A Grobman-Hartman theorem for general nonuniform exponential dichotomies

For a nonautonomous dynamics with discrete time given by a sequence of linear operators Am, we establish a version of the Grobman-Hartman theorem in Banach spaces for a very general nonuniformly hyperbolic dynamics. More precisely, we consider a sequence of linear operators whose products exhibit stable and unstable behaviors with respect to arbitrary growth rates ecρ(n), determined by a sequence ρ(n). For all sufficiently small Lipschitz perturbations Am+fm we construct topological conjugacies between the dynamics defined by this sequence and the dynamics defined by the operators Am. We also show that all conjugacies are Hölder continuous. We note that the usual exponential behavior is included as a very special case when ρ(n)=n, but many other asymptotic behaviors are included such as the polynomial asymptotic behavior when ρ(n)=logn.     

Control of chaotic behaviour and prevention of extinction using constant proportional feedback

We study a strategy to control the dynamics of one dimensional discrete maps known as the proportional feedback control method. We completely characterize the maps for which it is possible to stabilize the unstable or even chaotic dynamics towards an asymptotically stable equilibrium employing this method.Additionally, under conditions commonly assumed in modelling population dynamics, we show that the strategy drives the system to the optimal situation from a practical point of view, that is, to a global stable equilibrium since in that case the basin of attraction covers all the possible initial conditions. We also show that in some situations the strategy can be used to prevent the extinction of the population when controlling some models with the Allee effect.     

On global existence of solutions to a cross-diffusion system

We consider a strongly coupled nonlinear parabolic system which arises in population dynamics in n-dimensional domains (n?1). We prove the global existence of classical solutions to the system for n<10.     

Rate of Convergence Towards Semi-Relativistic Hartree Dynamics

We consider the semi-relativistic system of N gravitating Bosons with gravitation constant G. The time evolution of the system is described by the relativistic dispersion law, and we assume the mean-field scaling of the interaction where N → ∞ and G → 0 while GN = λ fixed. In the super-critical regime of large λ, we introduce the regularized interaction where the cutoff vanishes as N → ∞. We show that the difference between the many-body semi-relativistic Schrödinger dynamics and the corresponding semi-relativistic Hartree dynamics is at most of order N ^?1 for all λ, i.e., the result covers the sub-critical regime and the super-critical regime. The N dependence of the bound is optimal.     

Mathematical analysis of a virus dynamics model with general incidence rate and cure rate

The rate of infection in many virus dynamics models is assumed to be bilinear in the virus and uninfected target cells. In this paper, the dynamical behavior of a virus dynamics model with general incidence rate and cure rate is studied. Global dynamics of the model is established. We prove that the virus is cleared and the disease dies out if the basic reproduction number R₀≤1 while the virus persists in the host and the infection becomes endemic if R₀>1.     

Global analysis of an environmental disease transmission model linking within-host and between-host dynamics

In this paper, a multi-scale mathematical model for environmentally transmitted diseases is proposed which couples the pathogen-immune interaction inside the human body with the disease transmission at the population level. The model is based on the nested approach that incorporates the infection-age-structured immunological dynamics into an epidemiological system structured by the chronological time, the infection age and the vaccination age. We conduct detailed analysis for both the within-host and between-host disease dynamics. Particularly, we derive the basic reproduction number R₀ for the between-host model and prove the uniform persistence of the system. Furthermore, using carefully constructed Lyapunov functions, we establish threshold-type results regarding the global dynamics of the between-host system: the disease-free equilibrium is globally asymptotically stable when R₀ < 1, and the endemic equilibrium is globally asymptotically stable when R₀ > 1. We explore the connection between the within-host and between-host dynamics through both mathematical analysis and numerical simulation. We show that the pathogen load and immune strength at the individual level contribute to the disease transmission and spread at the population level. We also find that, although the between-host transmission risk correlates positively with the within-host pathogen load, there is no simple monotonic relationship between the disease prevalence and the individual pathogen load.     

Pullback attractors and extremal complete trajectories for non-autonomous reaction-diffusion problems

We analyse the dynamics of the non-autonomous nonlinear reaction-diffusion equation

ut−Δu=f(t,x,u),     

Center manifolds under nonuniform hyperbolicity - maximal regularity

We establish the existence of (invariant) center manifolds with maximal Cr regularity for a nonautonomous dynamics with discrete time. We consider the general case of perturbations of a nonuniform exponential trichotomy. Our proof uses the fiber contraction principle and allows linear perturbations without any further effort.     

Existence for the α-patch model and the QG sharp front in Sobolev spaces

We consider a family of contour dynamics equations depending on a parameter α with 0<α?1. The vortex patch problem of the 2-D Euler equation is obtained taking α→0, and the case α=1 corresponds to a sharp front of the QG equation. We prove local-in-time existence for the family of equations in Sobolev spaces.     

Glauber dynamics for the mean-field Ising model: cut-off, critical power law, and metastability

We study the Glauber dynamics for the Ising model on the complete graph, also known as the Curie–Weiss Model. For β < 1, we prove that the dynamics exhibits a cut-off: the distance to stationarity drops from near 1 to near 0 in a window of order n centered at [2(1 ? β)]^?1 n log n. For β = 1, we prove that the mixing time is of order n ^3/2. For β > 1, we study metastability. In particular, we show that the Glauber dynamics restricted to states of non-negative magnetization has mixing time O(n log n).     

