case when the amplitude of the oscillation is larger than its frequency. We applied the WKB method and the multi-scale analysis on the zero-dimensional climate model which has an asymmetric structure of the saddle-node bifurcation. The approximation of the PDF shows that the asymmetry of the shape of the bifurcation diagram plays an important role in the probability of early tipping.
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