Dynamical systems

In Press
Ghil, Michael, and Valerio Lucarini. “The Physics of Climate Variability and Climate Change” (In Press). arxiv Abstract
The climate system is a forced, dissipative, nonlinear, complex and heterogeneous system that is out of thermodynamic equilibrium. The system exhibits natural variability on many scales of motion, in time as well as space, and it is subject to various external forcings, natural as well as anthropogenic. This paper reviews the observational evidence on climate phenomena and the governing equations of planetary-scale flow, as well as presenting the key concept of a hierarchy of models as used in the climate sciences. Recent advances in the application of dynamical systems theory, on the one hand, and of nonequilibrium statistical physics, on the other, are brought together for the first time and shown to complement each other in helping understand and predict the system's behavior. These complementary points of view permit a self-consistent handling of subgrid-scale phenomena as stochastic processes, as well as a unified handling of natural climate variability and forced climate change, along with a treatment of the crucial issues of climate sensitivity, response, and predictability.
Ghil, Michael, and Eric Simonnet. “Geophysical Fluid Dynamics, Nonautonomous Dynamical Systems, and the Climate Sciences.” In Mathematical Approach to Climate Change and its Impacts: MAC2I, edited by Piermarco Cannarsa, Daniela Mansutti, and Antonello Provenzale, 3–81. Springer International Publishing, 2020. Abstract
This contribution introduces the dynamics of shallow and rotating flows that characterizes large-scale motions of the atmosphere and oceans. It then focuses on an important aspect of climate dynamics on interannual and interdecadal scales, namely the wind-driven ocean circulation. Studying the variability of this circulation and slow changes therein is treated as an application of the theory of nonautonomous dynamical systems. The contribution concludes by discussing the relevance of these mathematical concepts and methods for the highly topical issues of climate change and climate sensitivity.
Ghil, Michael. “Coupled Climate–Economics Modeling and Data Analysis: EnBCs and Fluctuation–Dissipation Theory.” CliMathParis 2019, Course IV: Coupled Climate–Ecology–Economy Modeling, Institut Henri Poincaré, Paris, France, 2019. CliMathParis 2019 Abstract

Ghil, Michael. “Low-Frequency Climate Variability: Markov Chains and Nonlinear Oscillations.” CliMathParis 2019, Associated Workshop I: The 9th International Workshop on Climate Informatics, Ecole Normale Supérieure & Institut Henri Poincaré, Paris, France, 2019. Workshop website Abstract

Ghil, Michael. “Nonautonomous and Random Dynamical Systems in the Climate Sciences.” CliMathParis 2019, Workshop 1: Nonlinear and stochastic methods in climate and geophysical fluid dynamics, Institut Henri Poincaré, Paris, France, 2019. Workshop website Abstract

Marangio, L., J. Sedro, S. Galatolo, A. Di Garbo, and Michael Ghil. “Arnold Maps with Noise: Differentiability and Non-monotonicity of the Rotation Number.” Journal of Statistical Physics (2019).
Walwer, Damian, Michael Ghil, and Eric Calais. “Oscillatory nature of the Okmok volcano's deformation.” Earth and Planetary Science Letters 506 (2019): 76–86.
Pierini, Stefano, Mickaël D. Chekroun, and Michael Ghil. “The onset of chaos in nonautonomous dissipative dynamical systems: a low-order ocean-model case study.” Nonlinear Processes in Geophysics 25, no. 3 (2018): 671–692.
Boers, Niklas, Michael Ghil, and Denis-Didier Rousseau. “Ocean circulation, ice shelf, and sea ice interactions explain Dansgaard-Oeschger cycles.” Proceedings of the National Academy of Sciences 115, no. 47 (2018): E11005–E11014.
Ghil, Michael, Andreas Groth, Dmitri Kondrashov, and Andrew W. Robertson. “Extratropical sub-seasonal–to–seasonal oscillations and multiple regimes: The dynamical systems view.” In The Gap between Weather and Climate Forecasting: Sub-Seasonal to Seasonal Prediction, edited by Andrew W. Robertson and Frederic Vitart, 119-142. 1st ed. Elsevier, 2018. 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.

Duane, G. S., C. Grabow, F. Selten, and Michael Ghil, ed. Synchronization in Large Networks and Continuous Media – Data, Models, and Supermodels. Focus Issue in Chaos. 27th ed. American Institute of Physics, Melville, NY, 2017.
Ghil, Michael. “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, 2017. Workshop website Abstract

Ghil, Michael. “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, 2017. Conference website Abstract

Ghil, Michael. “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, 2017. Workshop website Abstract
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Ghil, Michael. “The wind-driven ocean circulation: Applying dynamical systems theory to a climate problem.” Discrete and Continuous Dynamical Systems - A 37, no. 1 (2017): 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.

Ghil, Michael. “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, 2016. Abstract

Pierini, S., Michael Ghil, and Mickaël D. Chekroun. “Exploring the pullback attractors of a low-order quasigeostrophic ocean model: The deterministic case.” Journal of Climate 29, no. 11 (2016): 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. “Low-dimensional Galerkin approximations of nonlinear delay differential equations.” Discrete and Continuous Dynamical Systems - Series S 36, no. 8 (2016): 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.

Colon, Célian, David Claessen, and Michael Ghil. “Bifurcation analysis of an agent-based model for predator–prey interactions.” Ecological Modelling 317 (2015): 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. Approximation of Stochastic Invariant Manifolds: Stochastic Manifolds for Nonlinear SPDEs I. New York: Springer Briefs in Mathematics, Springer, 2015. 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.