周期性变量分解的多目标进化算法研究

大规模全局优化问题是目前多目标进化算法领域的研究热点和难点.随着优化问题决策变量增多,计算复杂度剧增.针对上述问题,本文给出周期性分解策略,将合作协同融合到多目标粒子群算法中,给出了周期性变量分解的多目标粒子群优化算法(PDMOPSO).将该算法在标准测试函数ZDT1、ZDT3、ZDT6以及DTLZ1、DTLZ2变量扩展后进行仿真实验,并用ε指标和超体积指标对算法性能进行对比分析.实验结果表明,该算法在解决大规模变量的多目标函数中,变量维度越高,比经典多目标优化算法具有更好的多样性与收敛性,同时计算复杂度显著降低.

On the Existence of Periodic Solutions of a Class Nonlinear Differential Systems

One method to show the existence of ω-periodic system is given. This method is based on the ultimately boundedness of the solution of the systems. By using comparing theorem and discussing some one dimensional equations the main results are obtained.

Periodic Solution and Global Stability of a Kind of Nonlinear Differential Equation

In this paper, it is discussed the model of a kind of nonlinear differential, equation d s d t=1-s-x 1s 0δQ 2(m 1s 0sk 1+s 0s-k) d x 1 d t=x 1Q(m 1s 0sk 1+s 0s-k)-x 1-x 2m 2x 1/Qk 2+x 1/Q d x 2 d t=x 2Q m 2x 1/Qk 2+x 1/Q-x 2.It is proved that the system is exist at least one stable periodic solution on under the following condition:m 2x * 2(k 1δ+s 0δ-Qλ 2-x * 2) 2】m 1δk 1(k 2+Q 2λ 2) 2.Furthermore, ifm 2x * 2(k 1δ+s 0δ-Qλ 2-x * 2) 2【m 1δk 1(k 2-Q 2λ 2) 2mold true them equilibrium point (s *,x * 1,x * 2)∈ set Ω is global asymptotic stable.

Thermal Induced Periodic Phase Separation in Polymer Blends

Simulations were carried out for studying the periodic phase separation of a symmetric binary polymer blend on the basis of Cahn-Hilliard-Cook theory. The time dependent interaction parameter x（τ） was assumed to undergo a step-wise oscillation. The hierarchic structures composed of both large and small domains were obtained. The mechanism of the periodic formation of hierarchic structures was also demonstrated.

Existence of Many and Infinitely Many Periodic Solutions for Some Types of Differential Delay Equations

J.Kaplan and J.Yorke's method is extended to establish the exis- tence of many and infinitely many periodic solutions for the DDEs (t) =±f(x(t-1))±f(x(t-2))and (t)=±f(x(t-1).

ON THE EXISTENCE OF PERIODIC SOLUTIONS FOR THE THIRD-ORDER NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS

Abstract In this paper,the existence of periodic solution for the third order nonlinear ordinary differential equation of the form x…+f()+g()+h(x)=p(t) is considered,where f,g,h and p are the continuous functions,and p(t+T)=p(t). By using the Leray Schauder degree method,some sufficient conditions to guarantee the existence of T periodic solution for the equation are obtained.Some previous results are extended and improved.

New Exact Travelling Wave Solutions for Generalized Zakharov-Kuzentsov EquationsUsing General Projective Riccati Equation Method

Applying the generalized method, which is a direct and unified algebraic method for constructing multipletravelling wave solutions of nonlinear partial differential equations (PDEs), and implementing in a computer algebraicsystem, we consider the generalized Zakharov-Kuzentsov equation with nonlinear terms of any order. As a result, wecan not only successfully recover the previously known travelling wave solutions found by existing various tanh methodsand other sophisticated methods, but also obtain some new formal solutions. The solutions obtained include kink-shapedsolitons, bell-shaped solitons, singular solitons, and periodic solutions.

A Computational Approach to the New Type Solutions of Whitham-Broer-KaupEquation in Shallow Water

Based on computerized symbolic computation,a new method and its algorithm are proposed for searching for exact travelling wave solutions of the nonlinear partial differential equations.Making use of our approach,we investigate the Whitham-Broer-Kaup equation in shallow water and obtain new families of exact solutions,which include soliton-like solutions and periodic solutions.As its special cases,the solutions of classical long wave equations and modified Boussinesq equations can also be found.

