Stability of Difference Systems with Finite Delay

In this paper, the authors establish some theorems that can ascertain the zero solutions of systemsx(n+1)=f(n,x n)(1)are uniformly stable,asymptotically stable or uniformly asymptotically stable. In the obtained theorems, ΔV is not required to be always negative, where ΔV(n,x n)≡V(n+1,x(n+1)) -V(n,x(n))=V(n+1,f(n,x n))-V(n,x(n)), especially, in Theorem 1, ΔV may be even positive, which greatly improve the known results and are more convenient to use.

EXISTENCE OF ALMOST PERIODIC SOLUTIONS FOR DIFFERENCE SYSTEMS

In this work, we first define the notions of almost periodic sequences, asymptotically almost periodic sequences, as well as uniformly almost periodic sequences,and reveal their basic properties. Then for the almost periodic difference systems of general form we establish the criteria of existence for almost periodic solutions.Especially, several existence theorems are proved in terms of discrete Liapunov functions.

BOUNDEDNESS OF INFINITE DELAY DIFFERENCE SYSTEMS IN TERMS OF TWO MEASURES

In this paper, by means of Liapunov functionals, we establish the criteria of uniform boundedness and uniform ultimate boundedness for infinite delay difference systems in terms of two measures. Moreover, an example is given to illustrate the obtained results.

ON WELL-CONDITIONED BOUNDARY VALUE PROBLEMS FOR SYSTEMS OF SECOND ORDER DIFFERENCE EQUATIONS

In this paper well-conditioning of boundary value problems for systems of second order difference equa-tions is studied.First,a sufficient condition for the existence of a unique bounded solution (for large enough number of steps) of an associated homogeneous system is given.Finally,a sufficient condition for well-condi-tioning,intrinsically related to the problem data is proposed.

Effect of Long-Term Application of K Fertilizer and Wheat Straw to Soil on Crop Yield and Soil K Under Different Planting Systems

Effect of application of K fertilizer and wheat straw to soil on crop yield and status of soil K in the plough layer under different planting systems was studied. The experiments on long-term application of K fertilizer and wheat straw to soil in Hebei fluvo aquic soil and Shanxi brown soil in northern China were begun in 1992. The results showed that K fertilizer and straw could improve the yields of wheat and maize with the order of NPK ＋ St 〉 NPK 〉 NP ＋ St 〉 NP, and treatment of K fertilizer made a significant difference to NP, and the efficiency of K fertilizer in maize was higher than in wheat under rotation system of Hebei. In contrast with Shanxi, the wastage of soil potassium was a more serious issue in the rotation system in Hebei, only treatment of NPK ＋ St showed a surplus of potassium and the others showed a wane. K fertilizer and straw could improve the content of water-soluble K, nonspecifically adsorbed K, non-exchangeable K, mineral K, and total K in contrast to NP; however, K fertilizer and straw reduce the proportion of mineral K and improve proportion of other forms of potassium in the two locating sites. Compared with the beginning of orientation, temporal variability character of soil K content and proportion showed a difference between the two soil types; furthermore, there was a decrease in the content of mineral K and total K simultaneously in the two locating sites. As a whole, the effect of K fertilizer applied to soil directly excelled to wheat straw to soil. Wheat straw to soil was an effective measure to complement potassium to increase crop yield and retard the decrease of soil K.

Synchronization of chaotic systems with different orders

A controller is designed to realize the synchronization between chaotic systems with different orders. The structure of the controller, the error equations and the Lyapunov functions are determined based on stability theory. Hyperchaotic Chen system and Rossler system are taken for example to demonstrate the method to be effective and feasible. Simulation results show that all the state wriables of Rossler system can be synchronized with those of hyperchaotic Chen system by using only one controller, and the error signals approach zero smoothly and quickly.

Adaptive generalized projective synchronization of two different chaotic systems with unknown parameters

This paper presents a general method of the generalized projective synchronization and the parameter identification between two different chaotic systems with unknown parameters. This approach is based on Lyapunov stability theory, and employs a combination of feedback control and adaptive control. With this method one can achieve the generalized projective synchronization and realize the parameter identifications between almost all chaotic （hyperchaotic） systems with unknown parameters. Numerical simulations results are presented to demonstrate the effectiveness of the method.

Generalized Synchronization Between Different Fractional-Order Chaotic Systems

In this paper,using scalar feedback controller and stability theory of fractional-order systems,a gener-alized synchronization method for different fractional-order chaotic systems is established.Simulation results show theeffectiveness of the theoretical results.

On the Behavior of Solutions of the System of Rational Difference Equations x_(n+1)=x_(n-1)/y_(n)c_(n-1)-1,y_(n+1)=y_(n-1)/x_(n)y_(n-1)-1,z_(n+1)=x_(n)/y_(n)z_(n-1)

In this paper, we investigate the solutions of the system of difference equations xn+1=xn-1/ynxn-1-1,yn+1=yn-1/xnyn-1-1,zn+1=xn/ynzn-1 where x0,x-1,y0,y-1,z0,z-1∈R.

DIFFERENCE DISCRETE SYSTEM OF EULER-BEAM WITH ARBITRARY SUPPORTS AND SIGN-OSCILLATORY PROPERTY OF STIFFNESS MATRICES

The difference discrete system of Euler-beam with arbitrary supports was constructed by using the two order central difference formulas. This system is equivalent to the spring-mass-rigidrod model. By using the theory of oscillatory matrix, the signoscillatory property of stiffness matrices of this system was proved, and the necessary and sufficient condition for the system to be positive was obtained completely.

