Linearizability of the polynomial differential systems with a resonant singular point

Integrability and linearizability of polynomial differential systems are studied. The computation of generalized period constants is a way to find necessary conditions for linearizable systems for any rational resonance ratio. A method to compute generalized period constants is given. The algorithm is recursive and easy to realize with computer algebraic system. As the application, we discuss linearizable conditions for several Lotka-Volterra systems, and where this is the first time that the linearizability is considered for 3:−4 and 3:−5 resonances.     

Noncommutative Integrability from Classical to Quantum Mechanics

We propose in this work a definition of integrable quantum system, which is based upon the correspondence with the concept of noncommutative integrability for classical mechanical systems. We then determine sufficient conditions under which, given an integrable classical system, it is possible to construct an integrable quantum system by means of a quantization procedure based on the symmetrized product of operators. As a first example of application of such an approach, we will consider the possible cases of noncommutative integrability for systems with rotational symmetry in an n-dimensional Euclidean configuration space.     

Permanence and global stability for nonautonomous N-species Lotka-Volterra competitive system with impulses and infinite delays

In this paper, we consider a class of nonautonomous N-species Lotka-Volterra competitive systems with impulses and infinite delays. By developing the methods given in Teng (2002) [25], we give sufficient conditions for permanence and global attractivity of the system.     

Local and global stability for Lotka-Volterra systems with distributed delays and instantaneous negative feedbacks

This paper addresses the local and global stability of n-dimensional Lotka-Volterra systems with distributed delays and instantaneous negative feedbacks. Necessary and sufficient conditions for local stability independent of the choice of the delay functions are given, by imposing a weak nondelayed diagonal dominance which cancels the delayed competition effect. The global asymptotic stability of positive equilibria is established under conditions slightly stronger than the ones required for the linear stability. For the case of monotone interactions, however, sharper conditions are presented. This paper generalizes known results for discrete delays to systems with distributed delays. Several applications illustrate the results.     

n阶Lotka-Volterra系统永久持续生存的新判据(英文)

本文首先研究了n阶Lotka Volterra系统的非负平衡点之间的关系 .然后在此基础上研究了该系统的永久持续生存问题 ,得到了若干判别n阶Lotka Volterra系统永久持续生存的充要条件 .     

The criteria of ultimate boundedness for nonautonomous Lotka-Volterra tree systems

In this paper, by introducing a concept called the degree of species, we obtain a set of sufficient conditions for the ultimate boundedness of nonautonomous n-species Lotka-Volterra tree systems. As a consequence, we also obtain the criteria of the existence of a globally stable equilibrium point for the autonomous Lotka-Volterra tree system. The criteria in this paper are in explicit forms of the parameters, and thus, are easily verifiable.     

Lotka-Volterra equations in three dimensions satisfying the Kowalevski-Painlevé property

We examine a class of Lotka-Volterra equations in three dimensions which satisfy the Kowalevski-Painlevé property. We restrict our attention to Lotka-Volterra systems defined by a skew symmetric matrix. We obtain a complete classification of such systems. The classification is obtained using Painlevé analysis and more specifically by the use of Kowalevski exponents. The imposition of certain integrality conditions on the Kowalevski exponents gives necessary conditions. We also show that the conditions are sufficient.     

Sharp conditions for global stability of Lotka-Volterra systems with distributed delays

We give a criterion for the global attractivity of a positive equilibrium of n-dimensional non-autonomous Lotka-Volterra systems with distributed delays. For a class of autonomous Lotka-Volterra systems, we show that such a criterion is sharp, in the sense that it provides necessary and sufficient conditions for the global asymptotic stability independently of the choice of the delay functions. The global attractivity of positive equilibria is established by imposing a diagonal dominance of the instantaneous negative feedback terms, and relies on auxiliary results showing the boundedness of all positive solutions. The paper improves and generalizes known results in the literature, namely by considering systems with distributed delays rather than discrete delays.     

Multidimensional linearizable system of <Emphasis Type="Italic">n</Emphasis>-wave-type equations

We propose a linearizable version of a multidimensional system of n-wave-type nonlinear partial differential equations (PDEs). We derive this system using the spectral representation of its solution via a procedure similar to the dressing method for nonlinear PDEs integrable by the inverse scattering transform method. We show that the proposed system is completely integrable and construct a particular solution.     

ON CRITERIA FOR GLOBAL STABILITY OF N-DIMENSIONAL LOTKA-VOLTERRA SYSTEMS

This paper gives the conditions for the existence of a globally stable equi-librium of rz-dimensional Lotka-Volterra systems in the following cases: Lotka-Volterra chain systems and Lotka-Volterra modei between one and multispecies. The conditions obtained in this paper are much weaker than those in [6] and more easily verifiable in application. So the results can be applied to more general Lotka-Volterra models. At the same time, the existence and stability conditions of positive equilibrium points of the above systems are given.     

