DCDS-B
Preface
Zhaosheng Feng Jinzhi Lei
This issue of Discrete and Continuous Dynamical Systems–Series B, is dedicated to our professor and friend, Qishao Lu, on the occasion of his 70th birthday and in honor of his important and fundamental contributions to the fields of applied mathematics, theoretical mechanics and computational neurodynamics. His pleasant personality and ready helpfulness have won our hearts as his admirers, students, and friends.

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CPAA
Periodic solutions for $p$-Laplacian systems of Liénard-type
Wenbin Liu Zhaosheng Feng
In this paper, we study the existence of periodic solutions for $n-$dimensional $p$-Laplacian systems by means of the topological degree theory. Sufficient conditions of the existence of periodic solutions for $n-$dimensional $p$-Laplacian systems of Liénard-type are presented.
keywords: topological degree Liénard equation. existence $p$--Laplacian system periodic solution
DCDS-S
Duffing-van der Pol-type oscillator systems
Zhaosheng Feng
In this paper, under certain parametric conditions we are concerned with the first integrals of the Duffing-van der Pol-type oscillator system, which include the van der Pol oscillator and the damped Duffing oscillator etc as particular cases. After applying the method of differentiable dynamics to analyze the bifurcation set and bifurcations of equilibrium points, we use the Lie symmetry reduction method to find two nontrivial infinitesimal generators and use them to construct canonical variables. Through the inverse transformations we obtain the first integrals of the original oscillator system under the given parametric conditions, and some particular cases such as the damped Duffing equation and the van der Pol oscillator system are included accordingly.
keywords: autonomous system Lie symmetry infinitesimal generator bifurcation Van der Pol oscillator first integral Duffing equation diffeomorphism.
CPAA
Preface
Zhaosheng Feng Wei Feng
This issue of Communications on Pure and Applied Analysis, comprises a collection in the general area of nonlinear systems and analysis, and related applications in mathematical biology and engineering. During the past few decades people have seen an enormous growth of the applicability of dynamical systems and the new developments of related dynamical concepts. This has been driven by modern computer power as well as by the discovery of advanced mathematical techniques. Scientists in all disciplines have come to realize the power and beauty of the geometric and qualitative techniques developed during this period. More importantly, they have been able to apply these techniques to a various nonlinear problems ranging from physics and engineering to biology and ecology, from the smallest scales of theoretical particle physics up to the largest scales of cosmic structure. The results have been truly exciting: systems which once seemed completely intractable from an analytical point of view can now be studied geometrically and qualitatively. Chaotic and random behavior of solutions of various systems is now understood to be an inherent feature of many nonlinear systems, and the geometric and numerical methods developed over the past few decades contributed significantly in those areas.
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PROC
Traveling waves to a reaction-diffusion equation
Zhaosheng Feng
In this paper, we study a nonlinear reaction–diffusion equation for its traveling waves. This equation can be regarded as a generalization of the Fisher equation and is used as a nonlinear model, in the one-dimensional situation, for studying insect and animal dispersal with growth dynamics. Applying the Lie symmetry method, we obtain two traveling wave solutions under certain parametric conditions and express them in terms of elliptic functions.
keywords: Fisher equation decomposition method. approximate solution differential operator Traveling wave equilibrium point
DCDS-B
Exact solution for a two-dimensional KDV-Burgers-type equation with nonlinear terms of any order
Zhaosheng Feng Qingguo Meng
In this paper, we study a two-dimensional Burgers--Korteweg-de Vries-type equation with higher-order nonlinearities. A class of solitary wave solution is obtained by means of the Divisor Theorem which is based on the ring theory of commutative algebra. Our result indicates that the presentation of traveling wave solution in [J. Phys. A (Math. Gen.) 35 (2002) 8253--8265] is incorrect; an explanation as to why this is so is given.
keywords: kink-profile wave. Burgers-KdV equation divisor theorem Bendixson Theorem solitary wave
CPAA
Begehr-Hile operator and its applications to some differential equations
Hua Liu Zhaosheng Feng
In the present paper, we are concerned with the integral hierarchy operator defined by Begehr and Hile in 1997. We show that the Begehr--Hile operator $T_{m,n}$ can be interpreted as the iteration of $T$ and $\bar {T}$ under certain conditions. Applications are also illustrated to some differential equations and singular integral equations in the complex plane.
keywords: Begehr-Hile operator Pompeiu operator singular integral operator complex differential equation.
DCDS
Isolated singularity for semilinear elliptic equations
Lei Wei Zhaosheng Feng
In this paper, we study a class of semilinear elliptic equations with the Hardy potential. By means of the super-subsolution method and the comparison principle, we explore the existence of a minimal positive solution and a maximal positive solution. Through a scaling technique, we obtain the asymptotic property of positive solutions near the origin. Finally, the nonexistence of a positive solution is proven when the parameter is larger than a critical value.
keywords: eigenvalue minimal positive solution. Hardy potential super-subsolution method
JIMO
A two-step algorithm for layout optimization of structures with discrete variables
Lianshuan Shi Enmin Feng Huanchun Sun Zhaosheng Feng
This paper presents a mathematical model for Layout optimization of structure with discrete variables. The optimization procedure is composed of two kinds of sub-procedures of optimization: the topological optimization and the shape optimization. In each one, a comprehensive algorithm is used to treat the problem. The two kinds of optimization procedures are used in turn until convergence appears. After the dimension of the structure is reduced, the delimiting combinatorial algorithm is used to search for the better objective value. A couple of classical examples are presented to show the efficiency of the method. Numerical results indicate that the method is efficient and the optimal results are satisfactory.
keywords: delimiting and combinatorial algorithm structural optimization layout optimization Discrete variable relative difference quotient.
DCDS-S
Preface
Zhaosheng Feng Wei Feng
As we all know, many biological and physical systems, such as neuronal systems and disease systems, are featured by certain nonlinear and complex patterns in their elements and networks. These phenomena carry significant biological and physical information and regulate down-stream mechanism in many instances. This issue of Discrete and Continuous Dynamical Systems, Series S, comprises a collection of recent works in the general area of nonlinear differential equations and dynamical systems, and related applications in mathematical biology and engineering. The common themes of this issue include theoretical analysis, mathematical models, computational and statistical methods on dynamical systems and differential equations, as well as applications in fields of neurodynamics, biology, and engineering etc.
    Research articles contributed to this issue explore a large variety of topics and present many of the advances in the field of differential equations, dynamical systems and mathematical modeling, with emphasis on newly developed theory and techniques on analysis of nonlinear systems, as well as applications in natural science and engineering. These contributions not only present valuable new results, ideas and techniques in nonlinear systems, but also formulate a few open questions which may stimulate further study in this area. We would like to thank the authors for their excellent contributions, the referees for their tireless efforts in reviewing the manuscripts and making suggestions, and the chief editors of DCDS-S for making this issue possible. We hope that these works will help the readers and researchers to understand and make future progress in the field of nonlinear analysis and mathematical modeling.
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