Two-sided approximation to periodic solutions of ordinary differential equations

Summary A two-sided approximation to the periodic orbit of an autonomous ordinary differential equation system is considered. First some results about variational equation systems for periodic solutions are obtained in Sect. 2. Then it is proved that if the periodic orbit is convex and stable, the explicit difference solution approximates the periodic orbit from the outer part and the implicit one from the inner part respectively. Finally a numerical example is given to illustrate our result and to point out that the numerical solution no longer has a one-sided approximation property, if the periodic orbit is not convex.The Work is supported by the National Natural Science Foundation of China     

Almost periodic solutions of neutral impulsive systems with periodic time-dependent perturbed delays

A neutral impulsive system with a small delay of the argument of the derivative and another delay which differs from a constant by a periodic perturbation of a small amplitude is considered. If the corresponding system with constant delay has an isolated ω-periodic solution and the period of the delay is not rationally dependent on ω, then under a nondegeneracy assumption it is proved that in any sufficiently small neighbourhood of this orbit the perturbed system has a unique almost periodic solution.     

On the spectrum for the artificial compressible system

Stability of stationary solutions of the incompressible Navier–Stokes system and the corresponding artificial compressible system is considered. Both systems have the same sets of stationary solutions and the incompressible system is obtained from the artificial compressible one in the zero limit of the artificial Mach number ? which is a singular limit. It is proved that if a stationary solution of the incompressible system is asymptotically stable and the velocity field of the stationary solution satisfies an energy-type stability criterion by variational method with admissible functions being only potential flow parts of velocity fields, then it is also stable as a solution of the artificial compressible one for sufficiently small ?. The result is applied to the Taylor problem.     

屈曲特征值问题的边界元方法及收敛性分析

讨论了屈曲特征值问题的定解条件,建立了相应的具约束的积分方程组及带Lagrange乘子的边界变分方程,给出了解的存在唯一性定理。建立了相应的边界元方法并讨论了近似解的误差估计。文末给出了数值算例。     

Periodic Solutions for Nonlinear Diferential Equations

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非线性扰动广义NNV系统的孤立子渐近行波解

采用了一个简单而有效的技巧,研究一类非线性扰动广义NNV(Nizhnik-Novikov-Veselov)系统.首先用待定系数法得到一个相应典型系统的孤立子解.其次构造一个广义泛函式,并对它进行变分计算,利用变分原理求出对应的Lagrange乘子,并由此构造一个特殊的变分迭代关系式.然后依次求出原非线性扰动广义NNV系统的孤立子渐近行波解.最后通过举例,说明了使用该方法得到的近似解具有简单而有效的优点.     

Local Entropy Conditions in Transonic Potential Flow Problems

The transonic potential flow problem is handled as a variational problem over a closed convex set which is given by a bound for the gas velocity and by a local entropy condition. It can be shown that the minimum problem has a solution though the functional need not be convex and the given set is not compact. Furthermore, the convergence of an approximation method (KATCHANOV'S method) for the solution to the corresponding variational inequality is proved.     

A nonintersection property for extremals of variational problems with vector-valued functions

In this work we study the structure of extremals of variational problems with vector-valued functions on [0,∞). We show that if an extremal is not periodic, then the corresponding curve in the phase space does not intersect itself.     

A spectral theory for linear differential systems

This paper is concerned with continuous and discrete linear skew-product dynamical systems including those generated by linear time-varying ordinary differential equations. The concept of spectrum is introduced for a linear skew-product dynamical system. In the case of a system of ordinary differential equations with constant coefficients the spectrum reduces to the real parts of the eigenvalues. In the general case continuous spectrum can occur and under certain conditions it consists of finitely many compact intervals of the real line, their number not exceeding the dimension of the system. A spectral decomposition theorem is proved which says that a certain naturally defined vector bundle is the sum of invariant subbundles, each one associated with a spectral subinterval. This partially generalizes the Jordan decomposition in the case of constant coefficients. A perturbation theorem is proved which says that nearby systems have spectra which are close. Almost periodic systems are given special attention.     

Global dynamics of a class of SEIRS epidemic models in a periodic environment

In this paper, we study a class of periodic SEIRS epidemic models and it is shown that the global dynamics is determined by the basic reproduction number R₀ which is defined through the spectral radius of a linear integral operator. If R₀<1, then the disease free periodic solution is globally asymptotically stable and if R₀>1, then the disease persists. Our results really improve the results in [T. Zhang, Z. Teng, On a nonautonomous SEIRS model in epidemiology Bull. Math. Biol. 69 (8) (2007) 2537-2559] for the periodic case. Moreover, from our results, we see that the eradication policy on the basis of the basic reproduction number of the time-averaged system may overestimate the infectious risk of the periodic disease. Numerical simulations which support our theoretical analysis are also given.     

Logic optimality for multi-objective optimization

Pareto dominance is one of the most basic concepts in multi-objective optimization. However, it is inefficient when the number of objectives is large because in this case it leads to an unmanageable number of Pareto solutions. In order to solve this problem, a new concept of logic dominance is defined by considering the number of improved objectives and the quantity of improvement simultaneously, where probabilistic logic is applied to measure the quantity of improvement. Based on logic dominance, the corresponding logic nondominated solution is defined as a feasible solution which is not dominated by other ones based on this new relationship, and it is proved that each logic nondominated solution is also a Pareto solution. Essentially, logic dominance is an extension of Pareto dominance. Since there are already several extensions for Pareto dominance, some comparisons are given in terms of numerical examples, which indicates that logic dominance is more efficient. As an application of logic dominance, a house choice problem with five objectives is considered.     

