# Nonlinear Equations with Memory

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.

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.

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.

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 *δ*^{18}O 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 *δ*^{18}O 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 *δ*^{18}O and dust time series; and (iv) non-Markovian contributions that represent short-term memory effects.

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.

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.

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.