参外联合激励下的簇发振荡及其机理

当周期激励频率远小于系统固有频率时，会存在快慢耦合效应，与单项激励不同，参外联合激励不仅会导致快子系统平衡曲线和分岔行为的复杂化，也会产生一些特殊的非线性现象，为此，本文以两耦合Hodgkin-Huxley细胞模型为例，引入周期参外联合激励，探讨在频域不同尺度耦合时该系统的簇发振荡的特点及其分岔机制．通过建立相应的快慢子系统，得到慢变参数变化下的快子系统的各种分岔模式以及相应的分岔行为，结合转换相图，揭示耦合系统随激励幅值变化时的动力学行为及其机理．研究表明，在激励幅值较小时，系统表现为概周期振荡，两频率分别近似于快子系统平衡曲线由Hopf分岔引起的两稳定极限环的振荡频率．概周期解随激励幅值的增加进入簇发振荡，导致这些簇发振荡的主要原因是在慢变参数变化的部分区间内，存在唯一稳定的平衡曲线，使得系统的轨迹逐渐趋向该平衡曲线，产生沉寂态，并随着慢变参数的变化，由分岔进入激发态．同时，快子系统中参与簇发振荡的稳定吸引子随激励幅值的变化也会不同，导致不同形式的簇发振荡．另外，与单项激励下的情形不同，联合激励时快子系统的部分稳定吸引子掩埋在其它稳定吸引子内，从而失去对簇发振荡的影响．     

Complex dynamics in the Leslie–Gower type of the food chain system with multiple delays

In this paper, we present a Leslie–Gower type of food chain system composed of three species, which are resource, consumer, and predator, respectively. The digestion time delays corresponding to consumer-eat-resource and predator-eat-consumer are introduced for more realistic consideration. It is called the resource digestion delay (RDD) and consumer digestion delay (CDD) for simplicity. Analyzing the corresponding characteristic equation, the stabilities of the boundary and interior equilibrium points are studied. The food chain system exhibits the species coexistence for the small values of digestion delays. Large RDD/CDD may destabilize the species coexistence and induce the system dynamic into recurrent bloom or system collapse. Further, the present of multiple delays can control species population into the stable coexistence. To investigate the effect of time delays on the recurrent bloom of species population, the Hopf bifurcation and periodic solution are investigated in detail in terms of the central manifold reduction and normal form method. Finally, numerical simulations are performed to display some complex dynamics, which include multiple periodic solution and chaos motion for the different values of system parameters. The system dynamic behavior evolves into the chaos motion by employing the period-doubling bifurcation.     

Spatial Patterns for reaction-diffusion systems with conditions described by inclusions

We consider a reaction-diffusion system of the activator-inhibitor type with boundary conditions given by inclusions. We show that there exists a bifurcation point at which stationary but spatially nonconstant solutions (spatial patterns) bifurcate from the branch of trivial solutions. This bifurcation point lies in the domain of stability of the trivial solution to the same system with Dirichlet and Neumann boundary conditions, where a bifurcation of this classical problem is excluded.     

The Chebyshev Hyperplane Optimization Problem

We consider the following problem. Given a finite set of pointsy ^j in we want to determine a hyperplane H such that the maximum Euclidean distance betweenH and the pointsy ^j is minimized. This problem(CHOP) is a non-convex optimization problem with a special structure. Forexample, all local minima can be shown to be strongly unique. We present agenericity analysis of the problem. Two different global optimizationapproaches are considered for solving (CHOP). The first is a Lipschitzoptimization method; the other a cutting plane method for concaveoptimization. The local structure of the problem is elucidated by analysingthe relation between (CHOP) and certain associated linear optimizationproblems. We report on numerical experiments.     

A Numerical Comparison of Some Modified Controlled Random Search Algorithms

In this paper we propose a new version of the Controlled Random Search(CRS) algorithm of Price. The new algorithmhas been tested on thirteen global optimization test problems. Numericalexperiments indicate that the resulting algorithm performs considerablybetter than the earlier versions of the CRS algorithms. The algorithm,therefore, could offer a reasonable alternative to many currently availablestochastic algorithms, especially for problems requiring direct searchtype methods. Also a classification of the CRS algorithms is made based onglobal technique – local technique and the relative performance ofclasses is numerically explored.     

A Global Optimization Algorithm for Multivariate Functions with Lipschitzian First Derivatives

In this paper we propose a new multi-dimensional methodto solve unconstrained global optimization problems with Lipschitzianfirst derivatives. The method is based on apartition scheme that subdivides the search domain into a set of hypercubesin the course of optimization. This partitioning is regulated by thedecision rule that provides evaluation of the "importance"of each generated hypercube and selection of some partition element to performthe next iteration. Sufficient conditions of global convergence for the newmethod are investigated. Results of numerical experiments are alsopresented.     

Crossed Modules and Quantum Groups in Braided Categories

Let A be a Hopf algebra in a braided category . Crossed modules over A are introduced and studied as objects with both module and comodule structures satisfying a compatibility condition. The category of crossed modules is braided and is a concrete realization of a known general construction of a double or center of a monoidal category. For a quantum braided group the corresponding braided category of modules is identified with a full subcategory in . The connection with cross products is discussed and a suitable cross product in the class of quantum braided groups is built. Majid–Radford theorem, which gives equivalent conditions for an ordinary Hopf algebra to be such a cross product, is generalized to the braided category. Majid's bosonization theorem is also generalized.