Dynamical systems

Chekroun, Mickaël D., and Honghu Liu. “Finite-horizon parameterizing manifolds, and applications to suboptimal control of nonlinear parabolic PDEs.” Acta Applicandae Mathematicae 135, no. 1 (2015): 81–144.
Groth, Andreas, Patrice Dumas, Michael Ghil, and Stéphane Hallegatte. “Impacts of natural disasters on a dynamic economy.” In Extreme Events : Observations, Modeling, and Economics, edited by Eric Chavez, Michael Ghil, and Jaime Urrutia-Fucugauchi, 343–360. American Geophysical Union and Wiley-Blackwell, 2015. Abstract

This chapter presents a modeling framework for macroeconomic growth dynamics; it is motivated by recent attempts to formulate and study “integrated models” of the coupling between natural and socioeconomic phe­ nomena. The challenge is to describe the interfaces between human activities and the functioning of the earth system. We examine the way in which this interface works in the presence of endogenous business cycle dynam­ ics, based on a nonequilibrium dynamic model. Recent findings about the macroeconomic response to natural disasters in such a nonequilibrium setting have shown a more severe response to natural disasters during expan­ sions than during recessions. These findings raise questions about the assessment of climate change damages or natural disaster losses that are based purely on long-term growth models. In order to compare the theoretical findings with observational data, we analyze cyclic behavior in the U.S. economy, based on multivariate singular spectrum analysis. We analyze a total of nine aggregate indicators in a 52 year interval (1954–2005) and demon­ strate that the behavior of the U.S. economy changes significantly between intervals of growth and recession, with higher volatility during expansions.

Chekroun, Mickaël D., Honghu Liu, and S. Wang. Stochastic Parameterizing Manifolds and Non-Markovian Reduced Equations: Stochastic Manifolds for Nonlinear SPDEs II. New York: Springer Briefs in Mathematics, Springer, 2015. Publisher's Version Abstract

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.

Ghil, Michael, and I. Zaliapin. “Understanding ENSO variability and its extrema: A delay differential equation approach.” In Extreme Events: Observations, Modeling and Economics, edited by M. Chavez, Michael Ghil, and J. Urrutia-Fucugauchi, 63–78. American Geophysical Union & Wiley, 2015. Abstract

The El-Nino/Southern-Oscillation (ENSO) phenomenon is the most prominent signal of seasonal-to-interannual climate variability. The past 30 years of research have shown that ENSO dynamics is governed, by and large, by the interplay of the nonlinear mechanisms, and that their simplest version can be studied in autonomous or forced delay differential equation (DDE) models. This chapter briefly reviews the results of Ghil et al., Zaliapin and Ghil, and Ghil and Zaliapin and pursues their DDE model analysis by focusing on multiple model solutions for the same parameter values and the dynamics of local extrema. It first introduces the DDE model of ENSO variability, reviews the main theoretical results concerning its solutions, and comments on the appropriate numerical integration methods. Novel results on multiple solutions and their extrema are reported and illustrated. After discussing the model's pullback attractor, the chapter explores parameter dependence in the model over its entire 3D parameter space.

Chekroun, Mickaël D., J. David Neelin, Dmitri Kondrashov, James C. McWilliams, and Michael Ghil. “Rough parameter dependence in climate models and the role of Ruelle-Pollicott resonances.” Proceedings of the National Academy of Sciences 111, no. 5 (2014): 1684-1690. Abstract

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.

Ghil, Michael. “Lecture 2: Toward a Mathematical Theory of Climate Sensitivity.” Workshop on Mathematics of Climate Change, Related Hazards and Risks, CIMAT, Guanajuato, Mexico, 2013. Abstract

Lecture 2: Toward a Mathematical Theory of Climate Sensitivity
Ghil, Michael. “Lecture 3 : The Coupled Dynamics of Climate and Economics.” Workshop on Mathematics of Climate Change, Related Hazards and Risks, CIMAT, Guanajuato, Mexico, 2013. Abstract

Lecture 3 : The Coupled Dynamics of Climate and Economics

Ghil, Michael. “The Complex Physics of Climate Change: Nonlinearity and Stochasticity.” Workshop on Critical Transitions in Complex Systems, Imperial College London, United Kingdom, 2012. Conference website Abstract

