Global asymptotic behaviour for a Nicholson model with patch structure and multiple delays

For a Nicholson’s blowflies model with patch structure and multiple discrete delays, we study some aspects of its global dynamics. Conditions for the absolute global asymptotic stability of both the trivial equilibrium and a positive equilibrium (when it exists) are given. The existence of positive heteroclinic solutions connecting the two equilibria is also addressed. We further consider a diffusive Nicholson-type model with patch structure, and establish a criterion for the existence of positive travelling wave solutions, for large wave speeds. Several applications illustrate the results, improving some criteria in the recent literature.     

Clustering and asymptotic behavior in opinion formation

We investigate the long time behavior of models of opinion formation. We consider the case of compactly supported interactions between agents which are also non-symmetric, including for instance the so-called Krause model. Because of the finite range of interaction, convergence to a unique consensus is not expected in general. We are nevertheless able to prove the convergence to a final equilibrium state composed of possibly several local consensus. This result had so far only been conjectured through numerical evidence. Because of the non-symmetry in the model, the analysis is delicate and is performed in two steps: First using entropy estimates to prove the formation of stable clusters and then studying the evolution in each cluster. We study both discrete and continuous in time models and give rates of convergence when those are available.     

Markov structures and decay of correlations for non-uniformly expanding dynamical systems

We consider non-uniformly expanding maps on compact Riemannian manifolds of arbitrary dimension, possibly having discontinuities and/or critical sets, and show that under some general conditions they admit an induced Markov tower structure. Moreover, the decay of the return time function can be controlled in terms of the time generic points need to achieve some uniform expanding behavior. As a consequence we obtain some rates for the decay of correlations of those maps and conditions for the validity of the Central Limit Theorem.     

From rates of mixing to recurrence times via large deviations

A classic approach in dynamical systems is to use particular geometric structures to deduce statistical properties, for example the existence of invariant measures with stochastic-like behaviour such as large deviations or decay of correlations. Such geometric structures are generally highly non-trivial and thus a natural question is the extent to which this approach can be applied. In this paper we show that in many cases stochastic-like behaviour itself implies that the system has certain non-trivial geometric properties, which are therefore necessary as well as sufficient conditions for the occurrence of the statistical properties under consideration. As a by product of our techniques we also obtain some new results on large deviations for certain classes of systems which include Viana maps and multidimensional piecewise expanding maps.     

Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion

Whether or not classical solutions of the 2D incompressible MHD equations without full dissipation and magnetic diffusion can develop finite-time singularities is a difficult issue. A major result of this paper establishes the global regularity of classical solutions for the MHD equations with mixed partial dissipation and magnetic diffusion. In addition, the global existence, conditional regularity and uniqueness of a weak solution is obtained for the 2D MHD equations with only magnetic diffusion.     

The higher-order Picone identity and comparison of half-linear differential equations of even order

An identity of the Picone type for higher-order half-linear ordinary differential operators of the form and where p_j and P_j, j=0,…,n, are continuous functions defined on [a,b] and , is derived and then the Sturmian comparison theory for the corresponding 2nth-order equations l_α[x]=0 and L_α[y]=0 based on this identity is developed.     

Construction of a local and global Lyapunov function for discrete dynamical systems using radial basis functions

The basin of attraction of an asymptotically stable fixed point of the discrete dynamical system given by the iteration x_n+1=g(x_n) can be determined through sublevel sets of a Lyapunov function. In Giesl [On the determination of the basin of attraction of discrete dynamical systems. J. Difference Equ. Appl. 13(6) (2007) 523–546] a Lyapunov function is constructed by approximating the solution of a difference equation using radial basis functions. However, the resulting Lyapunov function is non-local, i.e. it has no negative discrete orbital derivative in a neighborhood of the fixed point. In this paper we modify the construction method by using the Taylor polynomial and thus obtain a Lyapunov function with negative discrete orbital derivative both locally and globally.     

A simple proof of the Grobman–Hartman theorem for nonuniformly hyperbolic flows

We give a simple and direct proof of the Grobman–Hartman theorem for nonautonomous differential equations obtained from perturbing a nonuniform exponential dichotomy. In particular, we do not need to pass through discrete time and obtain the result as a consequence of a corresponding result for maps. To the best of our knowledge, this is the first direct approach for nonuniform exponential dichotomies. We also show that the conjugacies are continuous in time and Hölder continuous in space. In addition, we describe the dependence of the conjugacies on the perturbation, and we obtain a reversibility result for the conjugacies of reversible differential equations. We emphasize that the additional work required to consider nonuniform exponential dichotomies is substantial.     

Isochronicity conditions for some planar polynomial systems

We study the isochronicity of centers at O∈R² for systems , , where A,B∈R[x,y], which can be reduced to the Liénard type equation. Using the so-called C-algorithm we have found 27 new multiparameter isochronous centers.     

