Controllability and stabilization of periodic switched Boolean control networks with application to asynchronous updating

This paper investigates the periodic switching point controllability and stabilization of periodic switched Boolean control networks (PSBCNs), and applies the obtained results to the stabilization of deterministic asynchronous Boolean control networks (DABCNs). Firstly, using the algebraic state space representation of PSBCNs, a kind of periodic switching point controllability matrix is constructed, based on which, a necessary and sufficient condition is presented for the periodic switching point reachability and controllability of PSBCNs. Secondly, using the reachable set of PSBCNs, a constructive procedure is proposed to design time-variant state feedback controllers for the periodic switching point stabilization of PSBCNs. Finally, by converting the dynamics of DABCNs into the form of PSBCNs, the time-variant state feedback stabilization problem of DABCNs is solved.     

Bifurcations and New Traveling Wave Solutions for the Nonlinear Dispersion Drinfel'd-Sokolov ($D(m,n)$) System

In this paper, we employ the theory of the planar dynamical system to investigate the dynamical behavior and bifurcations of solutions of the traveling systems of the $D(m,n)$ equation. On the basis of the previous work of the reference \cite{zhang}, we obtain the solitary cusp waves solutions (peakons and valleyons), breaking wave solutions (compactons) and other periodic cusp wave solutions. Morever, we make a summary of exact traveling wave solutions to the $D(m,n)$ system including all the solutions which have been found from the references \cite{Deng,Xie,zhang}.     

How Nature Adapts Chemical Structures to Curved Surfaces

Guided by the symmetry in a natural way, periodic potential surfaces partition the space in solid crystalline compounds. The arrangement of atoms, clusters and molecules obviously follows the (in general) curved shape of these ‘space partitioners’. In structures, the atoms therefore choose only a very limited subset of the infinite set of possible positions. In collective structures the periodic surfaces separate areas of different interactions between atoms, clusters, and molecules. In a certain sense, they can be considered as inner surfaces, a knowledge of which reveals insights into the organization of crystalline matter. There are many good indications that the weakly bonded electrons in the highest occupied orbitals are preferably localized in the region of the space partitioners. Dynamic processes as well can be correlated to the shape of the periodic surfaces. Moreover, the surfaces are didactically very helpful in making accessible the complicated three-dimensional relations in collective structures because the main features are projected onto (although curved) two-dimensional creations. The application of periodic potential surfaces to such a variety of compounds as quartz, brass and alpha-amylose underscores their general significance. Simple qualitative considerations already reveal the manifold relations to animate and inanimate nature through to mathematics, art and architecture. It appears that, in a very universal sense, the adaption of structures to a collective order finds a natural solution through curvature.     

Periodic behavior of cellular automata

I propose an explanation of the observation of a globally synchronized behavior of deterministic cellular automata and coupled map lattices, together with local fluctuations.     

On spectral properties of periodic polyharmonic matrix operators

We consider a matrix operatorH = (-Δ)^l +V inR ⁿ, wheren ≥ 2,l ≥ 1, 4l > n + 1, andV is the operator of multiplication by a periodic inx matrixV(x). We study spectral properties ofH in the high energy region. Asymptotic formulae for Bloch eigenvalues and the corresponding spectral projections are constructed. The Bethe-Sommerfeld conjecture, stating that the spectrum ofH can have only a finite number of gaps, is proved.     

Existence and global attractivity of periodic solution for an impulsive delay differential equation with Allee effect

Sufficient conditions are obtained for the existence and global attractivity of positive periodic solution of an impulsive delay differential equation with Allee effect. The results of this paper improve and generalize noticeably the known theorems in the literature.     

Floquet theory for left-definite Sturm-Liouville problems

We determine the essential spectrum of left-definite Sturm-Liouville problems with periodic coefficients and a weight function which changes sign.     

Multiple periodic solutions for second order systems with changing sign potential

This paper deals with the multiplicity of solutions of a second order nonautonomous system. We extend a previous result of the author relaxing the assumptions on the sign of the potential.     

Bifurcation and stability analysis of a neural network model with distributed delays

In this paper, the dynamics of a generalized two-neuron model with self-connections and distributed delays are investigated, together with the stability of the equilibrium. In particular, the conditions under which the Hopf bifurcation occurs at the equilibrium are obtained for the weak kernel. This means that a family of periodic solutions bifurcates from the equilibrium when the bifurcation parameter exceeds a critical value. Explicit algorithms for determining the stability of the bifurcating periodic solutions and the direction of the Hopf bifurcation are derived by using the theory of normal form and center manifold [20]. Some numerical simulations are given to illustrate the effectiveness of the results found. The obtained results are new and they complement previously known results.This work was supported by the National Natural Science Foundation of China under Grants 60574043 and 60373067, the Natural Science Foundation of Jiangsu Province, China under Grants BK2003053.     

具有无穷时滞中立型泛函积分微分方程周期解的存在性

本文讨论具有无穷时滞中立型泛函积分微分方程ddtx(t)-∫t-∞B(t,s)x(s)ds=A(t,x(t))x(t) ∫t-∞C(t,s)x(s)ds ∑i=l1gi(t,x(t-τi(t)))的周期解问题.通过巧妙的构造算子,利用线性系统的指数二分性和Kras-noselskii不动点定理得到了周期解的存在性.我们的结果推广了相关文献的主要结果.