The spectrum of the Markov semigroup of a diffusion process, referred to as the mixing spectrum, provides a detailed characterization of the dynamics of statistics such as the correlation function and the power spectrum. Stochastic analysis techniques for the study of the mixing spectrum and a rigorous reduction method have been presented in the first part (Chekroun et al. 2017) of this contribution. This framework is now applied to the study of a stochastic Hopf bifurcation, to characterize the statistical properties of nonlinear oscillators perturbed by noise, depending on their stability. In light of the H\"ormander theorem, it is first shown that the geometry of the unperturbed limit cycle, in particular its isochrons, i.e., the leaves of the stable manifold of the limit cycle generalizing the notion of phase, is essential to understand the effect of the noise and the phenomenon of phase diffusion. In addition, it is shown that the mixing spectrum has a spectral gap, even at the bifurcation point, and that correlations decay exponentially fast. Small-noise expansions of the eigenvalues and eigenfunctions are then obtained, away from the bifurcation point, based on the knowledge of the linearized deterministic dynamics and the characteristics of the noise. These formulas allow to understand how the interaction of the noise with the deterministic dynamics affect the decay of correlations. Numerical results complement the study of the mixing spectrum at the bifurcation point, revealing interesting scaling laws. The analysis of the Markov semigroup for stochastic bifurcations is thus promising in providing a complementary approach to the more geometric random dynamical systems. This approach is not limited to low-dimensional systems and the reduction method presented in part I will be applied to stochastic models relevant to climate dynamics in Part III.

## **Research Interests**

**Nonlinear and Stochastic Reduction:**Markovian and non-Markovinan stochastic closures. Parameterizing manifolds and small-scale parameterizations in spatially extended systems. Ruelle-Pollicott resonances and reduced mixing spectrum.**Random Dynamical Systems:**Stochastic invariant manifolds and their numerical approximations. Random attractors and time-dependent invariant measures arising in dissipative stochastic systems.**Data-Driven Stochastic Modeling:**Data-adaptive harmonic spectra and multilayer Stuart-Landau models.**Optimal Control of Evolution Equations:**Reduced-order models for the optimal control of PDEs and delay differential equations: analysis and computation.**Geophysical Fluid Dynamics and Climate Dynamics:**Natural climate variability. Low-order stochastic modelling of the El Niño-Southern Oscillation. Prediction of stochastic systems. Application of the geometric and ergodic theory of dynamical systems to climate dynamics.

**From Chekroun et al. (2011), Physica D, 240 (21): 1685-1700:**

**Vimeo movie**: https://vimeo.com/240039610

## Recent Publications

Galerkin approximations for the optimal control of nonlinear delay differential equations.” Radon Series on Computational and Applied Mathematics. arXiv's version Abstract

. Forthcoming. “ Pullback attractor crisis in a delay differential ENSO model.” Advances in Nonlinear Geosciences, A. Tsonis, 1-33. Springer, 1-33. Publisher's version Abstract

. 2018. “ Data-adaptive harmonic decomposition and stochastic modeling of Arctic sea ice.” Advances in Nonlinear Geosciences, A. Tsonis, 179-205. Springer, 179-205. Publisher's Version Abstract

. 2018. “ Data-adaptive harmonic spectra and multilayer Stuart-Landau models.” Chaos: An Interdisciplinary Journal of Nonlinear Science 27 (9): 093110. Publisher's version Abstract

. 2017. “ Galerkin approximations of nonlinear optimal control problems in Hilbert spaces.” Electronic Journal of Differential Equations 2017 (189): 1-40. Publisher's version Abstract

. 2017. “