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.%G eng %U https://arxiv.org/abs/1903.08598 %0 Journal Article %J Chaos %D 2021 %T Reduced-order models for coupled dynamical systems: Data-driven methods and the Koopman operator %A Santos Gutiérrez, Manuel %A Lucarini, Valerio %A Chekroun, Mickaël D. %A Ghil, Michael %X

Providing efficient and accurate parameterizations for model reduction is a key goal in many areas of science and technology. Here, we present a strong link between data-driven and theoretical approaches to achieving this goal. Formal perturbation expansions of the Koopman operator allow us to derive general stochastic parameterizations of weakly coupled dynamical systems. Such parameterizations yield a set of stochastic integrodifferential equations with explicit noise and memory kernel formulas to describe the effects of unresolved variables. We show that the perturbation expansions involved need not be truncated when the coupling is additive. The unwieldy integrodifferential equations can be recast as a simpler multilevel Markovian model, and we establish an intuitive connection with a generalized Langevin equation. This connection helps setting up a parallelism between the top-down, equation-based methodology herein and the well-established empirical model reduction (EMR) methodology that has been shown to provide efficient dynamical closures to partially observed systems. Hence, our findings, on the one hand, support the physical basis and robustness of the EMR methodology and, on the other hand, illustrate the practical relevance of the perturbative expansion used for deriving the parameterizations.

Parameterizations aim to reduce the complexity of high-dimensional dynamical systems. Here, a theory-based and a data-driven approach for the parameterization of coupled systems are compared, showing that both yield the same stochastic multilevel structure. The results provide very strong support to the use of empirical methods in model reduction and clarify the practical relevance of the proposed theoretical framework.

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.

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%B Advances in Nonlinear Biological Systems: Modeling and Optimal Control, J. Kotas (Ed.). %I AIMS Applied Mathematics Book series. ISBN-10 : 1-60133-025-1, ISBN-13 : 978-1-60133-025-3 %V 11 %P 35-72 %G eng %U https://arxiv.org/abs/2007.11785 %0 Journal Article %J Chaos: An Interdisciplinary Journal of Nonlinear Science %D 2020 %T Efficient reduction for diagnosing Hopf bifurcation in delay differential systems: Applications to cloud-rain models %A Chekroun, Mickaël D. %A Koren, Ilan %A Liu, Honghu %XBy 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.

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%B Chaos: An Interdisciplinary Journal of Nonlinear Science %V 30 %P 053130 %G eng %U https://doi.org/10.1063/5.0004697 %0 Journal Article %J Journal of Computational Physics %D 2020 %T Enriched numerical scheme for singularly perturbed barotropic quasi-geostrophic equations %A Chekroun, Mickaël D. %A Hong, Youngjoon %A Temam, Roger %XSingularly 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.

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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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A theory of Ruelle–Pollicott (RP) resonances for stochastic differential systems is presented. These resonances are defined as the eigenvalues of the generator (Kolmogorov operator) of a given stochastic system. By relying on the theory of Markov semigroups, decomposition formulas of correlation functions and power spectral densities (PSDs) in terms of RP resonances are then derived. These formulas describe, for a broad class of stochastic differential equations (SDEs), how the RP resonances characterize the decay of correlations as well as the signal’s oscillatory components manifested by peaks in the PSD. It is then shown that a notion reduced RP resonances can be rigorously defined, as soon as the dynamics is partially observed within a reduced state space *V*. These reduced resonances are obtained from the spectral elements of reduced Markov operators acting on functions of the state space *V*, and can be estimated from series. They inform us about the spectral elements of some coarse-grained version of the SDE generator. When the time-lag at which the transitions are collected from partial observations in *V*, is either sufficiently small or large, it is shown that the reduced RP resonances approximate the (weak) RP resonances of the generator of the conditional expectation in *V*, i.e. the optimal reduced system in *V* obtained by averaging out the contribution of the unobserved variables. The approach is illustrated on a stochastic slow-fast system for which it is shown that the reduced RP resonances allow for a good reconstruction of the correlation functions and PSDs, even when the time-scale separation is weak. The companions articles, Part II and Part III, deal with further practical aspects of the theory presented in this contribution. One important byproduct consists of the diagnosis usefulness of stochastic dynamics that RP resonances provide. This is illustrated in the case of a stochastic Hopf bifurcation in Part II. There, it is shown that such a bifurcation has a clear manifestation in terms of a geometric organization of the RP resonances along discrete parabolas in the left half plane. Such geometric features formed by (reduced) RP resonances are extractable from time series and allow thus for providing an unambiguous “signature” of nonlinear oscillations embedded within a stochastic background. By relying then on the theory of reduced RP resonances presented in this contribution, Part III addresses the question of detection and characterization of such oscillations in a high-dimensional stochastic system, namely the Cane–Zebiak model of El Niño-Southern Oscillation subject to noise modeling fast atmospheric fluctuations.

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%B Journal of Statistical Physics %V 179 %P 1366–1402 %G eng %U https://link.springer.com/article/10.1007/s10955-020-02535-x %0 Journal Article %J Journal of Statistical Physics %D 2020 %T Variational approach to closure of nonlinear dynamical systems: Autonomous case %A Chekroun, Mickaël D. %A Liu, Honghu %A McWilliams, James C. %X A general, variational approach to derive low-order reduced systems for nonlinear systems subject to an autonomous forcing, is introduced. The approach is based on the concept of optimal parameterizing manifold (PM) that substitutes the more classical notion of slow manifold or invariant manifold when breakdown of slaving occurs. An optimal PM provides the manifold that describes the average motion of the neglected scales as a function of the resolved scales and allows, in principle, for determining the best vector field of the reduced state space that describes e.g. the dynamics' slow motion. The underlying optimal parameterizations are approximated by dynamically-based formulas derived analytically from the original equations. These formulas are contingent upon the determination of only a few (scalar) parameters obtained from minimization of cost functionals, depending on training dataset collected from direct numerical simulation. In practice, a training period of length comparable to a characteristic recurrence or decorrelation time of the dynamics, is sufficient for the efficient derivation of optimized parameterizations. Applications to the closure of low-order models of Atmospheric Primitive Equations and Rayleigh-Bénard convection are then discussed. The approach is finally illustrated --- in the context of the Kuramoto-Sivashinsky turbulence --- as providing efficient closures without slaving for a cutoff scale

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%B Journal of Statistical Physics %V 179 %P 1073–1160 %G eng %U https://link.springer.com/article/10.1007%2Fs10955-019-02458-2 %0 Journal Article %J Chinese Annals of Mathematics, Series B %D 2019 %T Mathematical analysis of the Jin-Neelin model of El Nino-Southern-Oscillation %A Cao, Yining %A Chekroun, Mickaël D. %A Temam, Roger %A Huang, Aimin %X

The Jin-Neelin model for the El Niño–Southern Oscillation (ENSO for short) is considered for which the authors establish existence and uniqueness of global solutions in time over an unbounded channel domain. The result is proved for initial data and forcing that are sufficiently small. The smallness conditions involve in particular key physical parameters of the model such as those that control the travel time of the equatorial waves and the strength of feedback due to vertical-shear currents and upwelling; central mechanisms in ENSO dynamics.

From the mathematical view point, the system appears as the coupling of a linear shallow water system and a nonlinear heat equation. Because of the very different nature of the two components of the system, the authors find it convenient to prove the existence of solution by semi-discretization in time and utilization of a fractional step scheme. The main idea consists of handling the coupling between the oceanic and temperature components by dividing the time interval into small sub-intervals of length *k* and on each sub-interval to solve successively the oceanic component, using the temperature *T* calculated on the previous sub-interval, to then solve the sea-surface temperature (SST for short) equation on the current sub-interval. The passage to the limit as *k* tends to zero is ensured via a priori estimates derived under the aforementioned smallness conditions.

