# Publications

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.

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.

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 Sinaï–Ruelle–Bowen (SRB) measures.

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

There is a gap between single-species model predictions, and empirical studies, regarding the effect of habitat fragmentation per se, i.e., a process involving the breaking apart of habitat without loss of habitat. Empirical works indicate that fragmentation can have positive as well as negative effects, whereas, traditionally, single-species models predict a negative effect of fragmentation. Within the class of reaction-diffusion models, studies almost unanimously predict such a detrimental effect. In this paper, considering a single-species reaction-diffusion model with a removal – or similarly harvesting – term, in two dimensions, we find both positive and negative effects of fragmentation of the reserves, i.e., the protected regions where no removal occurs. Fragmented reserves lead to higher population sizes for time-constant removal terms. On the other hand, when the removal term is proportional to the population density, higher population sizes are obtained on aggregated reserves, but maximum yields are attained on fragmented configurations, and for intermediate harvesting intensities.

In this article, we present a new approach to averaging in non-Hamiltonian systems with periodic forcing. The results here do not depend on the existence of a small parameter. In fact, we show that our averaging method fits into an appropriate nonlinear equivalence problem, and that this problem can be solved formally by using the Lie transform framework to linearize it. According to this approach, we derive formal coordinate transformations associated with both first-order and higher-order averaging, which result in more manageable formulae than the classical ones.

Using these transformations, it is possible to correct the solution of an averaged system by recovering the oscillatory components of the original non-averaged system. In this framework, the inverse transformations are also defined explicitly by formal series; they allow the estimation of appropriate initial data for each higher-order averaged system, respecting the equivalence relation.

Finally, we show how these methods can be used for identifying and computing periodic solutions for a very large class of nonlinear systems with time-periodic forcing. We test the validity of our approach by analyzing both the first-order and the second-order averaged system for a problem in atmospheric chemistry.