Bifurcation diagrams in a class of Kolmogorov systems

We study a class of Kolmogorov systems of dimension two depending on two independent parameters. The local behavior of the system is described in terms of bifurcation diagrams which contain the bifurcation curves separating a node from a focus.     

Permanence and convergence in multi-species competition systems with delay

The permanence and global attractivity of positive equilibria are obtained for some multi-species Kolmogorov competition models with delay by embedding the system into a larger cooperative system with delay and then appealing to the theory of monotone dynamical systems.

     

Equilibria in some families of Kolmogorov systems

Kolmogorov systems constitute a general model for the dynamics of biological species. In that sense, they are generalizations of the Lotka–Volterra systems. Here, some classical results on existence, uniqueness, and global attraction of local equilibria that hold in Lotka–Volterra systems are generalized to Kolmogorov systems.     

On the optimality of the trigonometric system

We study a new phenomenon of the behaviour of widths with respect to the optimality of trigonometric system. It is shown that the trigonometric system is optimal in the sense of Kolmogorov widths in the case of “super-high” and “super-small” smoothness but is not optimal in the intermediate cases. Bernstein’s widths behave differently when compared with Kolmogorov in the case of “super-small” smoothness. However, in the case of “super-high” smoothness Kolmogorov and Bernstein widths behave similarly, i.e. are realized by trigonometric polynomials.     

Empirical study of stochasticity for deterministic chaotical dynamics of geometric progressions of residues

Kolmogorov discovered in 1933 that the empirical statistics of several independent values of any random variable differs from the true distribution function of this variable in some universal way: the random distribution of the distance of one of these statistics from the other verifies (asymptotically) some stochastic distribution law (called later “Kolmogorov’s distribution”). The present paper compares the Kolmogorov’s distribution with a similar object, provided by the chain of observations of a nonrandom, deterministic dynamical system, formed by the consecutive members of a geometrical progression. Say, the Kolmogorov’s distribution is observed for the distribution of the last pairs of digits of the powers of integer 3, that is, for the sequence 01,03,09,27,81,43,29,87,… (which is not random at all and does not verify the Kolmogorov’s theorem conditions).     

非自治单种群时滞Kolmogorov系统的持续性和灭绝性

研究非自治单种群时滞Kolmogorov系统,给出了该系统中种群持续和灭绝的充分条件,这些结果与相应的非自治单种群Kolmogorov系统的有关结果相似.     

R~3中一类Kolmogorov系统的空间周期解

本文对一类具有相同增长率的三种群广义的Ｋｏｌｍｏｇｏｒｏｖ系统的空间周期解 存在条件进行研究,得到存在非常数空间周期解的判定条件．     

一类单滞量Kolmogorov型捕食-食饵系统的稳定性分析

本文给出了一类单滞量Kolmogorov型捕食-食饵系统的正平衡态线性渐近稳定的充要条件,这些条件易于检验和应用．     

Dynamics of Kolmogorov systems of competitive type under the telegraph noise

This paper studies the dynamics of Kolmogorov systems of competitive type under the telegraph noise. The telegraph noise switches at random two Kolmogorov competition-type deterministic models. The aim of this work is to describe the omega-limit set of the system and investigates properties of stationary density.     

Carrying simplices in nonautonomous and random competitive Kolmogorov systems

The purpose of this paper is to investigate the asymptotic behavior of positive solutions of nonautonomous and random competitive Kolmogorov systems via the skew-product flows approach. It is shown that there exists an unordered carrying simplex which attracts all nontrivial positive orbits of the skew-product flow associated with a nonautonomous (random) competitive Kolmogorov system.     

Symplectic invariants near hyperbolic-hyperbolic points

We construct symplectic invariants for Hamiltonian integrable systems of 2 degrees of freedom possessing a fixed point of hyperbolic-hyperbolic type. These invariants consist in some signs which determine the topology of the critical Lagrangian fibre, together with several Taylor series which can be computed from the dynamics of the system.We show how these series are related to the singular asymptotics of the action integrals at the critical value of the energy-momentum map. This gives general conditions under which the non-degeneracy conditions arising in the KAM theorem (Kolmogorov condition, twist condition) are satisfied. Using this approach, we obtain new asymptotic formulae for the action integrals of the C. Neumann system. As a corollary, we show that the Arnold twist condition holds for generic frequencies of this system.      