Local planarity in one-dimensional continua

In this paper we address a problem posed by W. Lewis at the Second International Conference on Continuum Theory held at BUAP, Puebla, Mexico. Lewis asked for a characterization of local-planarity in inverse limit spaces of finite graphs in terms of the dynamics of the bonding maps. We give some sufficiency conditions and show that points at which our sufficiency conditions do not guarantee the space is locally planar, the problem requires a solution to the harder problem of characterizing planarity in inverse limits of graphs. We also examine the case of an inverse limit generated by a single map, f, on a single graph, G. Assuming that f has finitely many turning points and is non-contracting, we characterize local planarity in terms of the dynamics of f.     

Effective Dynamics for N-Solitons of the Gross–Pitaevskii Equation

We consider several solitons moving in a slowly varying external field. We present results of numerical computations which indicate that the effective dynamics obtained by restricting the full Hamiltonian to the finite-dimensional manifold of N-solitons (constructed when no external field is present) provides a remarkably good approximation to the actual soliton dynamics. This is quantified as an error of size h ² where h is the parameter describing the slowly varying nature of the potential. This also indicates that previous mathematical results of Holmer and Zworski (Int. Math. Res. Not. 2008: Art. ID runn026, 2008) for one soliton are optimal. For potentials with unstable equilibria, the Ehrenfest time, log(1/h)/h, appears to be the natural limiting time for these effective dynamics. We also show that the results of Holmer et?al. (arXiv:0912.5122, 2009) for two mKdV solitons apply numerically to a larger number of interacting solitons. We illustrate the results by applying the method with the external potentials used in the Bose?CEinstein soliton train experiments of Strecker et?al. (Nature 417:150?C153, 2002).     

On some stochastic coalescents

We consider infinite systems of macroscopic particles characterized by their masses. Each pair of particles with masses x and y coalesce at a given rate K(x, y). We assume that K satisfies a sort of Hölder property with index λ ∈ (0,1], and that the initial condition admits a moment of order λ. We show the existence of such infinite particle systems, as strong Markov processes enjoying a Feller property. We also show that the obtained processes are the only possible limits when making the number of particles tend to infinity in a sequence of finite particle systems with the same dynamics.     

Iterations of linear maps over finite fields

We study the dynamics of the evolution of Ducci sequences and the Martin-Odlyzko-Wolfram cellular automaton by iterating their respective linear maps on . After a review of an algebraic characterization of cycle lengths, we deduce the relationship between the maximal cycle lengths of these two maps from a simple connection between them. For n odd, we establish a conjugacy relationship that provides a more direct identification of their dynamics. We give an alternate, geometric proof of the maximal cycle length relationship, based on this conjugacy and a symmetry property. We show that the cyclic dynamics of both maps in dimension 2n can be deduced from their periodic behavior in dimension n. This link is generalized to a larger class of maps. With restrictions shared by both maps, we obtain a formula for the number of vectors in dimension 2n belonging to a cycle of length q that expresses this number in terms of the analogous values in dimension n.     

Non-recurrence of exp(z)/z

In this paper, we consider the dynamics of the map z→exp(z)/z on the punctured plane C*=C\{0}. We show that for almost every point z ∈ C*, the ω-limit set of z is equal to {0, ∞}. In particular, the map is not recurrent.     

Nonlinear dynamics and intermittency in a long-term copepod time series

We consider the nonlinear dynamics of a long-term copepod (small crustaceans) time series sampled weekly in the Mediterranean sea from 1967 to 1992. Such population dynamics display a high variability that we consider here in an interdisciplinary study, using tools borrowed from the field of statistical physics. We analyse the extreme events of male and female abundances, and of the total population, and show that they both have heavy tailed probability density functions (pdf). We provide hyperbolic fits of the form p(x) ∼ 1/x^μ+1, and estimate the value of μ using Hill’s estimator. We then study the ratio of male to female abundances, compared to the female abundances. Using conditional probability density functions and conditional averages, we show that this ratio is independent of the female density, when the latter is larger than a given threshold. This property is very useful for modelization. We also consider the product of male to female abundances, which can be ecologically related to the encounters. We show that this product is extremely intermittent, and link its pdf to the female pdf.     

Mean field limit of a dynamical model of polymer systems

This paper provides a mathematically rigorous foundation for self-consistent mean feld theory of the polymeric physics.We study a new model for dynamics of mono-polymer systems.Every polymer is regarded as a string of points which are moving randomly as Brownian motions and under elastic forces.Every two points on the same string or on two diferent strings also interact under a pairwise potential V.The dynamics of the system is described by a system of N coupled stochastic partial diferential equations(SPDEs).We show that the mean feld limit as N→∞of the system is a self-consistent McKean-Vlasov type equation,under suitable assumptions on the initial and boundary conditions and regularity of V.We also prove that both the SPDE system of the polymers and the mean feld limit equation are well-posed.     

Homogenization of random parabolic operator with large potential

We study the averaging problem for a divergence form random parabolic operators with a large potential and with coefficients rapidly oscillating both in space and time variables. We assume that the medium possesses the periodic microscopic structure while the dynamics of the system is random and, moreover, diffusive. A parameter α will represent the ratio between space and time microscopic length scales. A parameter β will represent the effect of the potential term. The relation between α and β is of great importance. In a trivial case the presence of the potential term will be “neglectable”. If not, the problem will have a meaning if a balance between these two parameters is achieved, then the averaging results hold while the structure of the limit problem depends crucially on α (with three limit cases: one classical and two given under martingale problems form). These results show that the presence of stochastic dynamics might change essentially the limit behavior of solutions.