Realization of generalized synchronization between different chaotic systems via scalar controller

In this paper, a very simple generalized synchronization method between different chaotic systems is presented. Only a scalar controller is used in this method. The method of obtaining the scalar controller from chaotic systems is established. The sufficient and necessary condition of generalized synchronization is obtained from a rigorous theory, and the sufficient and necessary condition of generalized synchronization is irrelative to chaotic system itself. Theoretical analyses and simulation results show that the method established in this paper is effective.

Synchronization of N different coupled chaotic systems with ring and chain connections

Synchronization of N different coupled chaotic systems with ring and chain connections is investigated. The New system, the Chen system, the Lii system, the Lorenz system, and the Rossler system are used as examples in verifying effectiveness of the method. Based on the Lyapunov stability theory, the form of the controller is designed and the area of the coupling coefficients is determined. Simulations indicate that global synchronization of the N different chaotic systems can be realized by choosing appropriate coupling coefficients by using the controller.

Synchronization between fractional-order chaotic systems and integer orders chaotic systems (fractional-order chaotic systems)

Based on the idea of tracking control and stability theory of fractional-order systems, a controller is designed to synchronize the fractional-order chaotic system with chaotic systems of integer orders, and synchronize the different fractional-order chaotic systems. The proposed synchronization approach in this paper shows that the synchronization between fractional-order chaotic systems and chaotic systems of integer orders can be achieved, and the synchronization between different fractional-order chaotic systems can also be realized. Numerical experiments show that the present method works very well.

EXISTENCE OF PERIODIC SOLUTIONS OF SYSTEM OF DIFFERENCE EQUATIONS

This paper studies the nonautonomous nonlinear system of difference equations △x（n） A（n）x（n） ＋ f（n,x（n））, n∈Z ,（＊）where x（n）∈R^N,A（n） = （aij（n））N×N is an N×N matrix, with aij∈C（R,R） for i,j = 1,2,3 ,N, and f = （f1,f2,... ,fN）^T ∈C（R×R^N,R^N）, satisfying A（t＋） = A（t）, f（t＋ω,z） = f（t, z） for any t∈R, （t, z） ∈R× RN and ∈is a positive integer. Sufficient conditions for the existence of ω-periodic solutions to equations （＊） are obtained.

On the Periodicity of Solutions of the System of Rational Difference Equations X_(n+1)=X(n-1)+Y_(n)/Y_(n)X_(n-1)-1,Y_(n-1)=Y_(n-1)+X_(n)/X_(n)Y_(n-1)-1

In this paper, we have investigated the periodicity of the solutions of the system of difference equations , where .

On the Development of Meso-Scale Heavy Rain Parcels in China

Heavy rains occur in China frequently, which often bring us floods and serious disasters in the summer half-year. The meso-scale heavy rain parcels (MHRP) in the mid-latitude are usually developed in following cases:I.By the approaching, meeting and / or overlapping of different weather systems, when two or more different rainfall systems are getting to conjugate, some MHRPs could be developed, such as: 1) a new cold/warm front or squall line approaches an old front or squall, even when the old one is somewhat decrepit; 2) at the places where two or more synoptic systems with different characteristics are meeting together, such as the meeting of tropical cyclone with the cold airs coming from the mid- and / or high-latitudes, or the low latitude vortex meeting with the westerly trough; 3) at the intersections of some different weather systems, such as the intersection of drylines, squall lines or fronts moving from different directions; and 4) by the overlapping of rainfall parcels produced continuously from a meso-generation centre.II.Resonance Effect and Tibetan Plateau Influence are two reasons why high frequency of heavy and torrential rains arround the meiyu front is discussed also.

Origin, Alternative Expressions of Newcomb-Benford Law and Deviations of Digit Frequencies

The Newcomb-Benford law, which describes the uneven distribution of the frequencies of digits in data sets, is by its nature probabilistic. Therefore, the main goal of this work was to derive formulas for the permissible deviations of the above frequencies (confidence intervals). For this, a previously developed method was used, which represents an alternative to the traditional approach. The alternative formula expressing the Newcomb-Benford law is re-derived. As shown in general form, it is numerically equivalent to the original Benford formula. The obtained formulas for confidence intervals for Benford’s law are shown to be useful for checking arrays of numerical data. Consequences for numeral systems with different bases are analyzed. The alternative expression for the frequencies of digits at the second decimal place is deduced together with the corresponding deviation intervals. In general, in this approach, all the presented results are a consequence of the positionality property of digital systems such as decimal, binary, etc.

THE EXISTENCE OF POSITIVE PERIODIC SOLUTION FOR A LOTKA-VOLTERRA DIFFERENCE SYSTEM

A general periodic Lotka-Volterra difference system is studied in this paper. A set of easily verifiable sufficient conditions that guarantee the existence of the positive periodic solutions is obtained.

Diference Systems of Sets Based on Cosets Partitions

Difference systems of sets （DSS） are combinatorial configurations that arise in connection with code synchronization. This paper proposes a new method to construct DSSs, which uses known DSSs to partition some of the cosets of Zv relative to subgroup of order k, where v = km is a composite number. As applications, we obtain some new optimal DSSs.