Isochronicity conditions for some planar polynomial systems II

We study the isochronicity of centers at O∈R² for systems where A,B∈R[x,y], which can be reduced to the Liénard type equation. When deg(A)?4 and deg(B)?4, using the so-called C-algorithm we found 36 new multiparameter families of isochronous centers. For a large class of isochronous centers we provide an explicit general formula for linearization. This paper is a direct continuation of a previous one with the same title [Islam Boussaada, A. Raouf Chouikha, Jean-Marie Strelcyn, Isochronicity conditions for some planar polynomial systems, Bull. Sci. Math. 135 (1) (2011) 89–112], but it can be read independently.     

Dichotomy spectra and Morse decompositions of linear nonautonomous differential equations

Recently, the existence of Morse decompositions for nonautonomous dynamical systems was shown for three different time domains: the past, the future and—in the linear case—the entire time. In this article, notions of exponential dichotomy are discussed with respect to the three time domains. It is shown that an exponential dichotomy gives rise to an attractor-repeller pair in the projective space, which is a building block of a Morse decomposition. Moreover, based on the notions of exponential dichotomy, dichotomy spectra are introduced, and it is proved that the corresponding spectral manifolds lead to Morse decompositions in the projective space.     

Novel delay-dependent asymptotic stability criteria for neural networks with time-varying delays

The problem of delay-dependent asymptotic stability criteria for neural networks (NNs) with time-varying delays is investigated. An improved linear matrix inequality based on delay-dependent stability test is introduced to ensure a large upper bound for time-delay. A new class of Lyapunov function is constructed to derive a novel delay-dependent stability criteria. Finally, numerical examples are given to indicate significant improvement over some existing results.     

On linear and nonlinear fourth-order eigenvalue problems with indefinite weight

We determine the principal eigenvalues of the linear indefinite weight problem Moreover, we investigate the existence of positive solutions for the corresponding nonlinear indefinite weight problem, where g:[0,1]→R is a continuous function which attains both positive and negative values, f∈C(R,R), and r is a parameter.     

Partial hyperbolicity far from homoclinic bifurcations

We prove that any diffeomorphism of a compact manifold can be C¹-approximated by a diffeomorphism which exhibits a homoclinic bifurcation (a homoclinic tangency or a heterodimensional cycle) or by a diffeomorphism which is partially hyperbolic (its chain-recurrent set splits into partially hyperbolic pieces whose centre bundles have dimensions less or equal to two). We also study in a more systematic way the central models introduced in Crovisier (in press) [10].     

Dubrovin's duality for F-manifolds with eventual identities

A vector field E on an F-manifold (M,°,e) is an eventual identity if it is invertible and the multiplication X?Y:=X°Y°E−1 defines a new F-manifold structure on M. We give a characterization of such eventual identities, this being a problem raised by Manin (2005) [12]. We develop a duality between F-manifolds with eventual identities and we show that this duality is compatible with the local irreducible decomposition of F-manifolds and preserves the class of Riemannian F-manifolds. We find necessary and sufficient conditions on the eventual identity which ensure that the classes of harmonic Higgs bundles, DChk-structures and weak CV-structures are preserved by our duality. Examples of such structures are given in the case of a semi-simple multiplication. We use eventual identities to construct compatible pairs of metrics.     

On open problems concerning distributional chaos for triangular maps

We show that in the class T of the triangular maps (x,y)?(f(x),g_x(y)) of the square there is a map of type 2^∞ with non-minimal recurrent points which is not DC3. We also show that every DC1 continuous map of a compact metric space has a trajectory which cannot be (weakly) approximated by trajectories of compact periodic sets. These two results make possible to answer some open questions concerning classification of maps in T with zero topological entropy, and contribute to an old problem formulated by A.N. Sharkovsky.     

Nonlinear second-order equations with functional implicit impulses and nonlinear functional boundary conditions

In this paper we present existence results for solutions of nonlinear second-order boundary value problems with impulses. Our impulses are applied at p points in the interval and given implicitly by nonlinear functions of the solution. Moreover we allow functional dependence on the solution. Our existence results follow from the existence of a pair of well ordered lower and upper solutions. We generalize earlier results of Cabada and Tomec?ek, allowing more general compatible boundary conditions, impulses and φ-Laplacian equations.     

Global bifurcation of polygonal relative equilibria for masses, vortices and dNLS oscillators

Given a regular polygonal arrangement of identical objects, turning around a central object (masses, vortices or dNLS oscillators), this paper studies the global bifurcation of relative equilibria in function of a natural parameter (central mass, central circulation or amplitude of the oscillation). The symmetries of the problem are used in order to find the irreducible representations, the linearization and, with the help of a degree theory, the symmetries of the bifurcated solutions.     

A necessary and sufficient condition for the global asymptotic stability of damped half-linear oscillators

We are concerned with the global asymptotic stability of the equilibrium of the half-linear differential equation with a damped term $$\left(\phi_p(x')\right)' + h(t)\phi_p(x') + (p-1)\phi_p(x) = 0. $$ A necessary and sufficient growth condition on h(t) is obtained by using the generalized Prüfer transformation and the generalized Riccati transformation. Equivalent planar systems to the above-mentioned equation are also considered in the proof of our main result.