%B Chinese Annals of Mathematics, Series B %V 40 %P 1–38 %G eng %U https://link.springer.com/article/10.1007/s11401-018-0115-3 %N 1 %0 Journal Article %J Journal of Statistical Physics %D 2019 %T Ruelle–Pollicott Resonances of Stochastic Systems in Reduced State Space. Part III: Application to the Cane–Zebiak Model of the El Niño–Southern Oscillation %A Tantet, Alexis %A Chekroun, Mickaël D. %A Neelin, J. David %A Dijkstra, Henk A. %X

The response of a low-frequency mode of climate variability, El Niño–Southern Oscillation, to stochastic forcing is studied in a high-dimensional model of intermediate complexity, the fully-coupled Cane–Zebiak model (Zebiak and Cane 1987), from the spectral analysis of Markov operators governing the decay of correlations and resonances in the power spectrum. Noise-induced oscillations excited before a supercritical Hopf bifurcation are examined by means of complex resonances, the reduced Ruelle–Pollicott (RP) resonances, via a numerical application of the reduction approach of the first part of this contribution (Chekroun et al. 2019) to model simulations. The oscillations manifest themselves as peaks in the power spectrum which are associated with RP resonances organized along parabolas, as the bifurcation is neared. These resonances and the associated eigenvectors are furthermore well described by the small-noise expansion formulas obtained by Gaspard (2002) and made explicit in the second part of this contribution (Tantet et al. 2019). Beyond the bifurcation, the spectral gap between the imaginary axis and the real part of the leading resonances quantifies the diffusion of phase of the noise-induced oscillations and can be computed from the linearization of the model and from the diffusion matrix of the noise. In this model, the phase diffusion coefficient thus gives a measure of the predictability of oscillatory events representing ENSO. ENSO events being known to be locked to the seasonal cycle, these results should be extended to the non-autonomous case. More generally, the reduction approach theorized in Chekroun et al. (2019), complemented by our understanding of the spectral properties of reference systems such as the stochastic Hopf bifurcation, provides a promising methodology for the analysis of low-frequency variability in high-dimensional stochastic systems.

%B Journal of Statistical Physics %V 179 %P 1449–1474 %G eng %U https://link.springer.com/article/10.1007/s10955-019-02444-8 %0 Journal Article %J Nonlinear Processes in Geophysics %D 2018 %T The onset of chaos in nonautonomous dissipative dynamical systems: a low-order ocean-model case study %A Pierini, Stefano %A Mickaël D. Chekroun %A Ghil, Michael %X Decline in the Arctic sea ice extent (SIE) is an area of active scientific research with profound socio-economic implications. Of particular interest are reliable methods for SIE forecasting on subseasonal time scales, in particular from early summer into fall, when sea ice coverage in the Arctic reaches its minimum. Here, we apply the recent data-adaptive harmonic (DAH) technique of Chekroun and Kondrashov, (2017), *Chaos*, **27** for the description, modeling and prediction of the Multisensor Analyzed Sea Ice Extent (MASIE, 2006–2016) data set. The DAH decomposition of MASIE identifies narrowband, spatio-temporal data-adaptive modes over four key Arctic regions. The time evolution of the DAH coefficients of these modes can be modelled and predicted by using a set of coupled Stuart–Landau stochastic differential equations that capture the modes’ frequencies and amplitude modulation in time. Retrospective forecasts show that our resulting multilayer Stuart–Landau model (MSLM) is quite skilful in predicting September SIE compared to year-to-year persistence; moreover, the DAH–MSLM approach provided accurate real-time prediction that was highly competitive for the 2016–2017 Sea Ice Outlook.

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%B Dynamics and Statistics of the Climate System %V 3 %P 1 %8 27 March 2018 %G eng %U https://academic.oup.com/climatesystem/article/3/1/dzy001/4925706 %N 1 %0 Journal Article %J Fluids %D 2018 %T Multiscale Stuart-Landau Emulators: Application to Wind-Driven Ocean Gyres %A Kondrashov, Dmitri %A Mickaël D. Chekroun %A Pavel Berloff %XThe multiscale variability of the ocean circulation due to its nonlinear dynamics remains a big challenge for theoretical understanding and practical ocean modeling. This paper demonstrates how the data-adaptive harmonic (DAH) decomposition and inverse stochastic modeling techniques introduced in (Chekroun and Kondrashov, (2017), Chaos, 27), allow for reproducing with high fidelity the main statistical properties of multiscale variability in a coarse-grained eddy-resolving ocean flow. This fully-data-driven approach relies on extraction of frequency-ranked time-dependent coefficients describing the evolution of spatio-temporal DAH modes (DAHMs) in the oceanic flow data. In turn, the time series of these coefficients are efficiently modeled by a family of low-order stochastic differential equations (SDEs) stacked per frequency, involving a fixed set of predictor functions and a small number of model coefficients. These SDEs take the form of stochastic oscillators, identified as multilayer Stuart–Landau models (MSLMs), and their use is justified by relying on the theory of Ruelle–Pollicott resonances. The good modeling skills shown by the resulting DAH-MSLM emulators demonstrates the feasibility of using a network of stochastic oscillators for the modeling of geophysical turbulence. In a certain sense, the original quasiperiodic Landau view of turbulence, with the amendment of the inclusion of stochasticity, may be well suited to describe turbulence.

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In this article it is proved that the dynamical properties of a broad class of semilinear parabolic problems are sensitive to arbitrarily small but smooth perturbations of the nonlinear term, when the spatial dimension is either equal to one or two. This topological instability is shown to result from a local deformation of the global bifurcation diagram associated with the corresponding elliptic problems. Such a deformation is shown to systematically occur via the creation of either a multiple-point or a new fold-point on this diagram when an appropriate small perturbation is applied to the nonlinear term. More precisely, it is shown that for a broad class of nonlinear elliptic problems, one can always find an arbitrary small perturbation of the nonlinear term, that generates a local S on the bifurcation diagram whereas the latter is e.g. monotone when no perturbation is applied; substituting thus a single solution by several ones. Such an increase in the local multiplicity of the solutions to the elliptic problem results then into a topological instability for the corresponding parabolic problem.

The rigorous proof of the latter instability result requires though to revisit the classical concept of topological equivalence to encompass important cases for the applications such as semi-linear parabolic problems for which the semigroup may exhibit non-global dissipative properties, allowing for the coexistence of blow-up regions and local attractors in the phase space; cases that arise e.g. in combustion theory. A revised framework of topological robustness is thus introduced in that respect within which the main topological instability result is then proved for continuous, locally Lipschitz but not necessarily C1 nonlinear terms, that prevent in particular the use of linearization techniques, and for which the family of semigroups may exhibit non-dissipative properties.

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%B Discrete and Continuous Dynamical Systems B, doi: 10.3934/dcdsb.2018075 %8 2018 %G eng %U http://aimsciences.org//article/doi/10.3934/dcdsb.2018075 %0 Journal Article %J Journal of Atmospheric and Solar-Terrestrial Physics %D 2018 %T Data-adaptive harmonic analysis and modeling of solar wind-magnetosphere coupling %A Kondrashov, Dmitri %A Mickaël D. Chekroun %XThe solar wind-magnetosphere coupling is studied by new data-adaptive harmonic (DAH) decomposition approach for the spectral analysis and inverse modeling of multivariate time observations of complex nonlinear dynamical systems. DAH identifies frequency-based modes of interactions in the combined dataset of Auroral Electrojet (AE) index and solar wind forcing. The time evolution of these modes can be very effi- ciently simulated by using systems of stochastic differential equations (SDEs) that are stacked per frequency and formed by coupled Stuart-Landau oscillators. These systems of SDEs capture the modes’ frequencies as well as their amplitude modulations, and yield, in turn, an accurate modeling of the AE index’ statistical properties.