Ghil, Michael. “What is a Tipping Point and Why Do We Care?EGU 2012, 2012. Abstract

Chekroun, Mickaël D., and Jean Roux. “Homeomorphisms group of normed vector space: Conjugacy problems and the Koopman operator.” Discrete and Continuous Dynamical Systems - Series A 33 (2012): 3957–3980.
Chekroun, Mickaël D., and N. E. Glatt-Holtz. “Invariant measures for dissipative dynamical systems: Abstract results and applications.” Communications in Mathematical Physics 316 (2012): 723–761.
Ghil, Michael. “Toward a Mathematical Theory of Climate Sensitivity.” International Congress on Industrial and Applied Mathematics (ICIAM), Vancouver, 2011. Abstract

Chekroun, Mickaël D., Eric Simonnet, and Michael Ghil. “Stochastic climate dynamics: Random attractors and time-dependent invariant measures.” Physica D 240, no. 21 (2011): 1685-–1700. Abstract
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 Sina\"ı-Ruelle-Bowen (SRB) measures.
Chekroun, Mickaël D., F. Di Plinio, N. E. Glatt-Holtz, and V. Pata. “Asymptotics of the Coleman-Gurtin model.” Discrete and Continuous Dynamical Systems - Series S 4, no. 2 (2011): 351–369.
Coluzzi, Barbara, Michael Ghil, Stéphane Hallegatte, and Gérard Weisbuch. “Boolean delay equations on networks in economics and the geosciences.” International Journal of Bifurcation and Chaos 21, no. 12 (2011): 3511–3548.
Dumas, Patrice, Michael Ghil, Andreas Groth, and Stéphane Hallegatte. “Dynamic coupling of the climate and macroeconomic systems.” Math. & Sci. hum. / Mathematics and Social Sciences (2011). Abstract

This review paper presents a modeling framework for macroeco- nomic growth dynamics that is motivated by recent attempts to formulate and study “integrated models” of the coupling between natural and socio-economic phenomena. The challenge is to describe the interfaces between human acti- vities and the functioning of the earth system. We examine the way that this interface works in the presence of endogenous business cycle dynamics, based on a non-equilibrium dynamic model, and review the macroeconomic response to natural disasters. Our model exhibits a larger response to natural disasters during expansions than during recessions, and we raise questions about the as- sessment of climate change damages or natural disaster losses that are based purely on long-term growth models. In order to compare the theoretical fin- dings with observational data, we present a new method for extracting cyclic behavior from the latter, based on multivariate singular spectral analysis.

Ghil, Michael, P. Yiou, S. Hallegatte, B. D. Malamud, P. Naveau, A. Soloviev, P. Friederichs, et al.Extreme events: dynamics, statistics and prediction.” Nonlinear Processes in Geophysics 18, no. 3 (2011): 295–350. Abstract

We review work on extreme events, their causes and consequences, by a group of Euro- pean and American researchers involved in a three-year project on these topics. The review covers theoretical aspects of time series analysis and of extreme value theory, as well as of the deteministic modeling of extreme events, via continuous and discrete dynamic models. The applications include climatic, seismic and socio-economic events, along with their prediction.

Groth, Andreas, and Michael Ghil. “Multivariate singular spectrum analysis and the road to phase synchronization.” Physical Review E 84 (2011): 036206. Abstract

We show that multivariate singular spectrum analysis (M-SSA) greatly helps study phase synchronization in a large system of coupled oscillators and in the presence of high observational noise levels. With no need for detailed knowledge of individual subsystems nor any a priori phase de?nition for each of them, we demonstrate that M-SSA can automatically identify multiple oscillatory modes and detect whether these modes are shared by clusters of phase- and frequency-locked oscillators. As an essential modi?cation of M-SSA, here we introduce variance-maximization (varimax) rotation of the M-SSA eigenvectors to optimally identify synchronized-oscillator clustering.

Chekroun, Mickaël D., Dmitri Kondrashov, and Michael Ghil. “Predicting stochastic systems by noise sampling, and application to the El Niño-Southern Oscillation.” Proceedings of the National Academy of Sciences 108, no. 29 (2011): 11766–11771. Abstract

Interannual 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.