Frequency Tests for the Existence and Stability of Bounded Solutions to Differential Equations of Higher Order

To study a vector-matrix differential equation of order n, the method of integral equations is used. When the Lipschitz condition holds, an existence and uniqueness theorem for a bounded solution and its estimates are obtained. This solution is almost periodic if the nonlinearity is almost periodic, and it is asymptotically Lyapunov stable if the matrix characteristic polynomial is a Hurwitz polynomial. Under a Lipschitztype condition, a theorem on the existence of at least one bounded solution is proved; among the bounded solutions, there is at least one recurrent solution if the nonlinearity is almost periodic. The equation is S-dissipative if the matrix characteristic polynomial is a Hurwitz polynomial.     

A successful algorithm for solving directed Hamiltonian path problems

In this paper we present a graph-theoretic polynomial algorithm which has positive probability of finding a Hamiltonian path in a given graph, if there is one; if the algorithm fails, it can be rerun with a randomly chosen starting solution, and there is again a positive probability it will find an answer. If there is no Hamiltonian path, the algorithm will always terminate with failure. We call this a Successful Algorithm because it has high (close to 1) empirical probability of success and it works in polynomial time. Some basic theoretical results concerning spanning arborescences of a graph are given. The concept of a ramification index is defined and it is shown that ramification index of a Hamiltonian path is zero. The algorithm starts with finding any spanning arborescence and by suitable pivots it endeavors to reduce the ramification index to zero. Probabilistic properties of the algorithm are discussed. Computational experience with graphs up to 30 000 nodes is included.     

On the almost periodic solutions of Lattices equations

By using the Ekeland variational principle and the calculus of variations in mean, we study the existence of almost periodic solutions of a class of advanced-retarded differential equation. We show that under some hypothesis, for any given almost periodic forcing term can be ‘perturbed’ so that the corresponding forced equation admits an almost periodic solution.     

A Linearization Method for Nondegenerate Variational Conditions

We employ recent results about constraint nondegeneracy in variational conditions to design and justify a linearization algorithm for solving such problems. The algorithm solves a sequence of affine variational inequalities, but the variational condition itself need not be a variational inequality: that is, its underlying set need not be convex. However, that set must be given by systems of differentiable nonlinear equations with additional polyhedral constraints. We show that if the variational condition has a solution satisfying nondegeneracy and a standard regularity condition, and if the linearization algorithm is started sufficiently close to that solution, the algorithm will produce a well defined sequence that converges Q-superlinearly to the solution.     

Inverse problem of the calculus of variations on Lie groups

This article studies the inverse problem of the calculus of variations for the special case of the geodesic flow associated to the canonical symmetric bi-invariant connection of a Lie group. Necessary background on the differential geometric structure of the tangent bundle of a manifold as well as the Fröhlicher-Nijenhuis theory of derivations is introduced briefly. The first obstructions to the inverse problem are considered in general and then as they appear in the special case of the Lie group connection. Thereafter, higher order obstructions are studied in a way that is impossible in general. As a result a new algebraic condition on the variational multiplier is derived, that involves the Nijenhuis torsion of the Jacobi endomorphism. The Euclidean group of the plane is considered as a working example of the theory and it is shown that the geodesic system is variational by applying the Cartan-Kähler theorem. The same system is then reconsidered locally and a closed form solution for the variational multiplier is obtained. Finally some more examples are considered that point up the strengths and weaknesses of the theory.     

The linearization of periodic Hamiltonian systems with one degree of freedom under the Diophantine condition

In this paper we are concerned with the periodic Hamiltonian system with one degree of freedom, where the origin is a trivial solution. We assume that the corresponding linearized system at the origin is elliptic, and the characteristic exponents of the linearized system are $\pm i\omega$ with ω be a Diophantine number, moreover if the system is formally linearizable, then it is analytically linearizable. As a result, the origin is always stable in the sense of Liapunov in this case.     

非线性系统周期解不动点迭代法

本文提出一种求解非线性系统周期解的数值方法。首先对非线性自治系统和非自治系统给出不同的点映射定义。其次指出用线性映射逼近原非线性映射,而线性映射是由非线性映射插值获得的。继而求取线性映射的不动点,作为原系统不动点的近似解。如不满足精度则作为下次映射的初始点。本文还提出了研究周期解稳定性的相应方法。     

The Penalty Functions Method and Multiplier Rules Based on the Mordukhovich Subdifferential

We show that the finite-dimensional Fritz John multiplier rule, which is based on the limiting/Mordukhovich subdifferential, can be proved by using differentiable penalty functions and the basic calculus tools in variational analysis. The corresponding Kuhn–Tucker multiplier rule is derived from the Fritz John multiplier rule by imposing a constraint qualification condition or the exactness of an ?₁ penalty function. Complementing the existing proofs, our proofs provide another viewpoint on the fundamental multiplier rules employing the Mordukhovich subdifferential.     

某类常微系统的一个基本性质

<正> 简介 考虑一n维紧致的C~∞型Riemann流形M~n(n≥2)上,由所有的C~1型常微系统就C~1度量作成的空间.这里为简便,一系统∈将暂称为A_o-系统,如果它只具有有限多个奇点和至多可数多个周期轨道;将暂称为A_1-系统,如果它只具有有限多个奇点和有限多个周期轨道.依据周知的结果,一般绝不是中每一系统都有任意小的邻