Dynamical systems

2018
Ghil, Michael, Andreas Groth, Dmitri Kondrashov, and Andrew W. Robertson. 2018. “Extratropical sub-seasonal–to–seasonal oscillations and multiple regimes: The dynamical systems view.” The Gap between Weather and Climate Forecasting: Sub-Seasonal to Seasonal Prediction, edited by Andrew W. Robertson and Frederic Vitart, 1st ed., 119-142. Elsevier. Publisher's Version Abstract

This chapter considers the sub-seasonal–to–seasonal (S2S) prediction problem as intrinsically more difficult than either short-range weather prediction or interannual–to–multidecadal climate prediction. The difficulty arises from the comparable importance of atmospheric initial states and of parameter values in determining the atmospheric evolution on the S2S time scale. The chapter relies on the theoretical framework of dynamical systems and the practical tools this framework helps provide to low-order modeling and prediction of S2S variability. The emphasis is on mid-latitude variability and the complementarity of the nonlinear-waves vs. multiple-regime points of view in understanding this variability. Empirical model reduction and the forecast skill of the models thus produced in real-time prediction are reviewed.

2017
Ghil, Michael. 2017. “The Mathematics of Climate Change and of its Impacts.” Workshop on "Mathematical Approaches to Climate Change Impacts - MAC2I" at the Istituto Nazionale di Alta Matematica "Francesco Severi" (INdAM), Italy. Workshop website Abstract

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Ghil, Michael. 2017. “Circulation Regimes for the Hitchhiker Through the Galaxy.” Physics school on Diversity of Planetary Circulation Regimes, in our Solar System and beyond, Les Houches, France, March 2017. Conference website Abstract

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Ghil, Michael. 2017. “The atmosphere and oceans as unsteady flows: Intrinsic variability and time-dependent forcing.” BIRS Workshop 17w5048 - Transport in Unsteady Flows: from Deterministic Structures to Stochastic Models and Back Again. Workshop website Abstract
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Ghil, Michael. 2017. “The wind-driven ocean circulation: Applying dynamical systems theory to a climate problem.” Discrete and Continuous Dynamical Systems - A 37 (1): 189-228. Abstract

The large-scale, near-surface flow of the mid-latitude oceans 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. This physical phenomenology is described mathematically by a hierarchy of systems of nonlinear partial differential equations (PDEs). We study the low-frequency variability of this wind-driven, double-gyre circulation in mid-latitude ocean basins, subject to time-constant, purely periodic and more general forms of time-dependent wind stress. Both analytical and numerical methods of dynamical systems theory are applied to the PDE systems of interest. Recent work has focused on the application of non-autonomous and random forcing to double-gyre models. We discuss the associated pullback and random attractors and the non-uniqueness of the invariant measures that are obtained. The presentation moves from observations of the geophysical phenomena to modeling them and on to a proper mathematical understanding of the models thus obtained. Connections are made with the highly topical issues of climate change and climate sensitivity.

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2016
Ghil, Michael. 2016. “A Mathematical Theory of Climate Sensitivity: A Tale of Deterministic & Stochastic Dynamical Systems.” 11th AIMS Conf. on Dynamical Systems, Differential Equations & Applications, Honoring Peter Lax’s 90th Birthday, Orlando, FL, July 2016. Abstract

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Pierini, S., Michael Ghil, and Mickaël D. Chekroun. 2016. “Exploring the pullback attractors of a low-order quasigeostrophic ocean model: The deterministic case.” Journal of Climate 29 (11): 4185-4202. Abstract

A 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.

Chekroun, Mickaël D., Michael Ghil, Honghu Liu, and Shouhong Wang. 2016. “Low-dimensional Galerkin approximations of nonlinear delay differential equations.” Discrete and Continuous Dynamical Systems - Series S 36 (8): 4133-4177. Abstract

This 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.

2015
Colon, Célian, David Claessen, and Michael Ghil. 2015. “Bifurcation analysis of an agent-based model for predator–prey interactions.” Ecological Modelling 317: 93 - 106. Publisher's Version Abstract

Abstract The Rosenzweig–MacArthur model is a set of ordinary differential equations (ODEs) that provides an aggregate description of the dynamics of a predator–prey system. When including an Allee effect on the prey, this model exhibits bistability and contains a pitchfork bifurcation, a Hopf bifurcation and a heteroclinic bifurcation. We develop an agent-based model (ABM) on a two-dimensional, square lattice that encompasses the key assumptions of the aggregate model. Although the two modelling approaches – \ODE\ and \ABM\ – differ, both models exhibit similar bifurcation patterns. The \ABM\ model's behaviour is richer and it is analysed using advanced statistical methods. In particular, singular spectrum analysis is used to robustly locate the transition between apparently random, small-amplitude fluctuations around a fixed point and stable, large-amplitude oscillations. Critical slowing down of model trajectories anticipates the heteroclinic bifurcation. Systematic comparison between the \ABM\ and the \ODE\ models’ behaviour helps one understand the predator–prey system better; it provides guidance in model exploration and allows one to draw more robust conclusions on the nature of predator–prey interactions.

Chekroun, Mickaël D., Honghu Liu, and S. Wang. 2015. Approximation of Stochastic Invariant Manifolds: Stochastic Manifolds for Nonlinear SPDEs I. New York: Springer Briefs in Mathematics, Springer. Publisher's Version Abstract

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.

Ghil, Michael, Mickaël D. Chekroun, and Gabor Stepan. 2015. “A collection on `Climate Dynamics: Multiple Scales and Memory Effects'.” Proceedings of the Royal Society A 471 (20150097). Royal Society London. Publisher's Version
Chekroun, Mickaël D., and Honghu Liu. 2015. “Finite-horizon parameterizing manifolds, and applications to suboptimal control of nonlinear parabolic PDEs.” Acta Applicandae Mathematicae 135 (1): 81–144.
Groth, Andreas, Patrice Dumas, Michael Ghil, and Stéphane Hallegatte. 2015. “Impacts of natural disasters on a dynamic economy.” Extreme Events : Observations, Modeling, and Economics, edited by Eric Chavez, Michael Ghil, and Jaime Urrutia-Fucugauchi, 343–360. American Geophysical Union and Wiley-Blackwell. 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.

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Chekroun, Mickaël D., Honghu Liu, and S. Wang. 2015. Stochastic Parameterizing Manifolds and Non-Markovian Reduced Equations: Stochastic Manifolds for Nonlinear SPDEs II. New York: Springer Briefs in Mathematics, Springer. 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. 2015. “Understanding ENSO variability and its extrema: A delay differential equation approach.” Extreme Events: Observations, Modeling and Economics, edited by M. Chavez, Michael Ghil, and J. Urrutia-Fucugauchi, 63–78. American Geophysical Union & Wiley. 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.

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2014
Chekroun, Mickaël D., J. David Neelin, Dmitri Kondrashov, James C. McWilliams, and Michael Ghil. 2014. “Rough parameter dependence in climate models and the role of Ruelle-Pollicott resonances.” Proceedings of the National Academy of Sciences 111 (5): 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.

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

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

Lecture 3 : The Coupled Dynamics of Climate and Economics
2012

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

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