蛙卵有丝分裂模型的定性分析

本文对M.T.Borisuk和J.J.Tyson在[1]中所提出的一个有关蛙卵有丝分裂的平面三次系统模型证明了大范围周期解的存在性，给出了周期解所在的空间区域和所对应的参数区域及周期解不存在的空间区域和参数区域，所得结果严格地证明了[1]中给出的数值结果，最后我们证明在[1]的数值结果所用的参数下极限环的唯一性。     

Localization of invariant compacta of autonomous systems

A class of problems that may be characterized as localization problems are becoming increasingly popular in qualitative theory of differential equations [1–15]. The specific formulations differ, but geometrically all search for phase space subsets with desired properties, e.g., contain certain solutions of the system of differential equations. Such problems include construction of positive invariant sets that contain certain separatrices of the Lorenz system [1], analysis of asymptotic behavior of solutions of the Lorenz system and determination of sets that contain the Lorenz attractor [2–5, 14], as well as determination of sets containing all periodic trajectories [6–13], separatrices, and other trajectories [10, 11]. Such sets may be naturally called localizing sets and it is obviously interesting to study methods and results that produce exact or nearly exact localizing sets for each phase space structure. In this article we focus on localization of the invariant compact sets in the phase space of a differential equation system, specifically the problem of finding phase space subsets that contain all the invariant compacta of the system. Invariant compact sets are equilibria, periodic trajectories, separatrices, limit cycles, invariant tori, and other sets and their finite unions. These sets and their properties largely determine the phase space structure and the qualitative behavior of solutions of the differential equation system.     

Solitary wave families of a generalized microstructure PDE

Wave propagation in a generalized microstructure PDE, under the Mindlin relations, is considered. Limited analytic results exist for the occurrence of one family of solitary wave solutions of these equations. Since solitary wave solutions often play a central role in the long-time evolution of an initial disturbance, we consider such solutions here (via normal form approach) within the framework of reversible systems theory. Besides confirming the existence of the known family of solitary waves, we find a continuum of delocalized solitary waves (or homoclinics to small-amplitude periodic orbits). On isolated curves in the relevant parameter region, the delocalized waves reduce to genuine embedded solitons. The new family of solutions occur in regions of parameter space distinct from the known solitary wave solutions and are thus entirely new. Directions for future work are also mentioned.     

Existence and multiplicity results for nonlinear periodic boundary value problems

This paper studies the existence of positive solutions for periodic boundary value problems. The criteria for the existence, nonexistence and multiplicity of positive solutions are established by using the Global continuation theorem, fixed point index theory and approximate method. The results obtained herein generalize and complement some previous findings of [J.R. Graef, L. Kong, H. Wang, Existence, multiplicity, and dependence on a parameter for a periodic boundary value problem, J. Differential Equations 245 (2008) 1185–1197] and some other known results.     

A conjecture on the stability of the periodic solutions of Ricker’s equation with periodic parameters

In this work, we propose a conjecture about the stability of the periodic solutions of the Ricker equation with periodic parameters, which goes beyond the existing theory, and for the special case of period-two parameters we analytically show the conjecture is true. For this case we show that the stability region in parameter space obtained from the conjecture is larger than a previously proposed stability region. The period-three case is investigated numerically and similar extensions are realized. This suggests that the current theory cited in this paper, while giving sufficient conditions for stability is far from optimal.     

On the global and periodic regular flows of viscoelastic fluids with a differential constitutive law

This paper is concerned with the existence and uniqueness of global, periodic and stationary solutions for flows of incompressible viscoelastic fluids for which the extra-stress tensor satisfies a differential constitutive law. More precisely, we prove that the results obtained by C Guillopé and J.C. Saut [5] remain true without any restriction on the smallness of the retardation parameter.     

On almost periodic solutions of logistic delay differential equations with almost periodic time dependence

In this paper, we study almost periodic logistic delay differential equations. The existence and module of almost periodic solutions are investigated. In particular, we extend some results of Seifert in [G. Seifert, Almost periodic solutions of certain differential equations with piecewise constant delays and almost periodic time dependence, J. Differential Equations 164 (2000) 451–458].     

