共振情形下拟周期Hamiltonian系统的线性化

受Rsmann的工作的启发,本文考虑实解析拟周期Hamiltonian系统{■=i(ρz+(f)/(z)(z,■,θ))■=ω,其中P≥0,z∈C,θ∈T^d且f=O_3(|z|).假设ω是Diophantine向量,且P与ω共振.本文证明如果系统是形式可线性化的,则它也是解析可线性化的.从而该系统的不动点是Lyapunov稳定的.

非线性拟周期方程的Floquet理论

主要研究由如下的微分方程所组成的自治系统: x=Ax+f(x,θ),θ=ω,其中θ∈R^m,ω=(ω_1,…,ω_m)∈R^m,x∈R^n,A∈R^(n×n)是一个双曲的常矩阵,f是一个关于两个变元都是C~∞的函数,并且关于变元θ的各个分量都是2π周期的,还满足当x→0时,f=O(‖x‖~2).通过研究原系统的正规形,证明了在适当条件下原系统可以变为自治系统x=Ax+g(x),θ=ω.另外,证明过程自然蕴含了Chen定理在拟周期情形下的推广.

Optimal quasi-periodic maintenance policies for two-unit series system

To investigate the effects of various random factors on the preventive maintenance （PM） decision-making of one type of two-unit series system, an optimal quasi-periodic PM policy is introduced. Assume that PM is perfect for unit 1 and only mechanical service for unit 2 in the model. PM activity is randomly performed according to a dynamic PM plan distributed in each implementation period. A replacement is determined based on the competing results of unplanned and planned replacements. The unplanned replacement is trigged by a catastrophic failure of unit 2, and the planned replacement is executed when the PM number reaches the threshold N. Through modeling and analysis, a solution algorithm for an optimal implementation period and the PM number is given, and optimal process and parametric sensitivity are provided by a numerical example. Results show that the implementation period should be decreased as soon as possible under the condition of meeting the needs of practice, which can increase mean operating time and decrease the long-run cost rate.

KAM tori for generalized Boussinesq equation

One-dimensional generalized Boussinesq equation u tt-u xx＋（f（u）＋u xx）xx=0.with periodic boundary condition is considered, where f（u） = u3. First, the above equation is written as a Hamiltonian system, and then by choosing the eigenfunctions of the linear operator as bases, the Hamiltonian system in the coordinates is expressed. Because of the intricate resonance between the tangential frequencies and normal frequencies, some quasi-periodic solutions with special structures are considered. Secondly, the regularity of the Hamiltonian vector field is verified and then the fourth-order terms are normalized. By the Birkhoff normal form, the non- degeneracy and non-resonance conditions are obtained. Applying the infinite dimensional Kolmogorov-Arnold-Moser （KAM） theorem, the existence of finite dimensional invariant tori for the equivalent Hamiltonian system is proved. Hence many small-amplitude quasi-periodic solutions for the above equation are obtained.

The Application of BBCH Scale for Codification and Illustrations of the Floral Stages of Caribbean Fine Cacao Theobroma cacao L.

Cocoa （Theobroma cacao L.） is the major input into the chocolate manufacturing industry, particularly the fine or flavor of the indigenous variety Trinitario, which is the hybrid of Crillo and Forastero. One key problem encountered by researchers is the inability to differentiate the various phenological stages of the cocoa reproductive cycle and to assess or predict the yield. This study sets out to map the various stages of the reproductive biology of cacao and to develop a set of photographic illustrations that appropriately describe the floral biology and development stages of Caribbean free cocoa as a tool for agronomist and farmers. It used a modified version of the codification of the ＂extended BBCH＂ scale （Biologische Bundesantalt, Bundessortenamt and Chemische Industrie, Germany）, which is a numerical system that differentiates between principal, secondary and tertiary growth stages. The phases and all floral stages to fruit development are described and illustrated in a set of photographs, which can now be used for training and referencing purposes. This is the first attempt to undertake and document this study.

BOUNDEDNESS OF SOLUTIONS FOR SUPERLINEAR REVERSIBLE SYSTEMS

This paper is concerned with the boundedness of solutions for second order differential equations x + f(x, t)x + g(x, t) = 0, which are neither dissipative nor conservative, and where the functions f and g are odd in x and even in t, which are 1-periodic in t, and the function g satisfies g(x,t/x+, as|x| - +. Using the KAM theory for reversible systems, the author proves the existence of invariant tori and thus the boundedness of all the solutions and the existence of quasiperiodic solutions and subharmonic solutions.

Stochastic dynamic simulation of the 100-kyr cycles in climate system

Based upon the stochastic resonance theory,the formation mechanism of 100-kyr cycles in climate system is numerically studied in the perspective of stochastic dynamics.In this study,firstly we combine the idealized albedo model with the geological evidence and observation in climate system to construct a new albedo model.Secondly,a bistable nonlinear system is constructed by introducing the albedo model into zero-dimensional energy balance model.Finally,based on this new system,with the solar radiation cycles and stochastic perturbation simultaneously taken into account,the variation of 100-kyr cycles is analyzed by numerical simulations.The results show that,when the noise intensity reaches a certain value,the stochastic resonance can be triggered.However,the noise intensity in this level does not exist in the actual climate system.In order to explain the formation mechanism of 100-kyr glacial-interglacial cycles forced by the weak solar radiation cycles,besides the solar radiation stochastic perturbation,the stochastic dynamic effects of the other "non-solar" radiation stochastic perturbation in the climate change processes should also be considered.The stochastic dynamic simulations taking the two types of stochastic perturbation into consideration show that,when the two types of appropriately observable stochastic perturbation are introduced,the stochastic resonance also can be generated.In this situation,the contribution rate of solar radiation stochastic perturbation is about 38%,which proves the importance of solar radiation stochastic perturbation in the formation of 100-kyr climate cycles.

THE EXISTENCE AND ASYMPTOTIC BEHAVIOR OF PERIODIC SOLUTIONS TO A QUASILINEAR PARABOLIC SYSTEM

In this paper, we investigate the existence, stability and global attractivity of T-periodic solutions for a class of quasilinear parabolic equations under Robin boundary conditions. We obtain that periodic solutions exist if the inter-specific competition rates are weak. The numerical simulations are also presented to illustrate our result.