MULTIPLICITY OF POSITIVE PERIODIC SOLUTIONS TO SECOND ORDER SINGULAR DIFFERENTIAL SYSTEMS

In this article, the author is devoted to establish the multiplicity of positive periodic solutions to second-order singular differential systems. It isproved that such a problem has at least two positive solutions under our reasonable conditions. The proof relies on a nonlinear alternative of Leray--Schauder type and Krasnoselskii fixed point theorem in cones.     

Existence of heteroclinic orbits of the Shil'nikov type in a 3D quadratic autonomous chaotic system

In this paper, the existence of heteroclinic orbits of Shil'nikov type in a three-dimensional quadratic autonomous system is proved. Four heteroclinic orbits and four critical points together constitute two cycles simultaneously. The dynamical behaviors of the system are also studied.     

Periodic solutions of a discrete Hamiltonian system with a change of sign in the potential

In this paper, an existence theorem is obtained for periodic solutions of a second-order discrete Hamiltonian system with a change of sign in the potential by the minimax methods in the critical point theory.     

Periodic solutions for a class of nonautonomous subquadratic second order Hamiltonian systems

Some existence theorems are obtained for periodic solutions of a class of unbounded nonautonomous nonconvex subquadratic second order Hamiltonian systems by using the minimax methods in critical point theory.     

类氢原子椭圆轨道的动能、势能和能级

利用椭圆的几何性质确定类氢原子的轨道，并计算其能级。     

PBVP of integrodifferential equations with Carathéodory functions

We study the periodic boundary value problems for nonlinear integro-differential equations of Volterra type with Carathéodory functions. For two situations relative to lower and upper solutions α and β∶α≤ß or ß≤α, the existence of solutions and the monotone iterative method for establishing extreme solutions are considered.     

Natural operators of smooth mappings of manifoldswith metric fields

In this paper we introduce the concepts of both a natural bundle and a natural operator generalized for the case of the category Mf_m × Mf_m of cartesian products of two manifolds and products of local diffeomorphisms. It is shown that any r-th order natural bundle over M × N has a structure of an associated bundle (P^rM × P^rN)Z G_{m^r × Gm^r}]. We consider prolongations of such associated bundles and their reduction with respect to a chosen subgroup. The existence of a bijective correspondence between natural operators of order k and the equivariant mappings of the corresponding type fibers are proved. A basis of invariants of arbitrary order is constructed for natural operators of smooth mappings of manifolds endowed with metric fields or connections, with values in a natural bundle of order one.     

四阶非线性微分方程周期解的存在性

应用构造Ｌｉａｐｕｎｏｖ函数的方法，给出了四阶非线性微分方程存在周期解的一组充分条件．     

Faithful and unfaithful students in time series learning

A model neuron with delay line feedback connections can learna time series generated by another neuron. In the case thatboth neurons have identical transfer functions, a model neuron(student) is capable of reproducing the instruction sequencegenerated by another neuron (teacher), but the parameters arenot uniquely determined by learning a quasi-periodic time series.A student that has completed the learning can be either faithfulor unfaithful, depending on whether it continues mimicking theteacher's time series over a long interval after the learningor whether it departs from the teacher and eventually generatesa time series that bears no resemblance to the teacher's. Inthe case that both neurons have different transfer functions,a student is generally incapable of reproducing the instructionsequence. Each student readjusts its parameters so as to minimizethe mean-squared deviation between the instruction signal andits own output, but this parameter set is not uniquely determined.In this unrealizable learning, the degeneracy among faithfulstudents is lifted and there is no more distinction betweenfaithful and unfaithful students.