Abstract Lyme borreliosis, tick-borne encephalitis, and human granulocytic anaplasmosis are remarkable examples of tick-borne diseases. Motivated by their notable epidemiological role, this paper explores the interplay among seasonality, diapause, and diffusion on the population dynamics of ticks. To reach this goal, we analyze a metapopulation model in a fragmented ecosystem of m-patches, with ticks as a motivating example. Specifically, we offer a new methodology to derive criteria of global attraction in nonautonomous metapopulation models with delay (without assuming monotonicity requirements). A strength of our approach is that the results apply to metapopulations with any number of patches and topology. From a practical point of view, our theoretical analysis allows us to corroborate some previous experimental work. Another important conclusion of this paper is that the presence of sinks can benefit the tick population, simplifying the dynamical behavior of the whole metapopulation and increasing the total population size. To assess the real repercussions of the results, we analyze our model with parameters derived from real observations.

Abstract In this paper, we prove the global wellposedness of the Gross–Pitaevskii equation with white noise potential in dimension 2, i.e. a cubic nonlinear Schrödinger equation with harmonic confining potential and spatial white noise multiplicative term. This problem is ill-defined and a Wick renormalisation is needed in order to give a meaning to solutions. In order to do this, we introduce a change of variables which transforms the original equation into one with less irregular terms. We construct a solution as a limit of solutions of the same equation but with a regularised noise. This convergence is shown by interpolating between a diverging bound in a high regularity Hermite–Sobolev space and a Cauchy estimate in L 2 ( R 2 ) .

Abstract We develop a tool in order to analyse the dynamics of differentiable flows with singularities. It provides an abstract model for the local dynamics that can be used in order to control the size of invariant manifolds. This work is the first part of the results announced in Crovisier and Yang (2017 arXiv:1702.05994).

Abstract Solutions in self-similar form presenting finite time extinction to the singular diffusion equation with gradient absorption ∂ t u − div ( | ∇ u | p − 2 ∇ u ) + | ∇ u | q = 0 in ( 0 , ∞ ) × R N are studied when N ⩾ 1 and the exponents (p, q) satisfy p c = 2 N N + 1 < p < 2 , p − 1 < q < p 2 . Existence and uniqueness of such a solution are established in dimension N = 1. In dimension N ⩾ 2 , existence of radially symmetric self-similar solutions is proved and a fine description of their behaviour as | x | → ∞ is provided.

Abstract In this paper, we investigate the nonlinear stability of Couette flow for the two-dimensional tropical climate system with zero thermal dissipation in the domain T × R . To overcome the difficulty caused by the absence of the thermal dissipation, we fully exploit the structure of the system and the enhanced dissipation created by the non-self-adjoint operator y ∂ x − Δ to obtain the expected stability result.

Abstract In this note we complete the analysis carried on in Caby et al (2023 Nonlinearity 36 3603–21) about the topological synchronisation of unimodal maps of the interval coupled in a master–slave configuration, by answering to the questions raised in that Paper. Namely, we compute the weak limits of the invariant measure of the coupled system as the coupling strength k ∈ ( 0 , 1 ) tends to 0 and to 1 and discuss the uniqueness of the invariant measure of its random dynamical system counterpart, proving that the convergence of the associated Markov chain to its unique stationary measure is geometric.

Abstract The concept of sliding Shilnikov connection has been recently introduced and represents an important notion in Filippov systems, because its existence implies chaotic behavior on an invariant subset of the system. The investigation of its properties has just begun, and understanding the topology and complexity of its invariant set is of interest. In this paper, we conduct a local analysis on the first return map associated to a sliding Shilnikov connection, which reveals a conformal iterated function system (CIFS) structure. By using the theory of CIFS, we estimate the Hausdorff dimension of the local invariant set of the first return map, showing, in particular, that is a positive number smaller than 1, and with one-dimensional Lebesgue measure equal to zero. Moreover, we prove that the closure of the local invariant set is a Cantor set and retains both the Hausdorff dimension and Lebesgue measure of the local invariant set. Furthermore, this closure consists of the local invariant set along with the set of all pre-images, under the first return map, of the visible fold-regular point contained in the connection.

Abstract Bayesian state estimation of a dynamical system utilising a stream of noisy measurements is important in many geophysical and engineering applications. In these cases, nonlinearities, high (or infinite) dimensionality of the state space, and sparse observations pose key challenges for deriving efficient and accurate data assimilation (DA) techniques. A number of DA algorithms used commonly in practice, such as the Ensemble Kalman Filter (EnKF) or Ensemble Square Root Kalman Filter (EnSRKF), suffer from serious drawbacks such as catastrophic filter divergence and filter instability. The analysis of stability and accuracy of these DA schemes has thus far focused either on finite-dimensional dynamics, or on complete observations in case of infinite-dimensional dissipative systems. We develop a unified framework for the analysis of several well-known and empirically efficient DA techniques derived from various Gaussian approximations of the Bayesian filtering schemes for geophysical-type dissipative dynamics with quadratic nonlinearities. We establish rigorous results on (time-asymptotic) accuracy and stability of these algorithms with general covariance and observation operators. The accuracy and stability results for EnKF and EnSRKF for dissipative PDEs are, to the best of our knowledge, completely new in this general setting. It turns out that a hitherto unexploited cancellation property involving the ensemble covariance and observation operators and the concept of covariance localization in conjunction with covariance inflation play a pivotal role in the accuracy and stability for EnKF and EnSRKF. Our approach also elucidates the links, via determining functionals, between the approximate-Bayesian and control-theoretic approaches to DA. We consider the ‘model’ dynamics governed by the two-dimensional incompressible Navier–Stokes equations (NSEs) and observations given by noisy measurements of averaged volume elements or spectral/modal observations of the velocity field. In this setup, several continuous-time DA techniques, namely the so-called 3DVar, EnKF and EnSRKF reduce to a stochastically forced NSEs. For the first time, we derive conditions for accuracy and stability of EnKF and EnSRKF. The derived bounds are given for the limit supremum of the expected value of the L 2 norm and of the H 1 Sobolev norm of the difference between the approximating solution and the actual solution as the time tends to infinity. Moreover, our analysis reveals an interplay between the resolution of the observations (roughly, the ‘richness’ of the observation space) associated with the observation operator underlying the DA algorithms and covariance inflation and localization which are employed in practice for improved filter performance.

Abstract In this paper, we consider the convergence rate with respect to the Wasserstein distance in the invariance principle for sequential dynamical systems. We utilize and modify the techniques previously employed for stationary sequences to address our non-stationary case. Under certain assumptions, we can apply our result to a class of dynamical systems, including sequential β n -transformations, piecewise uniformly expanding maps with additive noise in one-dimensional and multidimensional case, and so on.

Abstract We investigate the Cahn–Hilliard and the conserved Allen–Cahn equations with logarithmic type potential and conservative noise in a periodic domain. These features ensure that the order parameter takes its values in the physical range and, albeit the stochastic nature of the problems, that the total mass is conserved almost surely in time. For the Cahn–Hilliard equation, existence and uniqueness of probabilistically-strong solutions is shown up to the three-dimensional case. For the conserved Allen–Cahn equation, under a restriction on the noise magnitude, existence of martingale solutions is proved even in dimension three, while existence and uniqueness of probabilistically-strong solutions holds in dimension one and two. The analysis is carried out by studying the Cahn–Hilliard/conserved Allen–Cahn equations jointly, that is a linear combination of both the equations, which has an independent interest.