The Study On Practical Stability Of A Kind Of Difference Equations

Dealing with safe operation of different actual system, scientists establish different models(including differential equation models, and so on), and use the methods of mathematic to research conditions which insures the stability of the system, and make sure security and stability of system. The mathematical properties stable operation of a financial system reflected is Lyapunov stability or practical stability of the solution of mathematical models. It is well known that Lyapunov stability theory describes that compared with the ideal status, the operational status of the entire dynamic system change small when the initial value of system disturbs relatively small .But in applications it is difficult to make sure relatively small disturbance of the initial value of system. Practical stability requires the following condition. Comparing with the ideal status, the operational status of the system changes in the permit range while the initial perturbations are in the certain range. Practical stability theory is the theory of quantitative analysis of the system, it well solved the contradiction that the qualitative description of Lyapunov stability concept is not in accordance with the quantitative requirements in practice. So, the study of practical stability of difference equations not only enrich the theory of differential equations, but also have practical value.However, so far, the results on the practical stability of difference equations are relatively few at home and abroad.The major tasks of this paper are following: Firstly, this paper introduces the concepts of Lyapunov stability and practical stability of the systems, and it illustrates the research about the practical stability of the systems has very important significance by comparing the concepts. Secondly, a series of non-autonomous difference equations and autonomous difference equations were studied by Lyapunov function, monotonous criterion and k-functions, and we obtain some conditions to make the system practical stability and strong practical stability. These conditions are only relative with the system parameters and easily verified it is practical in actual project.Thirdly, some difference equations were studied by inequality principle, monotonous criterion and matrix analysis methods, etc. and obtained the conditions to ensure the system practical stability. The conditions are very convenient. Fourth, some of the conclusions are applied to the research of the practical stability of the financial markets, the practicability and feasibility about the results were illustrated by practical examples.