Dynamical systems

Hallegatte S, Ghil M. Natural disasters impacting a macroeconomic model with endogenous dynamics. Ecological Economics. 2008;68 (1-2) :582–592.Abstract

We investigate the macroeconomic response to natural disasters by using an endogenous business cycle (EnBC) model in which cyclical behavior arises from the investment-profit instability. Our model exhibits a larger response to natural disasters during expansions than during recessions. This apparently paradoxical result can be traced to the disasters amplifying pre-existing disequilibria during expansions, while the existence of unused resources during recessions damps the exogenous shocks. It thus appears that high-growth periods are also highly vulnerable to supply-side shocks. In our EnBC model, the average production loss due to a set of disasters distributed at random in time is highly sensitive to the dynamical characteristics of the impacted economy. Larger economic flexibility allows for a more efficient and rapid response to supply-side shocks and reduces production losses. On the other hand, too high a flexibility can lead to vulnerability phases that cause average production losses to soar. These results raise questions about the assessment of climate change damages or natural disaster losses that are based purely on long-term growth models.

Kravtsov S, Berloff P, Dewar WK, Ghil M, McWilliams JC. Dynamical origin of low-frequency variability in a highly nonlinear midlatitude coupled model. Journal of Climate. 2006;19 (24).Abstract

A novel mechanism of decadal midlatitude coupled variability, which crucially depends on the nonlinear dynamics of both the atmosphere and the ocean, is presented. The coupled model studied involves quasigeostrophic atmospheric and oceanic components, which communicate with each other via a constant-depth oceanic mixed layer. A series of coupled and uncoupled experiments show that the decadal coupled mode is active across parameter ranges that allow the bimodality of the atmospheric zonal flow to coexist with oceanic turbulence. The latter is most intense in the regions of inertial recirculation (IR). Bimodality is associated with the existence of two distinct anomalously persistent zonal-flow modes, which are characterized by different latitudes of the atmospheric jet stream. The IR reorganizations caused by transitions of the atmosphere from its high- to low-latitude state and vice versa create sea surface temperature anomalies that tend to induce transition to the opposite atmospheric state. The decadal–interdecadal time scale of the resulting oscillation is set by the IR adjustment; the latter depends most sensitively on the oceanic bottom drag. The period T of the nonlinear oscillation is 7–25 yr for the range of parameters explored, with the most realistic parameter values yielding T \approx 20 yr. Aside from this nonlinear oscillation, an interannual Rossby wave mode is present in all coupled experiments. This coupled mode depends neither on atmospheric bimodality, nor on ocean eddy dynamics; it is analogous to the mode found previously in a channel configuration. Its time scale in the model with a closed ocean basin is set by cross-basin wave propagation and equals 3–5 yr for a basin width comparable with the North Atlantic.

Chekroun MD, Liu H, Wang S. Approximation of Stochastic Invariant Manifolds: Stochastic Manifolds for Nonlinear SPDEs I. New York: Springer Briefs in Mathematics, Springer; 2015. Publisher's VersionAbstract

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

Ghil M, S. Coho JT, Bube K, Isaacson E. Dynamic Meteorology: Data Assimilation Methods. In: Bengtsson L, Ghil M, Källén E Applied Mathematical Sciences. Vol. 36. Dynamic Meteorology - Data Assimilation Methods. Springer-Verlag ; 1981. pp. 139–224.
Chekroun MD, Kondrashov D, Ghil M. Predicting stochastic systems by noise sampling, and application to the El Niño-Southern Oscillation. Proceedings of the National Academy of Sciences. 2011;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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