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We present and apply a novel method of describing and modeling complex multivariate datasets in the geosciences and elsewhere. Data-adaptive harmonic (DAH) decomposition identifies narrow-banded, spatio-temporal modes (DAHMs) whose frequencies are not necessarily integer multiples of each other. The evolution in time of the DAH coefficients (DAHCs) of these modes can be modeled using a set of coupled Stuart-Landau stochastic differential equations that capture the modes’ frequencies and amplitude modulation in time and space. This methodology is applied first to a challenging synthetic dataset and then to Arctic sea ice concentration (SIC) data from the US National Snow and Ice Data Center (NSIDC). The 36-year (1979–2014) dataset is parsimoniously and accurately described by our DAHMs. Preliminary results indicate that simulations using our multilayer Stuart-Landau model (MSLM) of SICs are stable for much longer time intervals, beyond the end of the twenty-first century, and exhibit interdecadal variability consistent with past historical records. Preliminary results indicate that this MSLM is quite skillful in predicting September sea ice extent.

%B Advances in Nonlinear Geosciences %7 A. Tsonis %I Springer %P 179-205 %G eng %U https://link.springer.com/chapter/10.1007/978-3-319-58895-7_10 %0 Book Section %B Hamilton-Jacobi-Bellman Equations. Numerical Methods and Applications in Optimal Control %D 2018 %T Galerkin approximations for the optimal control of nonlinear delay differential equations %A Chekroun, Mickaël D. %A Axel Kröner %A Liu, Honghu %XOptimal control problems of nonlinear delay equations (DDEs) are considered for which we propose a general Galerkin approximation scheme built from Koornwinder polynomials. Error estimates for the resulting Galerkin-Koornwinder approximations to the optimal control and the value function, are derived for a broad class of cost functionals and nonlinear DDEs. The approach is illustrated on a delayed logistic equation set not far away from its Hopf bifurcation point in the parameter space. In this case, we show that low-dimensional controls for a standard quadratic cost functional can be efficiently computed from Galerkin-Koornwinder approximations to reduce at a nearly optimal cost the oscillation amplitude displayed by the DDE's solution. Optimal controls computed from the Pontryagin's maximum principle (PMP) and the Hamilton-Jacobi-Bellman equation (HJB) associated with the corresponding ODE systems, are shown to provide numerical solutions in good agreement. It is finally argued that the value function computed from the corresponding reduced HJB equation provides a good approximation of that obtained from the full HJB equation.

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%B Hamilton-Jacobi-Bellman Equations. Numerical Methods and Applications in Optimal Control %7 D. Kalise, K. Kunisch, and Z. Rao %I De Gruyter %C Berlin, Boston %V 21 %P 61-96 %8 2018 %G eng %U https://www.degruyter.com/view/book/9783110543599/10.1515/9783110543599-004.xml %0 Book Section %B Advances in Nonlinear Geosciences %D 2018 %T Pullback attractor crisis in a delay differential ENSO model %A Chekroun, Mickaël D. %A Ghil, Michael %A Neelin, J. David %XWe study the pullback attractor (PBA) of a seasonally forced delay differential model for the El Ni\~no--Southern Oscillation (ENSO); the model has two delays, associated with a positive and a negative feedback. The control parameter is the intensity of the positive feedback and the PBA undergoes a crisis that consists of a chaos-to-chaos transition. Since the PBA is dominated by chaotic behavior, we refer to it as a strange PBA. Both chaotic regimes correspond to an overlapping of resonances but the two differ by the properties of this overlapping. The crisis manifests itself by a brutal change not only in the size but also in the shape of the PBA. The change is associated with the sudden disappearance of the most extreme warm (El Ni\~no) and cold (La Ni\~na) events, as one crosses the critical parameter value from below. The analysis reveals that regions of the strange PBA that survive the crisis are those populated by the most probable states of the system. These regions are those that exhibit robust foldings with respect to perturbations. The effect of noise on this phase-and-paramater space behavior is then discussed. It is shown that the chaos-to-chaos crisis may or may not survive the addition of small noise to the evolution equation, depending on how the noise enters the latter.

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%B Advances in Nonlinear Geosciences %7 A. Tsonis %I Springer %P 1-33 %8 2017 %G eng %U https://link.springer.com/chapter/10.1007/978-3-319-58895-7_1 %0 Journal Article %J Chaos: An Interdisciplinary Journal of Nonlinear Science %D 2017 %T Data-adaptive harmonic spectra and multilayer Stuart-Landau models %A Chekroun, Mickaël D. %A Kondrashov, Dmitri %X

Harmonic decompositions of multivariate time series are considered for which we adopt an integral operator approach with

periodic semigroup kernels. Spectral decomposition theorems are derived that cover the important cases of two-time statistics drawn from a mixing invariant measure.

The corresponding eigenvalues can be grouped per Fourier frequency, and are actually given, at each frequency, as the singular values of a cross-spectral matrix depending on the data. These eigenvalues obey furthermore a variational principle that allows us to define naturally a multidimensional power spectrum. The eigenmodes, as far as they are concerned, exhibit a data-adaptive character manifested in their phase which allows us in turn to define a multidimensional phase spectrum.

The resulting data-adaptive harmonic (DAH) modes allow for reducing the data-driven modeling effort to elemental models stacked per frequency, only coupled at different frequencies by the same noise realization. In particular, the DAH decomposition extracts time-dependent coefficients stacked by Fourier frequency which can be efficiently modeled---provided the decay of temporal correlations is sufficiently well-resolved---within a class of multilayer stochastic models (MSMs) tailored here on stochastic Stuart-Landau oscillators.

Applications to the Lorenz 96 model and to a stochastic heat equation driven by a space-time white noise, are considered. In both cases, the DAH decomposition allows for an extraction of spatio-temporal modes revealing key features of the dynamics in the embedded phase space. The multilayer Stuart-Landau models (MSLMs) are shown to successfully model the typical patterns of the corresponding time-evolving fields, as well as their statistics of occurrence.

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%B Chaos: An Interdisciplinary Journal of Nonlinear Science %V 27 %P 093110 %8 2017 %G eng %U http://aip.scitation.org/doi/full/10.1063/1.4989400 %N 9 %0 Journal Article %J Electronic Journal of Differential Equations %D 2017 %T Galerkin approximations of nonlinear optimal control problems in Hilbert spaces %A Chekroun, Mickaël D. %A Axel Kröner %A Liu, Honghu %X Nonlinear optimal control problems in Hilbert spaces are considered for which we derive approximation theorems for Galerkin approximations. Approximation theorems are available in the literature. The originality of our approach

relies on the identification of a set of natural assumptions that allows us to deal with a broad class of nonlinear evolution equations and cost functionals for which we derive convergence of the value functions associated with the optimal control problem of the Galerkin approximations. This convergence result holds for a broad class of nonlinear control strategies as well. In particular, we show that the framework applies to the optimal control of semilinear heat equations posed on a general compact manifold without boundary. The framework is then shown to apply to geoengineering and mitigation of greenhouse gas emissions formulated here in terms of optimal control of energy balance climate models posed on the sphere S^{2}.

Proxy records from Greenland ice cores have been studied for several decades, yet many open questions remain regarding the climate variability encoded therein. Here, we use a Bayesian framework for inferring inverse, stochastic-dynamic models from *δ*^{18}O and dust records of unprecedented, subdecadal temporal resolution. The records stem from the North Greenland Ice Core Project (NGRIP) and we focus on the time interval 59 ka–22 ka b2k. Our model reproduces the dynamical characteristics of both the *δ*^{18}O and dust proxy records, including the millennial-scale Dansgaard–Oeschger variability, as well as statistical properties such as probability density functions, waiting times and power spectra, with no need for any external forcing. The crucial ingredients for capturing these properties are (i) high-resolution training data; (ii) cubic drift terms; (iii) nonlinear coupling terms between the *δ*^{18}O and dust time series; and (iv) non-Markovian contributions that represent short-term memory effects.

