## Research Interests

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

## Recent Publications

Chekroun, Mickaël D., Jeroen S.W. Lamb, Christian J. Pangerl, and Martin Rasmussen. Submitted. “A Girsanov approach to slow parameterizing manifolds in the presence of noise”. arXiv's link Abstract
We consider a three-dimensional slow-fast system with quadratic nonlinearity and additive noise. The associated deterministic system of this stochastic differential equation (SDE) exhibits a periodic orbit and a slow manifold. The deterministic slow manifold can be viewed as an approximate parameterization of the fast variable of the SDE in terms of the slow variables. In other words the fast variable of the slow-fast system is approximately "slaved" to the slow variables via the slow manifold. We exploit this fact to obtain a two dimensional reduced model for the original stochastic system, which results in the Hopf-normal form with additive noise. Both, the original as well as the reduced system admit ergodic invariant measures describing their respective long-time behaviour. We will show that for a suitable metric on a subset of the space of all probability measures on phase space, the discrepancy between the marginals along the radial component of both invariant measures can be upper bounded by a constant and a quantity describing the quality of the parameterization. An important technical tool we use to arrive at this result is Girsanov's theorem, which allows us to modify the SDEs in question in a way that preserves transition probabilities. This approach is then also applied to reduced systems obtained through stochastic parameterizing manifolds, which can be viewed as generalized notions of deterministic slow manifolds.

Wang, Peng, James C. McWilliams, Yusuke Uchiyama, Mickaël D. Chekroun, and Daling Li Yi. 2020. “Effects of wave streaming and wave variations on nearshore wave-driven circulation.” J. Phys. Oceanograhy 50 (10): 3025-3041. Publisher's Version Abstract

Wave streaming is a near-bottom mean current induced by the bottom drag on surface gravity waves. Wave variations include the variations in wave heights, periods, and directions. Here we use numerical simulations to study the effects of wave streaming and wave variations on the circulation that is driven by incident surface waves. Wave streaming induces an inner-shelf Lagrangian overturning circulation, which links the inner shelf with the surf zone. Wave variations cause along shore-variable wave breaking that produces surf eddies; however, such eddies can be suppressed by wave streaming. Moreover, with passive tracers we show that wave streaming and wave variations together enhance the cross- shelf material transport.

Chekroun, Mickaël D., and Honghu Liu. 2020. “Optimal management of harvested population at the edge of extinction.” Advances in Nonlinear Biological Systems: Modeling and Optimal Control, J. Kotas (Ed.)., 11: 35-72. AIMS Applied Mathematics Book series. ISBN-10 : 1-60133-025-1, ISBN-13 : 978-1-60133-025-3. arXiv version Abstract

Optimal control of harvested population at the edge of extinction in an unprotected area, is considered. The underlying population dynamics is governed by a Kolmogorov-Petrovsky-Piskunov equation with a harvesting term and space-dependent coefficients while the control consists of transporting individuals from a natural reserve. The nonlinear optimal control problem is approximated by means of a Galerkin scheme. Convergence result about the optimal controlled solutions and error estimates between the corresponding optimal controls, are derived. For certain parameter regimes, nearly optimal solutions are calculated from a simple logistic ordinary differential equation (ODE) with a harvesting term, obtained as a Galerkin approximation of the original partial differential equation (PDE) model. A critical allowable fraction of the reserve's population is inferred from the reduced logistic ODE with a harvesting term. This estimate obtained from the reduced model allows us to distinguish sharply between survival and extinction for the full PDE itself, and thus to declare whether a control strategy leads to success or failure for the corresponding rescue operation while ensuring survival in the reserve's population. In dynamical terms, this result illustrates that although continuous dependence on the forcing may hold on finite-time intervals, a high sensitivity in the system's response may occur in the asymptotic time. We believe that this work, by its generality, establishes bridges interesting to explore between optimal control problems of ODEs with a harvesting term and their PDE counterpart.