TRAVELINGWAVE SOLUTIONS FOR A CLASS OF NONLINEAR DISPERSIVE EQUATIONS

The method of the phase plane is emploied to investigate the solitary and periodic traveling waves for a class of nonlinear dispersive partial differential equations. By using the bifurcation theory of dynamical systems to do qualitative analysis, all possible phase portraits in the parametric space for the traveling wave systems are obtained. It can be shown that the existence of a singular straight line in the traveling wave system is the reason why smooth solitary wave solutions converge to solitary cusp wave solution when parameters are varied. The different parameter conditions for the existence of solitary and periodic wave solutions of different kinds are rigorously determined.     

TRAVELING WAVE SOLUTIONS FOR A CLASS OF NONLINEAR DISPERSIVE EQUATIONS

The method of the phase plane is emploied to investigate the solitary and periodic travelingwaves for a class of nonlinear dispersive partial differential equations.By using the bifurcationtheory of dynamical systems to do qualitative analysis,all possible phase portraits in theparametric space for the traveling wave systems are obtained.It can be shown that the existenceof a singular straight line in the traveling wave system is the reason why smooth solitary wavesolutions converge to solitary cusp wave solution when parameters are varied.The differentparameter conditions for the existence of solitary and periodic wave solutions of different kindsare rigorously determined.     

New explicit solutions for the Klein-Gordon equation with cubic nonlinearity

In this paper, we investigate Klein-Gordon equation with cubic nonlinearity. All explicit expressions of the bounded travelling wave solutions for the equation are obtained by using the bifurcation method and qualitative theory of dynamical systems. These solutions contain bell-shaped solitary wave solutions, kink-shaped solitary wave solutions and Jacobi elliptic function periodic solutions. Moreover, we point out the region which these periodic wave solutions lie in. We present the relation between the bounded travelling wave solution and the energy level h. We find that these periodic wave solutions tend to the corresponding solitary wave solutions as h increases or decreases. Finally, for some special selections of the energy level h, it is shown that the exact periodic solutions evolute into solitary wave solution.     

Continuation of relative periodic orbits in a class of triatomic Hamiltonian systems

We study relative periodic orbits (i.e. time-periodic orbits in a frame rotating at constant velocity) in a class of triatomic Euclidean-invariant (planar) Hamiltonian systems. The system consists of two identical heavy atoms and a light one, and the atomic mass ratio is treated as a continuation parameter. Under some nondegeneracy conditions, we show that a given family of relative periodic orbits existing at infinite mass ratio (and parametrized by phase, rotational degree of freedom and period) persists for sufficiently large mass ratio and for nearby angular velocities (this result is valid for small angular velocities). The proof is based on a method initially introduced by Sepulchre and MacKay [J.-A. Sepulchre, R.S. MacKay, Localized oscillations in conservative or dissipative networks of weakly coupled autonomous oscillators, Nonlinearity 10 (1997) 679–713] and further developed by Muñoz-Almaraz et al. [F.J. Muñoz-Almaraz, et al., Continuation of periodic orbits in conservative and Hamiltonian systems, Physica D 181 (2003) 1–38] for the continuation of normal periodic orbits in Hamiltonian systems. Our results provide several types of relative periodic orbits, which extend from small amplitude relative normal modes [J.-P. Ortega, Relative normal modes for nonlinear Hamiltonian systems, Proc. Roy. Soc. Edinburgh Sect. A 133 (2003) 665–704] up to large amplitude solutions which are not restrained to a small neighborhood of a stable relative equilibrium. In particular, we show the existence of large amplitude motions of inversion, where the light atom periodically crosses the segment between heavy atoms. This analysis is completed by numerical results on the stability and bifurcations of some inversion orbits as their angular velocity is varied.     

On the solutions of a second order nonlinear system with almost periodic forcing

A second order almost periodic perturbed nonlinear system with small parameter is discussed in this paper. Based on some analytical techniques and the method of averaging, some sufficient conditions are obtained for existence of one almost periodic solution. Also some suitable conditions are given for existence of two almost periodic solutions. The obtained new results generalized the known results in Seifert [1], [3] and He [2], [4]. Finally, some applications are presented to illustrate that our results are a good generalization of the known results.     

