Some Results Of Stability For Nonlinear Impulsive Dynamic Systems On Time Scales

In this paper,we study the stability properties of the two nonlinear impulsive dynamic systems on time scales as follow:(1) impulsive perturbed dynamic systems on time scalesdynamic systems on time scales corresponding to the systems (â… )where(1)f(t,x)=F(t,x)+R(t,x),R(t,x) is the perturbation of x^â–³(t)=f{t,x);(2)x^â–³(t)is the delta derivative of x(t) at t.(2) impulsive hybrid dynamic systems on time scaleswhere x^â–³(t)is the delta derivative of x(t) at t.We get the results of stability in terms of two measures for systerm(â… ),the results of stability and strict stability in terms of two measures for systerm(â…¢). Examples are also discussed to illustrate the theorems, repectively.Dynamic systems on time scales, which can unify the continuous and discrete systems,have gained more and more important applications in each field of modern technology.Because of the extensive applications in economics, biology and medicine, many scholars have paid attention to this theory gradually. For example, it can be modeled insect populations that are continuous in some season by differential equations,however in other season the community have been in the state of the ovums' hatching or dormancy. So it only can be modeled insect populations by difference equations at this time. This kind of issue can be subject to the dynamic systems on time scales.At the same time, the impulsive phenomena widely exists in practical problems of many fields in modern technology, so the study of impulsive dynamic systems on time scales has gained vital practical significance and applied background.In chapter one, firstly, we introduce the basic concepts of time scale calculus. Secondly,we expound the basic ideas of variational Lyapunov function method on time scales with a new right generalized derivative on time scales, and we establish a new comparison theorem by employing this variational Lyapunov function method. At last, by using the comparison theorem, we study the stability of impulsive perturbed dynamic systems on time scales (â… ), and we obtain some results, such as (h₀,h)-stability, (ho,h)-asymptotic stability, (h₀,h)-practical stability, (h₀,h)-eventual stability and so on. An example is given to illustrate the application of the theorems. In particular, the theorems in this chapter are all the local theorems.In chapter two, by employing Lyapunov direct method, we study the stability of impulsive hybrid dynamic systems (â…¢) with a new right generalized derivative on time scales, and we also obtain some results, such as (h₀,h)stability, (h₀,h)-asymptotic stability,(h₀,h)-strict stability, (h₀,h)-strict asymptotic stability. An example is also discussed to illustrate the application of the theorems. In particular, we impose conditions everywhere on V(t,x)in this chapter, that is, the theorems in this chapter are all the global theorems.