Dynamical systems

2014
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.

2013
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
2012

PDF
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

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

PDF
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.
2011
Ghil, Michael. “Toward a Mathematical Theory of Climate Sensitivity.” International Congress on Industrial and Applied Mathematics (ICIAM), Vancouver, 2011. Abstract

Presentation
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.
PDF
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.

PDF
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.

PDF
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.

PDF
2010
Zaliapin, Ilya, and Michael Ghil. “A delay differential model of ENSO variability, Part 2: Phase locking, multiple solutions, and dynamics of extrema.” Nonlinear Processes in Geophysics 17, no. 2 (2010): 123–135.
Chekroun, Mickaël D., Michael Ghil, Jean Roux, and Ferenc Varadi. “Averaging of time-periodic systems without a small parameter.” Discrete and Continuous Dynamical Systems 14, no. 4 (2010): 753–782.
PDF
Roques, Lionel, and Mickaël D. Chekroun. “Does reaction-diffusion support the duality of fragmentation effect?Ecological Complexity 7, no. 1 (2010): 100–106.
PDF
Ghil, Michael, Peter Read, and Leonard Smith. “Geophysical flows as dynamical systems: the influence of Hide's experiments.” Astronomy & Geophysics 51, no. 4 (2010): 4–28.
PDF

Pages