[[{"fid":"1721","view_mode":"default","fields":{"format":"default"},"type":"media","field_deltas":{"1":{"format":"default"}},"attributes":{"class":"media-element file-default","data-delta":"1"}}]]

%B Earth System Dynamics %V 8 %P 1171–1190 %G eng %U https://www.earth-syst-dynam.net/8/1171/2017/ %R 10.5194/esd-2017-8 %0 Journal Article %J Computers & Fluids %D 2017 %T The emergence of fast oscillations in a reduced primitive equation model and its implications for closure theories %A Chekroun, Mickaël D. %A Liu, Honghu %A McWilliams, James C. %K Balance equations %XThe problem of emergence of fast gravity-wave oscillations in rotating, stratified flow is reconsidered. Fast inertia-gravity oscillations have long been considered an impediment to initialization of weather forecasts, and the concept of a “slow manifold” evolution, with no fast oscillations, has been hypothesized. It is shown on a reduced Primitive Equation model introduced by Lorenz in 1980 that fast oscillations are absent over a finite interval in Rossby number but they can develop brutally once a critical Rossby number is crossed, in contradistinction with fast oscillations emerging according to an exponential smallness scenario such as reported in previous studies, including some others by Lorenz. The consequences of this dynamical transition on the closure problem based on slow variables is also discussed. In that respect, a novel variational perspective on the closure problem exploiting manifolds is introduced. This framework allows for a unification of previous concepts such as the slow manifold or other concepts of “fuzzy” manifold. It allows furthermore for a rigorous identification of an optimal limiting object for the averaging of fast oscillations, namely the optimal parameterizing manifold (PM). It is shown through detailed numerical computations and rigorous error estimates that the manifold underlying the nonlinear Balance Equations provides a very good approximation of this optimal PM even somewhat beyond the emergence of fast and energetic oscillations.

[[{"fid":"315","view_mode":"default","type":"media","field_deltas":{"1":{}},"fields":{},"attributes":{"height":"356","width":"527","style":"display: block; margin-left: auto; margin-right: auto;","class":"media-element file-default","data-delta":"1"}}]]

%B Computers & Fluids %V 151 %P 3-22 %G eng %U http://www.sciencedirect.com/science/article/pii/S004579301630216X %R http://dx.doi.org/10.1016/j.compfluid.2016.07.005 %0 Journal Article %J Dynamics and Statistics of the Climate System %D 2016 %T A numerical framework to understand transitions in high-dimensional stochastic dynamical systems %A Dijkstra, Henk A %A Tantet, Alexis %A Viebahn, Jan %A Mulder, Erik %A Hebbink, Mariët %A Castellana, Daniele %A van den Pol, Henri %A Frank, Jason %A Baars, Sven %A Wubs, Fred %A Chekroun, Mickaël %A Kuehn, Christian %X

Dynamical systems methodology is a mature complementary approach to forward simulation which can be used to investigate many aspects of climate dynamics. With this paper, a review is given on the methods to analyze deterministic and stochastic climate models and show that these are not restricted to low-dimensional toy models, but that they can be applied to models formulated by stochastic partial differential equations. We sketch the numerical implementation of these methods and illustrate these by showing results for two canonical problems in climate dynamics.

%B Dynamics and Statistics of the Climate System %V 1 %P 1-27 %G eng %U https://doi.org/10.1093/climsys/dzw003 %N 1 %R 10.1093/climsys/dzw003 %0 Journal Article %J Journal of Differential Equations %D 2016 %T The Stampacchia maximum principle for stochastic partial differential equations and applications %A Chekroun, Mickaël D. %A Eunhee Park %A Temam, Roger %K Nonlinear multiplicative noise %X Abstract Stochastic partial differential equations (SPDEs) are considered, linear and nonlinear, for which we establish comparison theorems for the solutions, or positivity results a.e., and a.s., for suitable data. Comparison theorems for \SPDEs\ are available in the literature. The originality of our approach is that it is based on the use of truncations, following the Stampacchia approach to maximum principle. We believe that our method, which does not rely too much on probability considerations, is simpler than the existing approaches and to a certain extent, more directly applicable to concrete situations. Among the applications, boundedness results and positivity results are respectively proved for the solutions of a stochastic Boussinesq temperature equation, and of reaction–diffusion equations perturbed by a non-Lipschitz nonlinear noise. Stabilization results to a Chafee–Infante equation perturbed by a nonlinear noise are also derived. %B Journal of Differential Equations %V 260 %P 2926 - 2972 %G eng %U http://www.sciencedirect.com/science/article/pii/S0022039615005653 %N 3 %R http://dx.doi.org/10.1016/j.jde.2015.10.022 %0 Journal Article %J Phys. Rev. E %D 2016 %T Comment on ``Nonparametric forecasting of low-dimensional dynamical systems'' %A Kondrashov, Dmitri %A Chekroun, Mickaël D. %A Ghil, Michael %X[[{"fid":"291","view_mode":"default","type":"media","attributes":{"height":"215","width":"275","style":"float: right;","class":"media-element file-default"}}]]The comparison performed in Berry *et al.* [Phys. Rev. E **91**, 032915 (2015)] between the skill in predicting the El Niño-Southern Oscillation climate phenomenon by the prediction method of Berry *et al.* and the “past-noise” forecasting method of Chekroun *et al.* [Proc. Natl. Acad. Sci. USA **108**, 11766 (2011)] is flawed. Three specific misunderstandings in Berry *et al.* are pointed out and corrected.

A suite of empirical model experiments under the empirical model reduction framework are conducted to advance the understanding of [[{"fid":"314","view_mode":"default","type":"media","field_deltas":{},"attributes":{"height":"266","width":"359","style":"float: right;","class":"media-element file-default","data-delta":"1"},"fields":{}}]]ENSO diversity, nonlinearity, seasonality, and the memory effect in the simulation and prediction of tropical Pacific sea surface temperature (SST) anomalies. The model training and evaluation are carried out using 4000-yr preindustrial control simulation data from the coupled model GFDL CM2.1. The results show that multivariate models with tropical Pacific subsurface information and multilevel models with SST history information both improve the prediction skill dramatically. These two types of models represent the ENSO memory effect based on either the recharge oscillator or the time-delayed oscillator viewpoint. Multilevel SST models are a bit more efficient, requiring fewer model coefficients. Nonlinearity is found necessary to reproduce the ENSO diversity feature for extreme events. The nonlinear models reconstruct the skewed probability density function of SST anomalies and improve the prediction of the skewed amplitude, though the role of nonlinearity may be slightly overestimated given the strong nonlinear ENSO in GFDL CM2.1. The models with periodic terms reproduce the SST seasonal phase locking but do not improve the prediction appreciably. The models with multiple ingredients capture several ENSO characteristics simultaneously and exhibit overall better prediction skill for more diverse target patterns. In particular, they alleviate the spring/autumn prediction barrier and reduce the tendency for predicted values to lag the target month value.

