Research Interests

From Chekroun et al. (2011), Physica D, 240 (21): 1685-1700:

Vimeo movie: https://vimeo.com/240039610

Recent Publications

Kondrashov, Dmitri, and Mickaël D. Chekroun. In Press, 2018. “Data-adaptive harmonic analysis and modeling of solar wind-magnetosphere coupling.” Journal of Atmospheric and Solar-Terrestrial Physics. Publisher's Version Abstract

The solar wind-magnetosphere coupling is studied by new data-adaptive harmonic (DAH) decomposition approach for the spectral analysis and inverse modeling of multivariate time observations of complex nonlinear dynamical systems. DAH identifies frequency-based modes of interactions in the combined dataset of Auroral Electrojet (AE) index and solar wind forcing. The time evolution of these modes can be very effi- ciently simulated by using systems of stochastic differential equations (SDEs) that are stacked per frequency and formed by coupled Stuart-Landau oscillators. These systems of SDEs capture the modes’ frequencies as well as their amplitude modulations, and yield, in turn, an accurate modeling of the AE index’ statistical properties.

Chekroun, Mickaël D., Axel Kröner, and Honghu Liu. In Press, 2018. “Galerkin approximations for the optimal control of nonlinear delay differential equations.” Radon Series on Computational and Applied Mathematics. arXiv's version Abstract

Optimal control problems of nonlinear delay  equations (DDEs) are considered for which we propose a general Galerkin approximation scheme built from Koornwinder polynomials. Error estimates for the resulting Galerkin-Koornwinder approximations to the optimal control and the value function, are derived for a broad class of cost functionals and nonlinear DDEs. The approach is illustrated on a delayed logistic equation set not far away from its Hopf bifurcation point in the parameter space. In this case, we show that low-dimensional controls for a standard quadratic cost functional can be efficiently computed from Galerkin-Koornwinder approximations to reduce at a nearly optimal cost the oscillation amplitude displayed by the DDE's solution. Optimal controls computed from the Pontryagin's maximum principle (PMP) and the Hamilton-Jacobi-Bellman equation (HJB) associated with the corresponding ODE systems, are shown to provide numerical solutions in good agreement. It is finally argued that the value function computed from the  corresponding reduced HJB equation provides a good approximation of that obtained from the full HJB equation.

Value function solving the reduced HJB equation

Tantet, Alexis, Mickaël D. Chekroun, Henk A. Dijkstra, and J. David Neelin. Submitted. “Mixing Spectrum in Reduced Phase Spaces of Stochastic Differential Equations. Part II: Stochastic Hopf Bifurcation”. ArXiv's Version Abstract

The spectrum of the Markov semigroup of a diffusion process, referred to as the mixing spectrum, provides a detailed characterization of the dynamics of statistics such as the correlation function and the power spectrum. Stochastic analysis techniques for the study of the mixing spectrum and a rigorous reduction method have been presented in the first part (Chekroun et al. 2017) of this contribution. This framework is now applied to the study of a stochastic Hopf bifurcation, to characterize the statistical properties of nonlinear oscillators perturbed by noise, depending on their stability. In light of the H\"ormander theorem, it is first shown that the geometry of the unperturbed limit cycle, in particular its isochrons, i.e., the leaves of the stable manifold of the limit cycle generalizing the notion of phase, is essential to understand the effect of the noise and the phenomenon of phase diffusion. In addition, it is shown that the mixing spectrum has a spectral gap, even at the bifurcation point, and that correlations decay exponentially fast. Small-noise expansions of the eigenvalues and eigenfunctions are then obtained, away from the bifurcation point, based on the knowledge of the linearized deterministic dynamics and the characteristics of the noise. These formulas allow to understand how the interaction of the noise with the deterministic dynamics affect the decay of correlations. Numerical results complement the study of the mixing spectrum at the bifurcation point, revealing interesting scaling laws. The analysis of the Markov semigroup for stochastic bifurcations is thus promising in providing a complementary approach to the more geometric random dynamical systems. This approach is not limited to low-dimensional systems and the reduction method presented in part I will be applied to stochastic models relevant to climate dynamics in Part III.

