Nonlinear Equations with Memory

Chekroun, Mickaël D., Ilan Koren, and Honghu Liu. 2020. “Efficient reduction for diagnosing Hopf bifurcation in delay differential systems: Applications to cloud-rain models.” Chaos: An Interdisciplinary Journal of Nonlinear Science 30: 053130 . Publisher's Version Abstract

By means of Galerkin-Koornwinder (GK) approximations, an efficient reduction approach to the Stuart-Landau (SL) normal form and center manifold is presented for a broad class of nonlinear systems of delay differential equations (DDEs) that covers the cases of discrete as well as distributed delays. The focus is on the Hopf bifurcation as the consequence of the critical equilibrium's destabilization resulting from an eigenpair crossing the imaginary axis. The nature of the resulting Hopf bifurcation (super- or subcritical) is then characterized by the inspection of a Lyapunov coefficient easy to determine based on the model's coefficients and delay parameter.  We believe that our approach, which does not rely too much on functional analysis considerations but more on analytic calculations, is suitable  to concrete situations arising in physics applications.

Thus, using this GK approach to the Lyapunov coefficient and SL normal form, the occurrence of Hopf bifurcations in the cloud-rain delay models of Koren and Feingold (KF) on one hand, and Koren, Tziperman and Feingold (KTF), on the other, are analyzed. Noteworthy is the existence for the KF model of large regions of the parameter space for which subcritical and supercritical Hopf bifurcations coexist. These regions are determined in particular by the intensity of the KF model's nonlinear effects. ``Islands'' of supercritical Hopf bifurcations are shown to exist within a subcritical Hopf bifurcation ``sea;'' these islands being bordered by double-Hopf bifurcations occurring when the linearized dynamics at the critical equilibrium exhibit two pairs of purely imaginary eigenvalues. 

Chekroun, Mickaël D., Axel Kröner, and Honghu Liu. 2018. “Galerkin approximations for the optimal control of nonlinear delay differential equations.” Hamilton-Jacobi-Bellman Equations. Numerical Methods and Applications in Optimal Control, D. Kalise, K. Kunisch, and Z. Rao, 21: 61-96. Berlin, Boston: De Gruyter. Publisher'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

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. 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.



Boers, N., M. D. Chekroun, H. Liu, D. Kondrashov, D.-D. Rousseau, A. Svensson, M. Bigler, and M. Ghil. 2017. “Inverse stochastic-dynamic models for high-resolution Greenland ice-core records.” Earth System Dynamics 8: 1171–1190. Publisher's Version Abstract

Proxy records from Greenland ice cores have been studied for several decades, yet many open questions remain regarding the climate variability encoded therein. Here, we use a Bayesian framework for inferring inverse, stochastic-dynamic models from δ18O and dust records of unprecedented, subdecadal temporal resolution. The records stem from the North Greenland Ice Core Project (NGRIP) and we focus on the time interval 59 ka–22 ka b2k. Our model reproduces the dynamical characteristics of both the δ18O and dust proxy records, including the millennial-scale Dansgaard–Oeschger variability, as well as statistical properties such as probability density functions, waiting times and power spectra, with no need for any external forcing. The crucial ingredients for capturing these properties are (i) high-resolution training data; (ii) cubic drift terms; (iii) nonlinear coupling terms between the δ18O and dust time series; and (iv) non-Markovian contributions that represent short-term memory effects.

Chen, C., M. Cane, N. Henderson, D. Lee, D. Chapman, D. Kondrashov, and M. D. Chekroun. 2016. “Diversity, nonlinearity, seasonality and memory effect in ENSO simulation and prediction using empirical model reduction.” Journal of Climate 29 (5): 1809–1830. Publisher's Version Abstract

A suite of empirical model experiments under the empirical model reduction framework are conducted to advance the understanding of ENSO diversity, nonlinearity, seasonality, and the memory effect in the simulation and prediction of tropical Pacific sea surface temperature (SST) anomalies. The model training and evaluation are carried out using 4000-yr preindustrial control simulation data from the coupled model GFDL CM2.1. The results show that multivariate models with tropical Pacific subsurface information and multilevel models with SST history information both improve the prediction skill dramatically. These two types of models represent the ENSO memory effect based on either the recharge oscillator or the time-delayed oscillator viewpoint. Multilevel SST models are a bit more efficient, requiring fewer model coefficients. Nonlinearity is found necessary to reproduce the ENSO diversity feature for extreme events. The nonlinear models reconstruct the skewed probability density function of SST anomalies and improve the prediction of the skewed amplitude, though the role of nonlinearity may be slightly overestimated given the strong nonlinear ENSO in GFDL CM2.1. The models with periodic terms reproduce the SST seasonal phase locking but do not improve the prediction appreciably. The models with multiple ingredients capture several ENSO characteristics simultaneously and exhibit overall better prediction skill for more diverse target patterns. In particular, they alleviate the spring/autumn prediction barrier and reduce the tendency for predicted values to lag the target month value.

Chekroun, Mickaël D., Michael Ghil, Honghu Liu, and Shouhong Wang. 2016. “Low-dimensional Galerkin approximations of nonlinear delay differential equations.” Discrete and Continuous Dynamical Systems 36 (8): 4133-4177. Publisher's Version Abstract

This article revisits the approximation problem of systems of nonlinear delay differential equations (DDEs) by a set of ordinary differential equations (ODEs). We work in Hilbert spaces endowed with a natural inner product including a point mass, and introduce polynomials orthogonal with respect to such an inner product that live in the domain of the linear operator associated with the underlying DDE. These polynomials are then used to design a general Galerkin scheme for which we derive rigorous convergence results and show that it can be numerically implemented via simple analytic formulas. The scheme so obtained is applied to three nonlinear DDEs, two autonomous and one forced: (i) a simple DDE with distributed delays whose solutions recall Brownian motion; (ii) a DDE with a discrete delay that exhibits bimodal and chaotic dynamics; and (iii) a periodically forced DDE with two discrete delays arising in climate dynamics. In all three cases, the Galerkin scheme introduced in this article provides a good approximation by low-dimensional ODE systems of the DDE's strange attractor, as well as of the statistical features that characterize its nonlinear dynamics.