%B Journal of Climate %V 29 %P 1809–1830 %G eng %U http://journals.ametsoc.org/doi/abs/10.1175/JCLI-D-15-0372.1 %N 5 %0 Journal Article %J Journal of Climate %D 2016 %T Exploring the pullback attractors of a low-order quasigeostrophic ocean model: the deterministic case %A Pierini, S. %A M. Ghil %A M. Chekroun %XA low-order quasigeostrophic double-gyre ocean model is subjected to an aperiodic forcing that mimics time dependence dominated by interdecadal variability. This model is used as a prototype of an unstable and nonlinear dynamical system with time-dependent forcing to explore basic features of climate change in the presence of natural variability. The study relies on the theoretical framework of nonautonomous dynamical systems and of their pullback attractors (PBAs), that is, of the time-dependent invariant sets attracting all trajectories initialized in the remote past. The existence of a global PBA is rigorously demonstrated for this weakly dissipative nonlinear model. Ensemble simulations are carried out and the convergence to PBAs is assessed by computing the probability density function (PDF) of localization of the trajectories. A sensitivity analysis with respect to forcing amplitude shows that the PBAs experience large modifications if the underlying autonomous system is dominated by small-amplitude limit cycles, while less dramatic changes occur in a regime characterized by large-amplitude relaxation oscillations. The dependence of the attracting sets on the choice of the ensemble of initial states is then analyzed. Two types of basins of attraction coexist for certain parameter ranges; they contain chaotic and nonchaotic trajectories, respectively. The statistics of the former does not depend on the initial states whereas the trajectories in the latter converge to small portions of the global PBA. This complex scenario requires separate PDFs for chaotic and nonchaotic trajectories. General implications for climate predictability are finally discussed.

%B Journal of Climate %V 29 %P 4185-4202 %G eng %U http://journals.ametsoc.org/doi/abs/10.1175/JCLI-D-15-0848.1?af=R %N 11 %0 Journal Article %J Discrete and Continuous Dynamical Systems %D 2016 %T Low-dimensional Galerkin approximations of nonlinear delay differential equations %A Chekroun, Mickaël D. %A Ghil, Michael %A Liu, Honghu %A Wang, Shouhong %K El Niño–Southern Oscillation (ENSO) %XThis article revisits the approximation problem of systems of nonlinear delay differential equations (DDEs) by a set of ordinary differential equations (ODEs). We work in Hilbert spaces endowed with a natural inner product including a point mass, and introduce polynomials orthogonal with respect to such an inner product that live in the domain of the linear operator associated with the underlying DDE. These polynomials are then used to design a general Galerkin scheme for which we derive rigorous convergence results and show that it can be numerically implemented via simple analytic formulas. The scheme so obtained is applied to three nonlinear DDEs, two autonomous and one forced: (i) a simple DDE with distributed delays whose solutions recall Brownian motion; (ii) a DDE with a discrete delay that exhibits bimodal and chaotic dynamics; and (iii) a periodically forced DDE with two discrete delays arising in climate dynamics. In all three cases, the Galerkin scheme introduced in this article provides a good approximation by low-dimensional ODE systems of the DDE's strange attractor, as well as of the statistical features that characterize its nonlinear dynamics.

[[{"fid":"293","view_mode":"default","type":"media","field_deltas":{"1":{}},"fields":{},"attributes":{"height":"323","width":"606","style":"display: block; margin-left: auto; margin-right: auto;","class":"media-element file-default","data-delta":"1"}}]]

%B Discrete and Continuous Dynamical Systems %V 36 %P 4133-4177 %G eng %U http://aimsciences.org/journals/displayArticlesnew.jsp?paperID=12348 %N 8 %R 10.3934/dcds.2016.36.4133 %0 Conference Paper %B 2016 IEEE 55th Conference on Decision and Control (CDC) %D 2016 %T Post-processing finite-horizon parameterizing manifolds for optimal control of nonlinear parabolic PDEs %A Chekroun, M. D. %A Liu, H. %X[[{"fid":"310","view_mode":"default","type":"media","attributes":{"height":"258","width":"401","style":"float: right;","class":"media-element file-default"}}]]The goal of this article is to propose an efficient way of empirically improving suboptimal solutions designed from the recent method of finite-horizon parameterizing manifolds (PMs) introduced by Chekroun and Liu (Acta Appl. Math., 2015) and concerned with the (sub)optimal control of nonlinear parabolic partial differential equations (PDEs). Given a finite horizon [0, T ] and a reduced low-mode phase space, a finite-horizon PM provides an approximate parameterization of the high modes by the low ones so that the unexplained high-mode energy is reduced — in an L 2-sense — when this parameterization is applied. In Chekroun and Liu (Acta Appl. Math., 2015), various PMs were constructed analytically from the uncontrolled version of the underlying PDE that allow for the design of reduced systems from which low-dimensional suboptimal controllers can be efficiently synthesized. In this article, the analytic approach from Chekroun and Liu (Acta Appl. Math., 2015) is recalled and a post-processing procedure is introduced to improve the PM-based suboptimal controllers. It consists of seeking for a high-mode parametrization aiming to reduce the energy contained in the high modes of the PDE solution, when the latter is driven by a PM-based suboptimal controller. This is achieved by solving simple regression problems. The skills of the resulting empirically post-processed suboptimal controllers are numerically assessed for an optimal control problem associated with a Burgers-type equation.

%B 2016 IEEE 55th Conference on Decision and Control (CDC) %I IEEE %C Las Vegas, USA %P 1411-1416 %G eng %U http://ieeexplore.ieee.org/document/7798464/ %0 Journal Article %J Acta Applicandae Mathematicae %D 2015 %T Finite-Horizon Parameterizing Manifolds, and Applications to Suboptimal Control of Nonlinear Parabolic PDEs %A Chekroun, Mickaël D. %A Liu, Honghu %XThis article proposes a new approach based on finite-horizon parameterizing manifolds (PMs) for the design of low-dimensional suboptimal controllers to optimal control problems of nonlinear partial differential equations (PDEs) of parabolic type. Given a finite horizon [0,T] and a low-mode truncation of the PDE, a PM provides an approximate parameterization of the uncontrolled high modes by the controlled low ones so that the unexplained high-mode energy is reduced, in an L2-sense, when this parameterization is applied. Analytic formulas of such PMs are derived by application of the method of pullback approximation of the high-modes. These formulas allow for an effective derivation of reduced ODE systems, aimed to model the evolution of the low-mode truncation of the controlled state variable, where the high-mode part is approximated by the PM function applied to the low modes. A priori error estimates between the resulting PM-based low-dimensional suboptimal controller u_R* and the optimal controller u* are derived. These estimates demonstrate that the closeness of u_R* to u*? is mainly conditioned on two factors: (i) the parameterization defect of a given PM, associated respectively with u_R* and u*; and (ii) the energy kept in the high modes of the PDE solution either driven by u_R* or u* itself. The practical performances of such PM-based suboptimal controllers are numerically assessed for various optimal control problems associated with a Burgers-type equation. The numerical results show that a PM-based reduced system allows for the design of suboptimal controllers with good performances provided that the associated parameterization defects and energy kept in the high modes are small enough, in agreement with the rigorous results. The practical performances of such PM-based suboptimal controllers are numerically assessed for optimal control problems associated with a Burgers-type equation; the locally as well as globally distributed cases being both considered. The numerical results show that a PM-based reduced system allows for the design of suboptimal controllers with good performances provided that the associated parameterization defects and energy kept in the high modes are small enough, in agreement with the rigorous results.