Chekroun, Mickaël D. 2018. “Topological instabilities in families of semilinear parabolic problems subject to nonlinear perturbations.” Discrete and Continuous Dynamical Systems B, doi: 10.3934/dcdsb.2018075. Publisher's version Abstract

In this article it is proved that the dynamical properties of a broad class of semilinear parabolic problems are sensitive to arbitrarily small but smooth perturbations of the nonlinear term, when the spatial dimension is either equal to one or two. This topological instability is shown to result from a local deformation of the global bifurcation diagram associated with the corresponding elliptic problems. Such a deformation is shown to systematically occur via the creation of either a multiple-point or a new fold-point on this diagram when an appropriate small perturbation is applied to the nonlinear term. More precisely, it is shown that for a broad class of nonlinear elliptic problems, one can always find an arbitrary small perturbation of the nonlinear term, that generates a local S on the bifurcation diagram whereas the latter is e.g. monotone when no perturbation is applied; substituting thus a single solution by several ones. Such an increase in the local multiplicity of the solutions to the elliptic problem results then into a topological instability for the corresponding parabolic problem.
The rigorous proof of the latter instability result requires though to revisit the classical concept of topological equivalence to encompass important cases for the applications such as semi-linear parabolic problems for which the semigroup may exhibit non-global dissipative properties, allowing for the coexistence of blow-up regions and local attractors in the phase space; cases that arise e.g. in combustion theory. A revised framework of topological robustness is thus introduced in that respect within which the main topological instability result is then proved for continuous, locally Lipschitz but not necessarily C1 nonlinear terms, that prevent in particular the use of linearization techniques, and for which the family of semigroups may exhibit non-dissipative properties.

Chekroun, Mickaël D., Michael Ghil, and J. David Neelin. 2018. “Pullback attractor crisis in a delay differential ENSO model.” Advances in Nonlinear Geosciences, A. Tsonis, 1-33. Springer, 1-33. Publisher's version Abstract

We study the pullback attractor (PBA) of a seasonally forced delay differential model for the El Ni\~no--Southern Oscillation (ENSO); the model has two delays, associated with a positive and a negative feedback. The control parameter is the intensity of the positive feedback and the PBA undergoes a crisis that consists of a chaos-to-chaos transition. Since  the PBA is dominated by chaotic behavior, we refer to it as a strange PBA. Both chaotic regimes correspond to an overlapping of resonances but the two differ by the properties of this overlapping. The crisis manifests itself by a brutal change not only in the size but also in the shape of the PBA. The change is associated with the sudden disappearance of the most extreme warm (El Ni\~no) and cold (La Ni\~na) events, as one crosses the critical parameter value from below.  The analysis reveals that regions of the strange PBA that survive the crisis are those populated by the most probable states of the system. These regions are those that exhibit robust foldings with respect to perturbations.  The effect of noise on this phase-and-paramater space behavior is then discussed. It is shown that the chaos-to-chaos crisis may or may not survive the addition of small noise to the evolution equation, depending on how the noise enters the latter.

Kondrashov, Dmitri, Mickaël D. Chekroun, Xiaojun Yuan, and Michael Ghil. 2018. “Data-adaptive harmonic decomposition and stochastic modeling of Arctic sea ice.” Advances in Nonlinear Geosciences, A. Tsonis, 179-205. Springer, 179-205. Publisher's Version Abstract

We present and apply a novel method of describing and modeling complex multivariate datasets in the geosciences and elsewhere. Data-adaptive harmonic (DAH) decomposition identifies narrow-banded, spatio-temporal modes (DAHMs) whose frequencies are not necessarily integer multiples of each other. The evolution in time of the DAH coefficients (DAHCs) of these modes can be modeled using a set of coupled Stuart-Landau stochastic differential equations that capture the modes’ frequencies and amplitude modulation in time and space. This methodology is applied first to a challenging synthetic dataset and then to Arctic sea ice concentration (SIC) data from the US National Snow and Ice Data Center (NSIDC). The 36-year (1979–2014) dataset is parsimoniously and accurately described by our DAHMs. Preliminary results indicate that simulations using our multilayer Stuart-Landau model (MSLM) of SICs are stable for much longer time intervals, beyond the end of the twenty-first century, and exhibit interdecadal variability consistent with past historical records. Preliminary results indicate that this MSLM is quite skillful in predicting September sea ice extent.

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