Dynamical systems

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

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

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Chekroun, Mickaël D., Eric Simonnet, and Michael Ghil. 2011. “Stochastic climate dynamics: Random attractors and time-dependent invariant measures.” Physica D 240 (21): 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. 2011. “Asymptotics of the Coleman-Gurtin model.” Discrete and Continuous Dynamical Systems - Series S 4 (2): 351–369.
Coluzzi, Barbara, Michael Ghil, Stéphane Hallegatte, and Gérard Weisbuch. 2011. “Boolean delay equations on networks in economics and the geosciences.” International Journal of Bifurcation and Chaos 21 (12). World Scientific: 3511–3548.
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Dumas, Patrice, Michael Ghil, Andreas Groth, and Stéphane Hallegatte. 2011. “Dynamic coupling of the climate and macroeconomic systems.” Math. & Sci. hum. / Mathematics and Social Sciences. 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.

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Ghil, Michael, P. Yiou, S. Hallegatte, B. D. Malamud, P. Naveau, A. Soloviev, P. Friederichs, et al. 2011. “Extreme events: dynamics, statistics and prediction.” Nonlinear Processes in Geophysics 18 (3): 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. 2011. “Multivariate singular spectrum analysis and the road to phase synchronization.” Physical Review E 84: 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.

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Chekroun, Mickaël D., Dmitri Kondrashov, and Michael Ghil. 2011. “Predicting stochastic systems by noise sampling, and application to the El Niño-Southern Oscillation.” Proceedings of the National Academy of Sciences 108 (29): 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.

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2010
Zaliapin, Ilya, and Michael Ghil. 2010. “A delay differential model of ENSO variability, Part 2: Phase locking, multiple solutions, and dynamics of extrema.” Nonlinear Processes in Geophysics 17 (2): 123–135.
Chekroun, Mickaël D., Michael Ghil, Jean Roux, and Ferenc Varadi. 2010. “Averaging of time-periodic systems without a small parameter.” Discrete and Continuous Dynamical Systems 14 (4). American Institute of Mathematical Sciences (AIMS): 753–782.
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Roques, Lionel, and Mickaël D. Chekroun. 2010. “Does reaction-diffusion support the duality of fragmentation effect?” Ecological Complexity 7 (1). Elsevier: 100–106.
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Ghil, Michael, Peter Read, and Leonard Smith. 2010. “Geophysical flows as dynamical systems: the influence of Hide's experiments.” Astronomy & Geophysics 51 (4). Oxford University Press: 4–28.
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Zaliapin, Ilya, Efi Foufoula-Georgiou, and Michael Ghil. 2010. “Transport on river networks: A dynamic tree approach.” Journal of Geophysical Research: Earth Surface 115 (F2). Wiley Online Library.
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2009
Kravtsov, Sergey, Dmitri Kondrashov, and Michael Ghil. 2009. “Empirical model reduction and the modelling hierarchy in climate dynamics and the geosciences.” Stochastic physics and climate modelling. Cambridge University Press, Cambridge, 35–72. Abstract
Modern climate dynamics uses a two-fisted approach in attacking and solving the problems of atmospheric and oceanic flows. The two fists are: (i) observational analyses; and (ii) simulations of the geofluids, including the coupled atmosphere–ocean system, using a hierarchy of dynamical models. These models represent interactions between many processes that act on a broad range of spatial and time scales, from a few to tens of thousands of kilometers, and from diurnal to multidecadal, respectively. The evolution of virtual climates simulated by the most detailed and realistic models in the hierarchy is typically as difficult to interpret as that of the actual climate system, based on the available observations thereof. Highly simplified models of weather and climate, though, help gain a deeper understanding of a few isolated processes, as well as giving clues on how the interaction between these processes and the rest of the climate system may participate in shaping climate variability. Finally, models of intermediate complexity, which resolve well a subset of the climate system and parameterise the remainder of the processes or scales of motion, serve as a conduit between the models at the two ends of the hierarchy. We present here a methodology for constructing intermediate mod- els based almost entirely on the observed evolution of selected climate fields, without reference to dynamical equations that may govern this evolution; these models parameterise unresolved processes as multi- variate stochastic forcing. This methodology may be applied with equal success to actual observational data sets, as well as to data sets resulting from a high-end model simulation. We illustrate this methodology by its applications to: (i) observed and simulated low-frequency variability of atmospheric flows in the Northern Hemisphere; (ii) observed evo- lution of tropical sea-surface temperatures; and (iii) observed air–sea interaction in the Southern Ocean. Similar results have been obtained for (iv) radial-diffusion model simulations of Earth’s radiation belts, but are not included here because of space restrictions. In each case, the reduced stochastic model represents surprisingly well a variety of linear and nonlinear statistical properties of the resolved fields. Our methodology thus provides an efficient means of constructing reduced, numerically inexpensive climate models. These models can be thought of as stochastic–dynamic prototypes of more complex deterministic models, as in examples (i) and (iv), but work just as well in the situation when the actual governing equations are poorly known, as in (ii) and (iii). These models can serve as competitive prediction tools, as in (ii), or be included as stochastic parameterisations of certain processes within more complex climate models, as in (iii). Finally, the methodology can be applied, with some modifications, to geophysical problems outside climate dynamics, as illustrated by (iv).
Simonnet, Eric, Henk A. Dijkstra, and Michael Ghil. 2009. “Bifurcation analysis of ocean, atmosphere, and climate models.” Handbook of numerical analysis, edited by R. Temam and J. Tribbia, 14: 187–229. Elsevier.
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Bordi, Isabella, Klaus Fraedrich, Michael Ghil, and Alfonso Sutera. 2009. “Zonal flow regime changes in a GCM and in a simple quasigeostrophic model: The role of stratospheric dynamics.” Journal of the Atmospheric Sciences 66 (5): 1366–1383.
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2008
Ghil, Michael, Ilya Zaliapin, and Barbara Coluzzi. 2008. “Boolean delay equations: A simple way of looking at complex systems.” Physica D: Nonlinear Phenomena 237 (23). Elsevier: 2967–2986.
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