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
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.
In this paper, we study a model of insect and animal dispersal
where both density-dependent diffusion and nonlinear rate of
growth are present. We analyze the existence of bounded traveling
wave solution under certain parametric conditions by using the
qualitative theory of dynamical systems. An explicit traveling
wave solution is obtained by means of the first integral method.
Traveling wave solutions in parametric forms for three particular
cases are established by the Lie symmetry method.
This paper is concerned with traveling waves for temporally delayed, spatially discrete reaction-diffusion equations without quasi-monotonicity. We first establish the existence of non-critical traveling waves (waves with speeds c>c*, where c* is minimal speed). Then by using the weighted energy method with a suitably selected weight function, we prove that all noncritical traveling waves Φ(x+ct) (monotone or nonmonotone) are time-asymptotically stable, when the initial perturbations around the wavefronts in a certain weighted Sobolev space are small.
In this paper, we discuss the Liouville
integrability of the Burgers-Korteweg-de Vries equation under
certain parametric condition. An approximate solution is obtained by
means of the Adomian decomposition method.
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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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.
In this paper, we are concerned with a time periodic and diffusivepredator-prey model with disease transmission in the prey. Firstwe consider a $ SI $ model when the predator species is absent. Byintroducing the basic reproduction number for the $ SI $ model, weshow the sufficient conditions for the persistence and extinctionof the disease. When the presence of the predator is taken intoaccount, a number of sufficient conditions for the co-existence ofthe prey and predator species, the global extinction of predatorspecies and the global extinction of both the prey and predatorspecies are given.
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
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.