Relationship Between Soliton-like Solutions and Soliton Solutions to a Class of Nonlinear Partial Differential Equations

By using the generally projective Riccati equation method, a series of doubly periodic solutions (Jacobi elliptic function solution) for a class of nonlinear partial differential equations are obtained in a unified way. When the module m → 1, these solutions exactly degenerate to the soliton solutions of the equations. Then we reveal the relationship between the soliton-like solutions obtained by other authors and these soliton solutions of the equations.

Existence of Periodic Solutions About One Class Differential Equation

We discuss the existence and the number of periodic solutions of differential equation dx/dt=A_1(t)x+A_2(t)x^2+A_3(t)x^3/a_0(t)+a_1(t)x+a_2(t)x^2 (1) where A_i(t), a_j(t) (i=1,2,3;j=0,1,2) are continuous periodic functions. The results of this paper extend the work of paper [1].

Envelope Periodic Solutions to Coupled Nonlinear Equations

The envelope periodic solutions to some nonlinear coupled equations are obtained by means of the Jacobielliptic function expansion method. And these envelope periodic solutions obtained by this method can degenerate tothe envelope shock wave solutions and/or the envelope solitary wave solutions.

Periodic Solutions for a Class of Nonlinear Differential-Difference Equations

In this paper, by applying the Jacobi elliptic function expansion method, the periodic solutions for three nonlinear differential-difference equations are obtained.

On Periodic Solutions of a Class of Differential Equation with Periodic Coefficients

In the present paper we are concerned with the following differential equation with pcriodic coefficientsy’ = f(t,y) = A(t)ym + B(t)y + C(t) (m 2,m ∈N) (1)and obtain the theorems for the existence and the number of equation (1). Our results extend and improve the corresponding ones in [1,4].

Exact Periodic and Solitary-Wave Solutions of Multi-component Modified Korteweg-de Vries Equations

The present work extends the search of Jacobi elliptic function solutions for the multi-component modified Korteweg-de Vries equations. When the modulum m →1, those periodic solutions degenerate as the corresponding solitary wave and shock wave ones. Especially, exact solutions for the three-component system are presented in detail and graphically.

Periodic Solutions for Two Coupled Nonlinear-Partial Differential Equations

In this paper, by applying the Jacobi elliptic function expansion method, the periodic solutions for two coupled nonlinear partial differential equations are obtained.

Periodic Solutions for a Kind of Second Order Differential Equation with a Deviating Argument

By using the theory of coincidence degree, we study a kind of periodic solutions to second order differential equation with a deviating argument such as x″（t） ＋ f（x′（t）） ＋ h（x（t））x′（t） ＋ g（x（t - τ（t））） ≈ p（t）, some sufficient conditions on the existence of periodic solutions are obtained.

New Approach to Find Exact Solutions to Classical Boussinesq System

In this paper, based on a new system of three Riccati equations, we give a new method to construct more new exact solutions of nonlinear differential equations in mathematical physics. The classical Boussinesq system is chosen to illustrate our method. As a consequence, more families of new exact solutions are obtained, which include solitary wave solutions and periodic solutions.

New Extended Jacobi Elliptic Function Rational Expansion Method and Its Application

In this paper, an extended Jacobi elliptic function rational expansion method is proposed for constructing new forms of exact Jacobi elliptic function solutions to nonlinear partial differential equations by means of making a more general transformation. For illustration, we apply the method to the （2＋1）-dimensional dispersive long wave equation and successfully obtain many new doubly periodic solutions, which degenerate as soliton solutions when the modulus m approximates 1. The method can also be applied to other nonlinear partial differential equations.

Note on Existence of Periodic Solutions of One Class Differential Equations

In the present paper we investigate the number of periodic solutions of the following differential equationdydt=A 1(t)y+A 2(t)y 2+A 3(t)y 3a 0(t)+a 1(t)y+a 2(t)y 2(**)which was discussed in paper , and obtain the theorem by the method of cross-ratio of the solutions of (**) without the traditional condition assumption that the functions A i(t),a j(t) (i=1,2,3; j=0,1,2) are differential.

PERMANENCE AND PERIODIC SOLUTION IN AN INTEGRODIFFERENTIAL SYSTEM WITH DISCRETE DIFFUSION

Dynamical characteristics of an integrodifferential modelling competitive sys-tem with diffusion are investigated.In particular,we derive sufficient conditions for the permanence of species,existence of an attracting periodic solution to the periodic system.The results of Wang Ke in 1994 and 1998 are improved and extended.