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%B Acta Applicandae Mathematicae %V 135 %P 81–144 %G eng %U http://dx.doi.org/10.1007/s10440-014-9949-1 %N 1 %R 10.1007/s10440-014-9949-1 %0 Journal Article %J Physica D: Nonlinear Phenomena %D 2015 %T Data-driven non-Markovian closure models %A Kondrashov, Dmitri %A Chekroun, Mickaël D. %A Ghil, Michael %K Nonlinear stochastic closure model %X Abstract This paper has two interrelated foci: (i) obtaining stable and efficient data-driven closure models by using a multivariate time series of partial observations from a large-dimensional system; and (ii) comparing these closure models with the optimal closures predicted by the Mori–Zwanzig (MZ) formalism of statistical physics. Multilayer stochastic models (MSMs) are introduced as both a generalization and a time-continuous limit of existing multilevel, regression-based approaches to closure in a data-driven setting; these approaches include empirical model reduction (EMR), as well as more recent multi-layer modeling. It is shown that the multilayer structure of \MSMs\ can provide a natural Markov approximation to the generalized Langevin equation (GLE) of the \MZ\ formalism. A simple correlation-based stopping criterion for an EMR–MSM model is derived to assess how well it approximates the \GLE\ solution. Sufficient conditions are derived on the structure of the nonlinear cross-interactions between the constitutive layers of a given \MSM\ to guarantee the existence of a global random attractor. This existence ensures that no blow-up can occur for a broad class of \MSM\ applications, a class that includes non-polynomial predictors and nonlinearities that do not necessarily preserve quadratic energy invariants. The EMR–MSM methodology is first applied to a conceptual, nonlinear, stochastic climate model of coupled slow and fast variables, in which only slow variables are observed. It is shown that the resulting closure model with energy-conserving nonlinearities efficiently captures the main statistical features of the slow variables, even when there is no formal scale separation and the fast variables are quite energetic. Second, an \MSM\ is shown to successfully reproduce the statistics of a partially observed, generalized Lotka–Volterra model of population dynamics in its chaotic regime. The challenges here include the rarity of strange attractors in the model’s parameter space and the existence of multiple attractor basins with fractal boundaries. The positivity constraint on the solutions’ components replaces here the quadratic-energy–preserving constraint of fluid-flow problems and it successfully prevents blow-up. %B Physica D: Nonlinear Phenomena %V 297 %P 33 - 55 %G eng %U http://www.sciencedirect.com/science/article/pii/S0167278914002413 %R http://dx.doi.org/10.1016/j.physd.2014.12.005 %0 Book %D 2015 %T Approximation of Stochastic Invariant Manifolds : Stochastic Manifolds for Nonlinear SPDEs I %A Chekroun, Mickaël D. %A Liu, Honghu %A Wang, Shouhong %X[[{"fid":"285","view_mode":"default","type":"media","field_deltas":{"1":{}},"fields":{},"attributes":{"height":"232","width":"153","style":"float: right;","class":"media-element file-default","data-delta":"1"}}]]This first volume is concerned with the analytic derivation of explicit formulas for the leading-order Taylor approximations of (local) stochastic invariant manifolds associated with a broad class of nonlinear stochastic partial differential equations. These approximations take the form of Lyapunov-Perron integrals, which are further characterized in Volume II as pullback limits associated with some partially coupled backward-forward systems. This pullback characterization provides a useful interpretation of the corresponding approximating manifolds and leads to a simple framework that unifies some other approximation approaches in the literature. A self-contained survey is also included on the existence and attraction of one-parameter families of stochastic invariant manifolds, from the point of view of the theory of random dynamical systems.

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%I Springer Briefs in Mathematics, Springer %C New York %P pp. 127 %G eng %U http://www.springer.com/us/book/9783319124957 %0 Journal Article %J Mathematics of Climate and Weather Forecasting %D 2015 %T Numerical simulations of the humid atmosphere above a mountain %A A.Bousquet %A Chekroun, M. D. %A Y. Hong %A R. Temam %A J. Tribbia %XNew avenues are explored for the numerical study of the two dimensional inviscid hydrostatic primitive equations of the atmosphere with humidity and saturation, in presence of topography and subject to physically plausible boundary conditions for the system of equations. Flows above a mountain are classically treated by the so-called method of terrain following coordinate system. We avoid this discretization method which induces errors in the discretization of tangential derivatives near the topography. Instead we implement a first order finite volume method for the spatial discretization using the initial coordinates x and p. A compatibility condition similar to that related to the condition of incompressibility for the Navier- Stokes equations, is introduced. In that respect, a version of the projection method is considered to enforce the compatibility condition on the horizontal velocity field, which comes from the boundary conditions. For the spatial discretization, a modified Godunov type method that exploits the discrete finite-volume derivatives by using the so-called Taylor Series Expansion Scheme (TSES), is then designed to solve the equations. We report on numerical experiments using realistic parameters. Finally, the effects of a random small-scale forcing on the velocity equation is numerically investigated.

%B Mathematics of Climate and Weather Forecasting %V 1 %P 96-126 %G eng %U http://www.degruyter.com/view/j/mcwf.2015.1.issue-1/mcwf-2015-0005/mcwf-2015-0005.xml %N 1 %0 Book %D 2015 %T Stochastic Parameterizing Manifolds and Non-Markovian Reduced Equations : Stochastic Manifolds for Nonlinear SPDEs II %A Chekroun, Mickaël D. %A Liu, Honghu %A Wang, Shouhong %X[[{"fid":282,"view_mode":"default","type":"media","field_deltas":{"1":{}},"fields":{},"attributes":{"height":"232","width":"153","style":"float: right;","class":"media-element file-default","data-delta":"1"}}]]In this second volume, a general approach is developed to provide approximate parameterizations of the "small" scales by the "large" ones for a broad class of stochastic partial differential equations (SPDEs). This is accomplished via the concept of parameterizing manifolds (PMs), which are stochastic manifolds that improve, for a given realization of the noise, in mean square error the partial knowledge of the full SPDE solution when compared to its projection onto some resolved modes. Backward-forward systems are designed to give access to such PMs in practice. The key idea consists of representing the modes with high wave numbers as a pullback limit depending on the time-history of the modes with low wave numbers. Non-Markovian stochastic reduced systems are then derived based on such a PM approach. The reduced systems take the form of stochastic differential equations involving random coefficients that convey memory effects. The theory is illustrated on a stochastic Burgers-type equation.

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%I Springer Briefs in Mathematics, Springer %C New York %P pp. 129 %G eng %U http://www.springer.com/us/book/9783319125190 %0 Journal Article %J Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences %D 2014 %T Parameter estimation for energy balance models with memory %A Roques, Lionel %A Chekroun, Mickaël D. %A Cristofol, Michel %A Soubeyrand, Samuel %A Ghil, Michael %X We study parameter estimation for one-dimensional energy balance models with memory (EBMMs) given localized and noisy temperature measurements. Our results apply to a wide range of nonlinear, parabolic partial differential equations with integral memory terms. First, we show that a space-dependent parameter can be determined uniquely everywhere in the PDE’s domain of definition D, using only temperature information in a small subdomain E⊂D. This result is valid only when the data correspond to exact measurements of the temperature. We propose a method for estimating a model parameter of the EBMM using more realistic, error-contaminated temperature data derived, for example, from ice cores or marine-sediment cores. Our approach is based on a so-called mechanistic-statistical model that combines a deterministic EBMM with a statistical model of the observation process. Estimating a parameter in this setting is especially challenging, because the observation process induces a strong loss of information. Aside from the noise contained in past temperature measurements, an additional error is induced by the age-dating method, whose accuracy tends to decrease with a sample’s remoteness in time. Using a Bayesian approach, we show that obtaining an accurate parameter estimate is still possible in certain cases. %B Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences %I The Royal Society %V 470 %G eng %U http://rspa.royalsocietypublishing.org/content/470/2169/20140349 %N 2169 %R 10.1098/rspa.2014.0349 %0 Journal Article %J Proceeding of the National Academy of Sciences %D 2014 %T Rough parameter dependence in climate models and the role of Ruelle-Pollicott resonance %A Chekroun, M. D. %A Neelin, J. D. %A Kondrashov, D. %A J. C. McWilliams %A M. Ghil %X[[{"fid":"308","view_mode":"default","type":"media","field_deltas":{},"attributes":{"height":"164","width":"329","style":"float: right;","class":"media-element file-default","data-delta":"1"},"fields":{}}]]Despite the importance of uncertainties encountered in climate model simulations, the fundamental mechanisms at the origin of sensitive behavior of long-term model statistics remain unclear. Variability of turbulent flows in the atmosphere and oceans exhibits recurrent large-scale patterns. These patterns, while evolving irregularly in time, manifest characteristic frequencies across a large range of time scales, from intraseasonal through interdecadal. Based on modern spectral theory of chaotic and dissipative dynamical systems, the associated low-frequency variability may be formulated in terms of Ruelle-Pollicott (RP) resonances. RP resonances encode information on the nonlinear dynamics of the system, and an approach for estimating them—as filtered through an observable of the system—is proposed. This approach relies on an appropriate Markov representation of the dynamics associated with a given observable. It is shown that, within this representation, the spectral gap—defined as the distance between the subdominant RP resonance and the unit circle—plays a major role in the roughness of parameter dependences. The model statistics are the most sensitive for the smallest spectral gaps; such small gaps turn out to correspond to regimes where the low-frequency variability is more pronounced, whereas autocorrelations decay more slowly. The present approach is applied to analyze the rough parameter dependence encountered in key statistics of an El-Niño–Southern Oscillation model of intermediate complexity. Theoretical arguments, however, strongly suggest that such links between model sensitivity and the decay of correlation properties are not limited to this particular model and could hold much more generally.

