Graduate Students, Program in Applied Mathematics
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Speaker: Ben Stilin, Program in Applied Mathematics
Title: A Study of Spectral Properties of a Transfer Operator of an Intermittently Chaotic System
Abstract: We study a circle map with a neutral fixed point exhibiting intermittent chaos. We add small random noise and ask how fast the invariant density of the noisy system approaches that of the noiseless map in L^1, and how fast the spectral gap of the noisy transfer operator closes. The key step is to isolate a boundary layer near the neutral fixed point where deterministic drift and noise balance, so a continuous-time SDE is a good approximation. We then adjust the associated Fokker–Planck equation with a source term that accounts for mass entering this layer from the discrete map and use matched asymptotics to build a perturbative solution. From this, we obtain explicit convergence rates for the invariant densities and for the closing of the spectral gap.
We check these predictions numerically. For the invariant densities, we use a collocation method with a lightning-plus-polynomial basis. For the spectral gap, we use a Galerkin method with a non-uniform B-spline basis to discretize the noisy transfer operator, then compute its spectrum and track how the gap shrinks as the noise level goes to zero.