Kondrashov, Dmitri, Mickaël D. Chekroun, and Michael Ghil. 2015. “Data-driven non-Markovian closure models.” Physica D: Nonlinear Phenomena 297: 33 - 55. Publisher's Version Abstract
Abstract This paper has two interrelated foci: (i) obtaining stable and efficient data-driven closure models by using a multivariate time series of partial observations from a large-dimensional system; and (ii) comparing these closure models with the optimal closures predicted by the Mori–Zwanzig (MZ) formalism of statistical physics. Multilayer stochastic models (MSMs) are introduced as both a generalization and a time-continuous limit of existing multilevel, regression-based approaches to closure in a data-driven setting; these approaches include empirical model reduction (EMR), as well as more recent multi-layer modeling. It is shown that the multilayer structure of \MSMs\ can provide a natural Markov approximation to the generalized Langevin equation (GLE) of the \MZ\ formalism. A simple correlation-based stopping criterion for an EMR–MSM model is derived to assess how well it approximates the \GLE\ solution. Sufficient conditions are derived on the structure of the nonlinear cross-interactions between the constitutive layers of a given \MSM\ to guarantee the existence of a global random attractor. This existence ensures that no blow-up can occur for a broad class of \MSM\ applications, a class that includes non-polynomial predictors and nonlinearities that do not necessarily preserve quadratic energy invariants. The EMR–MSM methodology is first applied to a conceptual, nonlinear, stochastic climate model of coupled slow and fast variables, in which only slow variables are observed. It is shown that the resulting closure model with energy-conserving nonlinearities efficiently captures the main statistical features of the slow variables, even when there is no formal scale separation and the fast variables are quite energetic. Second, an \MSM\ is shown to successfully reproduce the statistics of a partially observed, generalized Lotka–Volterra model of population dynamics in its chaotic regime. The challenges here include the rarity of strange attractors in the model’s parameter space and the existence of multiple attractor basins with fractal boundaries. The positivity constraint on the solutions’ components replaces here the quadratic-energy–preserving constraint of fluid-flow problems and it successfully prevents blow-up.
Roques, Lionel, Mickaël D. Chekroun, Michel Cristofol, Samuel Soubeyrand, and Michael Ghil. 2014. “Parameter estimation for energy balance models with memory.” Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 470 (2169). The Royal Society. Publisher's Version Abstract
We study parameter estimation for one-dimensional energy balance models with memory (EBMMs) given localized and noisy temperature measurements. Our results apply to a wide range of nonlinear, parabolic partial differential equations with integral memory terms. First, we show that a space-dependent parameter can be determined uniquely everywhere in the PDE’s domain of definition D, using only temperature information in a small subdomain E⊂D. This result is valid only when the data correspond to exact measurements of the temperature. We propose a method for estimating a model parameter of the EBMM using more realistic, error-contaminated temperature data derived, for example, from ice cores or marine-sediment cores. Our approach is based on a so-called mechanistic-statistical model that combines a deterministic EBMM with a statistical model of the observation process. Estimating a parameter in this setting is especially challenging, because the observation process induces a strong loss of information. Aside from the noise contained in past temperature measurements, an additional error is induced by the age-dating method, whose accuracy tends to decrease with a sample’s remoteness in time. Using a Bayesian approach, we show that obtaining an accurate parameter estimate is still possible in certain cases.
Chekroun, Mickaël D., and Nathan E. Glatt-Holtz. 2012. “Invariant Measures for Dissipative Dynamical Systems: Abstract Results and Applications.” Communications in Mathematical Physics 316 (3): 723–761. Publisher's Version Abstract
In this work we study certain invariant measures that can be associated to the time averaged observation of a broad class of dissipative semigroups via the notion of a generalized Banach limit. Consider an arbitrary complete separable metric space X which is acted on by any continuous semigroup \S(t)\ t ≥ 0. Suppose that \S(t)\ t ≥ 0 possesses a global attractor \$\$\\backslashmathcal\A\\\$\$ . We show that, for any generalized Banach limit LIM T → ∞ and any probability distribution of initial conditions \$\$\\backslashmathfrak\m\\_0\\$\$ , that there exists an invariant probability measure \$\$\\backslashmathfrak\m\\\$\$ , whose support is contained in \$\$\\backslashmathcal\A\\\$\$ , such that \$\$\backslashint\_\X\ \backslashvarphi(x) \\backslashrm d\\backslashmathfrak\m\(x) = \backslashunderset\t \backslashrightarrow \backslashinfty\\\backslashrm LIM\\backslashfrac\1\\T\ \backslashint\_0^T \backslashint\_X \backslashvarphi(S(t) x) \\backslashrm d\\backslashmathfrak\m\\_0(x) \\backslashrm d\t,\$\$ for all observables $\phi$ living in a suitable function space of continuous mappings on X.
Chekroun, M. 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. Publisher's Version Abstract

This paper is concerned with the integrodifferential equation

arising in the Coleman-Gurtin's theory of heat conduction with hereditary memory, in presence of a nonlinearity of critical growth. Rephrasing the equation within the history space framework, we prove the existence of global and exponential attractors of optimal regularity and finite fractal dimension for the related solution semigroup, acting both on the basic weak-energy space and on a more regular phase space.