Classification of discrete systems on a square lattice

We consider the classification up to a M?bius transformation of real linearizable and integrable partial difference equations with dispersion defined on a square lattice by the multiscale reduction around their harmonic solution. We show that the A₁, A₂, and A₃ linearizability and integrability conditions constrain the number of parameters in the equation, but these conditions are insufficient for a complete characterization of the subclass of multilinear equations on a square lattice.     

Weierstrass integrability of differential equations

The integrability problem consists of finding the class of functions a first integral of a given planar polynomial differential system must belong to. We recall the characterization of systems which admit an elementary or Liouvillian first integral. We define Weierstrass integrability and we determine some Weierstrass integrable systems which are Liouvillian integrable. Inside this new class of integrable systems there are non-Liouvillian integrable systems.     

The Wiener integral with respect to second order processes with stationary increments

We prove that for any second order stochastic process X with stationary increments with continuous paths and continuous variance function, there exists a tempered measure μ (for which we give an explicit expression) related with the domain of the Wiener integral with respect to X as follows: the space of tempered distributions f such that the Fourier transform of f is square integrable with respect to μ is always a dense subset of the domain of the Wiener integral. Moreover, we provide sufficient conditions on μ in order that the domain of the integral is exactly this space of distributions. We apply our results to the fractional Brownian motion. In particular, it is proved that the domain of the Wiener integral with respect to the fractional Brownian motion with Hurst parameter H>1/2 contains distributions that are not given by locally integrable functions, this fact was suggested by Pipiras and Taqqu (2000) in [5]. We have also considered the example of the process given by Ornstein and Uhlenbeck as a model for the position of a Brownian particle.     

On Positive Hilbert–Schmidt Operators

We study classes of self-adjoint Hilbert–Schmidt operators, focusing on sufficient conditions for the operators to be positive. The integral kernels for which the conditions hold true encompass kernel functions that arise in the setting of elliptic Calogero–Moser type integrable N-particle systems, a context where the positivity property has crucial consequences.     

Average conditions for permanence and extinction in nonautonomous Lotka-Volterra system

An n species nonautonomous competitive Lotka-Volterra system is considered in this paper. The average conditions on the coefficients are given to guarantee that all but one of the species are driven to extinction. The generalization for the result is presented, that is, for each r?n the average conditions on the coefficients are provided to guarantee that r of the species in the system are permanent while the remaining n−r are driven to extinction. It is shown that these average conditions are improvement of those of Ahmad and Montes de Oca [Appl. Math. Comput. 90 (1998) 155-166] and Montes de Oca and Zeeman [Proc. Amer. Math. Soc. 124 (1996) 3677-3687] and [J. Math. Anal. Appl. 192 (1995) 360-370].     

L2 − L∞ filtering for Markovian jump systems with time-varying delays and partly unknown transition probabilities

This paper considers the L₂ − L_∞ filtering problem for Markovian jump systems. The systems under consideration involve time-varying delays, disturbance signal and partly unknown transition probabilities. The aim of this paper is to design a filter, which is suitable for exactly known and partly unknown transition probabilities, such that the filtering error system is stochastically stable and a prescribed L₂ − L_∞ disturbance attenuation level is guaranteed. By using the Lyapunov-Krasovskii functional, sufficient conditions are formulated in terms of linear matrix inequalities (LMIs). A numerical example is given to illustrate the effectiveness of the proposed main results. All these results are expected to be of use in the study of filter design for Markovian jump systems with partly unknown transition probabilities.     

Fractional monodromy of resonant classical and quantum oscillatorsMonodromie fractionnelle des oscillateurs résonnants classiques et quantiques

We introduce fractional monodromy for a class of integrable fibrations which naturally arise for classical nonlinear oscillator systems with resonance. We show that the same fractional monodromy characterizes the lattice of quantum states in the joint spectrum of the corresponding quantum systems. Results are presented on the example of a two-dimensional oscillator with resonance 1:(?1) and 1:(?2). To cite this article: N.N. Nekhoroshev et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 985–988.     

Abelian equations and rank problems for planar webs

We find an invariant characterization of planar webs of maximum rank. For 4-webs, we prove that a planar 4-web is of maximum rank three if and only if it is linearizable and its curvature vanishes. This result leads to the direct web-theoretical proof of the Poincaré theorem: A planar 4-web of maximum rank is linearizable. We also find an invariant intrinsic characterization of planar 4-webs of rank two and one and prove that in general such webs are not linearizable. This solves the Blaschke problem “to find invariant conditions for a planar 4-web to be of rank 1 or 2 or 3.” Finally, we find invariant characterization of planar 5-webs of maximum rank and prove than in general such webs are not linearizable. The text was submitted by the authors in English.     

Permanence and extinction analysis for a delayed periodic predator-prey system with Holling type II response function and diffusion

In this paper, it is studied that two species predator-prey Lotka-Volterra type dispersal system with delay and Holling type II response function, in which the prey species can disperse among n patches, while the density-independent predator species is confined to one of the patches and cannot disperse. Sufficient conditions of integrable form for the boundedness, permanence, extinction and the existence of positive periodic solution are established, respectively.