ON THE PERSISTENCE AND POSITIVE PERIODIC SOLUTION FOR PLANAR COMPETING LOTKA-VOLTERRA SYSTEMS

1IntroductionLetusa;onsi~~l'tile2-(1llTlellsiollalco'ttipetit;ion1,of-,ha--\./O]1,'lrra,s:\,s(t()lcfl.r]=.r,(l)j(t)--a,1(-t)xl--a,,(lt)x,),,.(II)ac=ac(6~(t-)--a21(t)x,--a22(t);rZ),xvhere(J:,.:x'2)eR;,~{(;t:l,.xZ)la:l20,IZ.20},l),(l),ail(t)fi{~j:lar(>conti…     

Periodic families of solutions for difference equations with a first integral

In the following paper we establish that a one-parameter family of N- periodic solutions out of the origin is guaranteed to exist when the dimension of the N- periodic solution space of the corresponding linear problem is unity. When this dimension is greater than unity we establish that one parameter families generically exist. These results are obtained by adapting the method of Hale3 to a N-periodic difference equation with a N-periodic first integral     

Existence of Global and Periodic Solutions for Delay Equation with Quasilinear Perturbation

Delay parabolic problems have been studied by many authors. Some authors investigated more general delay problem (refer to [1], [2]), some investigated concrete delay partial differential equations. Recently, we have done some work on delay parabolic problem. We discussed semilinear parabolic delay problem and obtained some results on the existence of solutions. In particular the results on existence of periodic solutions are characteristic (see [3], [4], [5], [6]). The purpose of this paper is to study delay equation with quasilinear perturbation. We present the existence of global and periodic solutions of abstract evolution equations in Section 2. The abstract results are used to obtain the existence of global and periodic solutions of delay parabolic problem with quasilinear perturbation in Section 3. We make preparation for our investigation and give a generalization of Gronwall inequality (Lemma 1.3) which is used in next section.     

Solitary-wave families of the Ostrovsky equation: An approach via reversible systems theory and normal forms

The Ostrovsky equation is an important canonical model for the unidirectional propagation of weakly nonlinear long surface and internal waves in a rotating, inviscid and incompressible fluid. Limited functional analytic results exist for the occurrence of one family of solitary-wave solutions of this equation, as well as their approach to the well-known solitons of the famous Korteweg–de Vries equation in the limit as the rotation becomes vanishingly small. Since solitary-wave solutions often play a central role in the long-time evolution of an initial disturbance, we consider such solutions here (via the normal form approach) within the framework of reversible systems theory. Besides confirming the existence of the known family of solitary waves and its reduction to the KdV limit, we find a second family of multihumped (or N-pulse) solutions, as well as a continuum of delocalized solitary waves (or homoclinics to small-amplitude periodic orbits). On isolated curves in the relevant parameter region, the delocalized waves reduce to genuine embedded solitons. The second and third families of solutions occur in regions of parameter space distinct from the known solitary-wave solutions and are thus entirely new. Directions for future work are also mentioned.     

A new proof of a theorem of J. Moser concerning the restricted problem of three bodies

In a recent paper [Barrar (1965)], we have shown that the result ofR. Arenstorf (1963) on the existence of periodic orbits of the second kind for the restricted problem of three bodies can be very readily obtained with the use of Delaunay or Poincaré variables. In the present paper we will show that the results ofJ. Moser (1953) can also be more readily obtained with the use of Poincaré variables.Moser, dealing with the restricted problem of three bodies, demonstrated the existence of periodic solutions that close after many revolutions and are near periodic solutions of the first kind.     

Pseudo almost periodic solutions of infinite class for some functional differential equations

The aim of this work is to investigate the existence and uniqueness of pseudo almost periodic solutions for some neutral partial functional differential equations in a Banach space when the delay is distributed using the variation of constants formula and the spectral decomposition of the phase space developed in Adimy et al. [M. Adimy, K. Ezzinbi, and A. Ouhinou, Variation of constants formula and almost periodic solutions for some partial functional differential equations with infinite delay, J. Math. Anal. Appl. 317(2) (2006), pp. 668–689]. Here, we assume that the undelayed part is not necessarily densely defined and satisfies the well-known Hille–Yosida condition, the delayed part is assumed to be pseudo almost periodic with respect to the first argument and Lipschitz continuous with respect to the second argument.