Kolmogorov complexity and set theoretical representations of integers

We reconsider some classical natural semantics of integers (namely iterators of functions, cardinals of sets, index of equivalence relations) in the perspective of Kolmogorov complexity. To each such semantics one can attach a simple representation of integers that we suitably effectivize in order to develop an associated Kolmogorov theory. Such effectivizations are particular instances of a general notion of “self‐enumerated system” that we introduce in this paper. Our main result asserts that, with such effectivizations, Kolmogorov theory allows to quantitatively distinguish the underlying semantics. We characterize the families obtained by such effectivizations and prove that the associated Kolmogorov complexities constitute a hierarchy which coincides with that of Kolmogorov complexities defined via jump oracles and/or infinite computations (cf. [6]). This contrasts with the well‐known fact that usual Kolmogorov complexity does not depend (up to a constant) on the chosen arithmetic representation of integers, let it be in any base n ≥ 2 or in unary. Also, in a conceptual point of view, our result can be seen as a mean to measure the degree of abstraction of these diverse semantics. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A constructive hypothesis test for the single-index models with two groups

Comparison of two-sample heteroscedastic single-index models, where both the scale and location functions are modeled as single-index models, is studied in this paper. We propose a test for checking the equality of single-index parameters when dimensions of covariates of the two samples are equal. Further, we propose two test statistics based on Kolmogorov–Smirnov and Cramér–von Mises type functionals. These statistics evaluate the difference of the empirical residual processes to test the equality of mean functions of two single-index models. Asymptotic distributions of estimators and test statistics are derived. The Kolmogorov–Smirnov and Cramér–von Mises test statistics can detect local alternatives that converge to the null hypothesis at a parametric convergence rate. To calculate the critical values of Kolmogorov–Smirnov and Cramér–von Mises test statistics, a bootstrap procedure is proposed. Simulation studies and an empirical study demonstrate the performance of the proposed procedures.     

Stochastic stabilization of first-passage failure of Rayleigh oscillator under Gaussian White-Noise parametric excitations

Stochastic stabilization of first-passage failure of Rayleigh oscillator under Gaussian White-Noise parametric excitation is studied. The equation of motion of the system is first reduced to an averaged Itô stochastic differential equation by using the stochastic averaging method. Then, a backward Kolmogorov equation governing the conditional reliability function of first-passage failure is established. The conditional reliability function, and the conditional probability density are obtained by solving the backward Kolmogorov equation with boundary conditions. Finally, the cost function and optimal control forces are determined by the requirements of stabilizing the system by evaluating the maximal Lyapunov exponent. The numerical results show that the procedure is effective and efficiency.     

Existence of Limit Cycles for a Cubic Kolmogorov System with a Hyperbolic Solution

When we discuss some problems in bio-mathematics,we often meet cubicKolmogorov systems.For general type of Kolmogorov systems,the qualitative analysisis very difficult. Usually,one discusses some special type of cubic Kolmogorovsystems,for example,cubic Kolmogorov systems with an algebraic curve solution.Firstly,the existence oflimitcycles fora cubic Kolmogorov system with quadratic curvesolution was studied.In article[1 ] ,though the author proved the cubic Kolmogorovsystem with a solution…     

EXISTENCE OF LIMIT CYCLES FOR A CUBIC KOLMOGOROV SYSTEM WITH A PARABOLIC SOLUTION WHICH DOES NOT CONTACT AND INTERSECT THE COORDINATE AXES

This paper is concerned with a cubic Kolmogorov system with a parabolicsolution which does not contact and intersect the coordinates. The conclusionis that such a system may possess limit cycles.     

一类三次Kolmogorov捕食者-食饵种群差分系统正周期解的存在性

研究了一类三次Kolmogorov捕食者-食饵种群差分系统正周期解的存在性,利用重合度理论中的连续性定理建立了存在正周期解的充分性依据,推广并改进了文献中的相关结果.     

Estimates in the kolmogorov theorem on conservation of conditionally periodic motions

A Hamiltonian system which differs from the integrable system by a small perturbation is considered. According to the Kolmogorov theorem /1–3/ the majority of invariant tori present in the unperturbed system do not decompose under a perturbation, and few of them become deformed. The following estimates are obtained below: for the perturbation under the usual conditions of nondegeneracy, the measure of the set of the decomposing tori and the deformation of the remaining tori are both estimate from above by the quantities of the order of √, and these estimates cannot be improved. The proof follows that /2,3/ of the Kolmogorov theorem with the intermediate estimates obtained more accurately. Similar estimates were obtained in the Moser theorem concerning the invariant curves of the mapping of a plane onto itself in /4/, and for the mapping in the multidimensional case, in /5/.     

Multiple Bifurcations of Critical Period for a Quartic Kolmogorov Model

Acta Mathematicae Applicatae Sinica, English Series - Our work is concerned with the bifurcation of critical period for a quartic Kolmogorov system. By computing the periodic constants carefully,...     

Inference in lognormal multidimensional diffusion processes with exogenous factors: Application to modelling in economics

The multidimensional lognormal diffusion process with exogenous factors is treated using the Kolmogorov equations, and the mean vector and covariance matrix are estimated using discrete sampling by the maximum-likelihood method. Also, this process is constructed as a solution of a multidimensional stochastic differential equation, and an estimation is made through the maximum-likelihood method to infer the parameters of the exogenous factors, this time using continuous sampling. Finally, a test for a hypothesis based on these parameters is constructed.