%B Proceeding of the National Academy of Sciences %V 111 %P 1684—1690 %G eng %U http://www.pnas.org/content/111/5/1684.full?sid=caae9302-0489-4096-86d1-61846dd5bbaa %N 5 %0 Journal Article %J Discrete and Continuous Dynamical Systems (DCDS-A) %D 2013 %T Homeomorphisms group of normed vector spaces : The conjugacy problem and the Koopman operator %A Chekroun, M. D. %A Roux, J. %X

This article is concerned with conjugacy problems arising in the homeomorphisms group, Hom(F), of unbounded subsets F of normed vector spaces E. Given two homeomorphisms f and g in Hom(F), it is shown how the existence of a conjugacy may be related to the existence of a common generalized eigenfunction of the associated Koopman operators. This common eigenfunction serves to build a topology on Hom(F), where the conjugacy is obtained as limit of a sequence generated by the conjugacy operator, when this limit exists. The main conjugacy theorem is presented in a class of generalized Lipeomorphisms.

%B Discrete and Continuous Dynamical Systems (DCDS-A) %V 33 %P 3957—3950 %G eng %U http://www.aimsciences.org/article/doi/10.3934/dcds.2013.33.3957 %N 9 %0 Journal Article %J Geophysical Research Letters %D 2013 %T Low-order stochastic model and “past-noise forecasting” of the Madden-Julian oscillation %A K. Kondrashov %A Chekroun, M. D. %A Robertson, A. W. %A M. Ghil %XThis paper presents a predictability study of the Madden-Julian Oscillation (MJO) that relies on combining empirical model reduction (EMR) with the “past-noise forecasting” (PNF) method. EMR is a data-driven methodology for constructing stochastic low-dimensional models that account for nonlinearity, seasonality and serial correlation in the estimated noise, while PNF constructs an ensemble of forecasts that accounts for interactions between (i) high-frequency variability (noise), estimated here by EMR, and (ii) the low-frequency mode of MJO, as captured by singular spectrum analysis (SSA). A key result is that—compared to an EMR ensemble driven by generic white noise—PNF is able to considerably improve prediction of MJO phase. When forecasts are initiated from weak MJO conditions, the useful skill is of up to 30 days. PNF also significantly improves MJO prediction skill for forecasts that start over the Indian Ocean.

%B Geophysical Research Letters %V 40 %P 5303—5310 %G eng %U http://onlinelibrary.wiley.com/doi/10.1002/grl.50991/abstract %N 19 %0 Journal Article %J Communications in Mathematical Physics %D 2012 %T Invariant Measures for Dissipative Dynamical Systems: Abstract Results and Applications %A Chekroun, Mickaël D. %A Glatt-Holtz, Nathan E. %X In this work we study certain invariant measures that can be associated to the time averaged observation of a broad class of dissipative semigroups via the notion of a generalized Banach limit. Consider an arbitrary complete separable metric space X which is acted on by any continuous semigroup \S(t)\ t ≥ 0. Suppose that \S(t)\ t ≥ 0 possesses a global attractor \$\$\\backslashmathcal\A\\\$\$ . We show that, for any generalized Banach limit LIM T → ∞ and any probability distribution of initial conditions \$\$\\backslashmathfrak\m\\_0\\$\$ , that there exists an invariant probability measure \$\$\\backslashmathfrak\m\\\$\$ , whose support is contained in \$\$\\backslashmathcal\A\\\$\$ , such that \$\$\backslashint\_\X\ \backslashvarphi(x) \\backslashrm d\\backslashmathfrak\m\(x) = \backslashunderset\t \backslashrightarrow \backslashinfty\\\backslashrm LIM\\backslashfrac\1\\T\ \backslashint\_0^T \backslashint\_X \backslashvarphi(S(t) x) \\backslashrm d\\backslashmathfrak\m\\_0(x) \\backslashrm d\t,\$\$ for all observables $\phi$ living in a suitable function space of continuous mappings on X. %B Communications in Mathematical Physics %V 316 %P 723–761 %G eng %U http://dx.doi.org/10.1007/s00220-012-1515-y %N 3 %R 10.1007/s00220-012-1515-y %0 Journal Article %J Ecological Complexity %D 2011 %T Probing chaos and biodiversity in a simple competition model %A Roques, Lionel %A Chekroun, Mickaël D. %K Simulated annealing %X Recent theoretical work has reported that chaos facilitates biodiversity. In this paper, we study the lowest-dimensional Lotka–Volterra competition model that exhibits chaotic trajectories, a model with four species. We observe that interaction and growth parameters leading respectively to extinction of three species, or coexistence of two, three or four species, are for the most part arranged in large regions with clear boundaries. Small islands of parameters that lead to chaos are also found. These regions where chaos occurs are, in the three cases presented here, situated at the interface between a non-chaotic four-species region and a region where extinction occurs. This implies a high sensitivity of biodiversity with respect to parameter variations in the chaotic regions. Additionally, in regions where extinction occurs which are adjacent to chaotic regions, the computation of local Lyapunov exponents reveals that a possible cause of extinction is the overly strong fluctuations in species abundances induced by local chaos at the beginning of the interval of study. For this model, we conclude that biodiversity is a necessary condition for chaos rather than a consequence of chaos, which can be seen as a signal of a high extinction risk. %B Ecological Complexity %V 8 %P 98 - 104 %G eng %U http://www.sciencedirect.com/science/article/pii/S1476945X10000693 %N 1 %R http://dx.doi.org/10.1016/j.ecocom.2010.08.004 %0 Journal Article %J Discrete and Continuous Dynamical Systems, Series S %D 2011 %T Asymptotics of the Coleman-Gurtin model %A Chekroun, M. D. %A Di Plinio, F. %A Glatt-Holtz, N. E. %A Pata, V. %XThis paper is concerned with the integrodifferential equation

arising in the Coleman-Gurtin's theory of heat conduction with hereditary memory, in presence of a nonlinearity of critical growth. Rephrasing the equation within the history space framework, we prove the existence of global and exponential attractors of optimal regularity and finite fractal dimension for the related solution semigroup, acting both on the basic weak-energy space and on a more regular phase space.