Chekroun, Mickaël D., Ilan Koren, and Honghu Liu. 2020. “Efficient reduction for diagnosing Hopf bifurcation in delay differential systems: Applications to cloud-rain models.” Chaos: An Interdisciplinary Journal of Nonlinear Science 30: 053130 . Publisher's Version Abstract

By means of Galerkin-Koornwinder (GK) approximations, an efficient reduction approach to the Stuart-Landau (SL) normal form and center manifold is presented for a broad class of nonlinear systems of delay differential equations (DDEs) that covers the cases of discrete as well as distributed delays. The focus is on the Hopf bifurcation as the consequence of the critical equilibrium's destabilization resulting from an eigenpair crossing the imaginary axis. The nature of the resulting Hopf bifurcation (super- or subcritical) is then characterized by the inspection of a Lyapunov coefficient easy to determine based on the model's coefficients and delay parameter.  We believe that our approach, which does not rely too much on functional analysis considerations but more on analytic calculations, is suitable  to concrete situations arising in physics applications.

Thus, using this GK approach to the Lyapunov coefficient and SL normal form, the occurrence of Hopf bifurcations in the cloud-rain delay models of Koren and Feingold (KF) on one hand, and Koren, Tziperman and Feingold (KTF), on the other, are analyzed. Noteworthy is the existence for the KF model of large regions of the parameter space for which subcritical and supercritical Hopf bifurcations coexist. These regions are determined in particular by the intensity of the KF model's nonlinear effects. Islands'' of supercritical Hopf bifurcations are shown to exist within a subcritical Hopf bifurcation sea;'' these islands being bordered by double-Hopf bifurcations occurring when the linearized dynamics at the critical equilibrium exhibit two pairs of purely imaginary eigenvalues.

Chekroun, Mickaël D., Youngjoon Hong, and Roger Temam. 2020. “Enriched numerical scheme for singularly perturbed barotropic quasi-geostrophic equations.” Journal of Computational Physics 416: 109493. Publisher's Version Abstract

Singularly perturbed barotropic Quasi-Geostrophic (QG) models are considered. A boundary layer analysis is presented and the convergence of solutions is studied. Based on the rigorous analysis of the underlying boundary layer problems, an enriched spectral method (ESM) is proposed to solve the QG models. It consists of adding to the Legendre basis functions, analytically-determined boundary layer elements called “correctors," with the aim of capturing most of the complex behavior occurring near the boundary with such elements. Through detailed numerical experiments, it is shown that high-accuracy is often reached by the ESM scheme with only a relatively low number N of basis functions, when compared to approximations based on spectral elements which typically display non-physical oscillations throughout the physical domain, for such values of N. The key to success relies on our analytically-based boundary layer elements, which, due to their highly nonlinear nature, are able to capture most of the steep gradients occurring in the problem’s solution, near the boundary. Our numerical results include multi-dimensional as well as time-dependent problems.

Tantet, Alexis, Mickaël D. Chekroun, Henk A. Dijkstra, and J. David Neelin. 2020. “Ruelle-Pollicott Resonances of Stochastic Systems in Reduced State Space. Part II: Stochastic Hopf Bifurcation.” Journal of Statistical Physics 179: 1403–1448. Publisher's Version Abstract

The spectrum of the generator (Kolmogorov operator) of a diffusion process, referred to as the Ruelle-Pollicott (RP) spectrum, provides a detailed characterization of correlation functions and power spectra of stochastic systems via decomposition formulas in terms of RP resonances; see Part I of this contribution (Chekroun et al. in Theory J Stat. https://doi.org/10.1007/s10955-020-02535-x, 2020). Stochastic analysis techniques relying on the theory of Markov semigroups for the study of the RP spectrum and a rigorous reduction method is presented in Part I Chekroun et al. (2020). This framework is here applied to study a stochastic Hopf bifurcation in view of characterizing the statistical properties of nonlinear oscillators perturbed by noise, depending on their stability. In light of the Hörmander 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 RP spectrum has a spectral gap, even at the bifurcation point, and that correlations decay exponentially fast. Explicit small-noise expansions of the RP 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 one to understand how the interaction of the noise with the deterministic dynamics affect the decay of correlations. Numerical results complement the study of the RP spectrum at the bifurcation point, revealing useful 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 system (RDS) approach. This approach is not limited to low-dimensional systems and the reduction method presented in Chekroun et al. (2020) is applied to a stochastic model relevant to climate dynamics in the third part of this contribution (Tantet et al. in J Stat Phys. https://doi.org/10.1007/s10955-019-02444-8, 2019).

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