Dynamical systems

Chekroun MD, Simonnet E, Ghil M. Stochastic climate dynamics: Random attractors and time-dependent invariant measures. Physica D. 2011;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.
Groth A, Ghil M. Multivariate singular spectrum analysis and the road to phase synchronization. Physical Review E. 2011;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.

Dumas P, Ghil M, Groth A, Hallegatte S. 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.

Pierini S, Ghil M, Chekroun MD. Exploring the pullback attractors of a low-order quasigeostrophic ocean model: The deterministic case. Journal of Climate. 2016;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.

Hallegatte S, Ghil M, Dumas P, Hourcade J-C. Business cycles, bifurcations and chaos in a neo-classical model with investment dynamics. Journal of Economic Behavior & Organization. 2008;67 (1) :57–77.Abstract

This paper presents a non-equilibrium dynamic model (NEDyM) that introduces investment dynamics and non-equilibrium effects into a Solow growth model. NEDyM can reproduce several typical economic regimes and, for certain ranges of parameter values, exhibits endogenous business cycles with realistic characteristics. The cycles arise from the investment-profit instability and are constrained by the increase in labor costs and the inertia of production capacity. For other parameter ranges, the model exhibits chaotic behavior. These results show that complex variability in the economic system may be due to deterministic, intrinsic factors, even if the long-term equilibrium is neo-classical in nature.