%B Discrete and Continuous Dynamical Systems, Series S %V 4 %P 351-369 %G eng %U http://www.aimsciences.org/article/doi/10.3934/dcdss.2011.4.351 %N 2 %0 Journal Article %J Proceeding of the National Academy of Sciences %D 2011 %T Predicting stochastic systems by noise sampling, and application to the El Niño-Southern Oscillation %A Chekroun, M. D. %A Kondrashov, D. %A M. Ghil %XInterannual and interdecadal prediction are major challenges of climate dynamics. In this article we develop a prediction method for climate processes that exhibit low-frequency variability (LFV). The method constructs a nonlinear stochastic model from past observations and estimates a path of the “weather” noise that drives this model over previous finite-time windows. The method has two steps: (*i*) select noise samples—or “snippets”—from the past noise, which have forced the system during short-time intervals that resemble the LFV phase just preceding the currently observed state; and (*ii*) use these snippets to drive the system from the current state into the future. The method is placed in the framework of pathwise linear-response theory and is then applied to an El Niño–Southern Oscillation (ENSO) model derived by the empirical model reduction (EMR) methodology; this nonlinear model has 40 coupled, slow, and fast variables. The domain of validity of this forecasting procedure depends on the nature of the system’s pathwise response; it is shown numerically that the ENSO model’s response is linear on interannual time scales. As a result, the method’s skill at a 6- to 16-month lead is highly competitive when compared with currently used dynamic and statistic prediction methods for the Niño-3 index and the global sea surface temperature field.

This article attempts a unification of the two approaches that have dominated theoretical climate dynamics since its inception in the 1960s: the nonlinear deterministic and the linear stochastic one. This unification, via the theory of random dynamical systems (RDS), allows one to consider the detailed geometric structure of the random attractors associated with nonlinear, stochastically perturbed systems. We report on high-resolution numerical studies of two idealized models of fundamental interest for climate dynamics. The first of the two is a stochastically forced version of the classical Lorenz model. The second one is a low-dimensional, nonlinear stochastic model of the El Niño–Southern Oscillation (ENSO). These studies provide a good approximation of the two models’ global random attractors, as well as of the time-dependent invariant measures supported by these attractors; the latter are shown to have an intuitive physical interpretation as random versions of Sinaï–Ruelle–Bowen (SRB) measures.

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

%B Physica D: Nonlinear Phenomena %V 240 %P 1685 - 1700 %G eng %U http://www.sciencedirect.com/science/article/pii/S016727891100145X %N 21 %R http://dx.doi.org/10.1016/j.physd.2011.06.005 %0 Journal Article %J Ecological Complexity %D 2010 %T Does reaction-diffusion support the duality of fragmentation effect? %A Roques, Lionel %A Chekroun, M. D. %K Conservation biology %X

There is a gap between single-species model predictions, and empirical studies, regarding the effect of habitat fragmentation per se, i.e., a process involving the breaking apart of habitat without loss of habitat. Empirical works indicate that fragmentation can have positive as well as negative effects, whereas, traditionally, single-species models predict a negative effect of fragmentation. Within the class of reaction-diffusion models, studies almost unanimously predict such a detrimental effect. In this paper, considering a single-species reaction-diffusion model with a removal – or similarly harvesting – term, in two dimensions, we find both positive and negative effects of fragmentation of the reserves, i.e., the protected regions where no removal occurs. Fragmented reserves lead to higher population sizes for time-constant removal terms. On the other hand, when the removal term is proportional to the population density, higher population sizes are obtained on aggregated reserves, but maximum yields are attained on fragmented configurations, and for intermediate harvesting intensities.

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%B Ecological Complexity %V 7 %P 100 - 106 %G eng %U http://www.sciencedirect.com/science/article/pii/S1476945X09000646 %N 1 %R http://dx.doi.org/10.1016/j.ecocom.2009.07.003 %0 Journal Article %J Physica D: Nonlinear Phenomena %D 2008 %T Climate dynamics and fluid mechanics: Natural variability and related uncertainties %A Ghil, Michael %A Chekroun, Mickaël D. %A Simonnet, Eric %K Arnol’d tongues %X The purpose of this review-and-research paper is twofold: (i) to review the role played in climate dynamics by fluid-dynamical models; and (ii) to contribute to the understanding and reduction of the uncertainties in future climate-change projections. To illustrate the first point, we review recent theoretical advances in studying the wind-driven circulation of the oceans. In doing so, we concentrate on the large-scale, wind-driven flow of the mid-latitude oceans, which is dominated by the presence of a larger, anticyclonic and a smaller, cyclonic gyre. The two gyres share the eastward extension of western boundary currents, such as the Gulf Stream or Kuroshio, and are induced by the shear in the winds that cross the respective ocean basins. The boundary currents and eastward jets carry substantial amounts of heat and momentum, and thus contribute in a crucial way to Earth’s climate, and to changes therein. Changes in this double-gyre circulation occur from year to year and decade to decade. We study this low-frequency variability of the wind-driven, double-gyre circulation in mid-latitude ocean basins, via the bifurcation sequence that leads from steady states through periodic solutions and on to the chaotic, irregular flows documented in the observations. This sequence involves local, pitchfork and Hopf bifurcations, as well as global, homoclinic ones. The natural climate variability induced by the low-frequency variability of the ocean circulation is but one of the causes of uncertainties in climate projections. The range of these uncertainties has barely decreased, or even increased, over the last three decades. Another major cause of such uncertainties could reside in the structural instability–in the classical, topological sense–of the equations governing climate dynamics, including but not restricted to those of atmospheric and ocean dynamics. We propose a novel approach to understand, and possibly reduce, these uncertainties, based on the concepts and methods of random dynamical systems theory. The idea is to compare the climate simulations of distinct general circulation models (GCMs) used in climate projections, by applying stochastic-conjugacy methods and thus perform a stochastic classification of \GCM\ families. This approach is particularly appropriate given recent interest in stochastic parametrization of subgrid-scale processes in GCMs. As a very first step in this direction, we study the behavior of the Arnol’d family of circle maps in the presence of noise. The maps’ fine-grained resonant landscape is smoothed by the noise, thus permitting their coarse-grained classification. %B Physica D: Nonlinear Phenomena %V 237 %P 2111 - 2126 %G eng %U http://www.sciencedirect.com/science/article/pii/S0167278908001139 %N 14–17 %R http://dx.doi.org/10.1016/j.physd.2008.03.036 %0 Journal Article %J SIAM Journal on Applied Mathematics %D 2007 %T On Population resilience to external perturbations %A L. Roques %A Chekroun, M. D. %X We study a spatially explicit harvesting model in periodic or bounded environments. The model is governed by a parabolic equation with a spatially dependent nonlinearity of Kolmogorov–Petrovsky–Piskunov type, and a negative external forcing term $-\delta$. Using sub- and supersolution methods and the characterization of the first eigenvalue of some linear elliptic operators, we obtain existence and nonexistence results as well as results on the number of stationary solutions. We also characterize the asymptotic behavior of the evolution equation as a function of the forcing term amplitude. In particular, we define two critical values $\delta^*$ and $\delta_2$ such that, if $\delta$ is smaller than $\delta^*$, the population density converges to a “significant" state, which is everywhere above a certain small threshold, whereas if $\delta$ is larger than $\delta_2$, the population density converges to a “remnant" state, everywhere below this small threshold. Our results are shown to be useful for studying the relationships between environmental fragmentation and maximum sustainable yield from populations. We present numerical results in the case of stochastic environments.

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In this article, we present a new approach to averaging in non-Hamiltonian systems with periodic forcing. The results here do not depend on the existence of a small parameter. In fact, we show that our averaging method fits into an appropriate nonlinear equivalence problem, and that this problem can be solved formally by using the Lie transform framework to linearize it. According to this approach, we derive formal coordinate transformations associated with both first-order and higher-order averaging, which result in more manageable formulae than the classical ones.

Using these transformations, it is possible to correct the solution of an averaged system by recovering the oscillatory components of the original non-averaged system. In this framework, the inverse transformations are also defined explicitly by formal series; they allow the estimation of appropriate initial data for each higher-order averaged system, respecting the equivalence relation.

Finally, we show how these methods can be used for identifying and computing periodic solutions for a very large class of nonlinear systems with time-periodic forcing. We test the validity of our approach by analyzing both the first-order and the second-order averaged system for a problem in atmospheric chemistry.