Exponential Dichotomy, Exponential Trichotomy And Invariant Manifolds On Time Scales

The theory of time scales was first introduced by Germanic mathematician Hilger in 1988. At first, the main purpose of introducing this concept is to unify discrete and continuous dynamic systems. As the research deepened, it was discovered that the theoretical study for time scales also enable us to understanding the nature difference of discrete and continuous analysis. The theory of time scales is very usefull in practice. It can better describe the actual problems. Such as population model [19], the economic model [2,3,104] and so on.In 2001, Potsche had discussed the concept of exponential dichotomy. Exponen-tial dichotomy is very important for the progress of qualitative theory. So it is very necessary to study exponential dichotomy. In this paper, we give in-depth study of exponential dichotomy on time scales.In order to make our paper more independent, in the first chapter, we firstly give some useful definitions and properties on time scales. The most of the contents about time scale which we introduce here come from the book "Dynamic Equations on Time Scales:An Introduction with Applications" which edited by Bohner and Peterson. This book is not only a primer for readers who are interested in time scales but also a tool for the theoretical study of time scales. More about the contents of time scale we refer the readers to this book. In Section 1.3 we recall the notions of exponential dichotomy, exponential trichotomy and invariant manifolds on time scales. Next, in Section 2.1, we give the concepts of strong exponential dichotomy and strict quadratic Lyapunov function. And then using these concepts we give the following two main theorems. Theorem 1 If equation xΔ=A(t)x has a strong exponential dichotomy on time scalesΤ,then it has a strict quadratic Lyapunov function.Theorem 2 If equation xΔ=A(t)x has a strict quadratic Lyapunov function and satisfies then equation xΔ= A(t)x has an exponential dichotomy.As an application of Section 2.2,we study the nonlinear instability of zero solution of equation xΔ=A(t)x+f(t,x)in Section 2.3.Specifically,we give the following theorem.Theorem 3 Assume that equation xΔ=A(t)x admits a strong exponential dichotomy. Consider the nonlinear equation xΔ=A(t)x+f(t,x).Here,f(t,x)satisfies that f(t,0)=0 for every t∈Τand f(t,x)is Lipschitz continuous with respect to x,i.e. there exists R>0 such thatAssume also that Q(σ(t))f(t,x)=Q(σ(t))f(t,Q(t)x),where Q(t)=I-P(t).If R is sufficiently small,then there exist constants N＞0 and d>0 such that whenever z(τ)∈Fτu.In the third chapter,we introduce the concepts 0f strict exponential trichotomy and(λ,μ)-quadratic Lyapunov function on time scales. Furthermore, we study the relationship between them and state the following theorems.Theorem 4 Suppose that equation xΔ=A(t)x has two(λ,μ)-quadratic Lyapunov functions V(t,x)with(r1,r2)and W(t,x)with(l1,l2) satisfying r10 is sufficiently small. Now, we have the following theorem.Theorem 6 Suppose that equation xΔ= A(t)x has a strict exponential trichotomy, then whenδ>0 is sufficiently small, we have that the linear perturbed equation xΔ= A(t)x+B(t)x also has an exponential trichotomy.In the first three chapters we have made some research about exponential di-chotomy and exponential trichotomy on a single time scale. Naturally, we will ask:if a dynamic system has an exponential dichotomy or trichotomy on a given time scale, then on the other time scale, when the time scale is sufficiently close to the given time scale, is the character still valid? In the fourth chapter we answer the question. Suppose that A(t) is an n×n matrix-valued function on time scaleΤ∪Τ'and that A(t) is rd-continuous and regressive. Let f(t,0)= 0 for any t∈Τ∪Τ'and f(t, x) be rd-continuous with respect to t and Lipschitz continuous with respect to x uniformly for t∈Τ∪Τ', i.e. there is a constant L> 0 such that for all x, y∈Rn and t∈Τ. We have the following theorem.Theorem 7 Suppose that equation xΔ= A(t)x has an exponential dichotomy on time scaleΤ. Then equation xΔ= A(t)x admits an exponential dichotomy on time scaleΤ' wheneverΤandΤ' are sufficiently close. Let P(t) and P'(t') be the projections associated to the exponential dichotomy on time scaleΤandΤ', respectively. Then Let B(t)be an n×n matrix-valued function on time scaleΤ'satisfying‖B(t)‖≤β, whereβis a constant satisfyingβ> 0.By Theorem 4.2 in[108]and the above theorem, we have the following corollary.Corollary 1 Suppose that equation xΔ=A(t)x has an exponential dichotomy on time scaleΤ.thenΤandΤ' are sufficiently close andβis sufficiently small,the linear perturbed equation xΔ=A(t)x+B(t)x admits an exponential dichotomy on time scale T'.For the exponential trichotomy we have the following theorem.Theorem 8 Suppose that equation xΔ=A(t)x has an exponential trichotomy on time scaleΤ.Then equation xΔ=A(t)x admits an exponential trichotomy on time scaleΤ' wheneverΤandΤ' are sufficiently close. Let Pi(t) and P'i(t') for i=1,2,3 be the projections associated to the exponential trichotomy on time scaleΤandΤ',respectively. Then‖P'i(t')-Pi(t)‖→0 as｜t-t'｜→0 for t'∈Τ',t∈Τ.In Section 2 of Chapter 4 we study the stable manifold theorem on time scales.We always suppose that the time scaleΤis bi-infinite,i.e.infΤ=-∞and supΤ=+∞. Firstly,we give the existence of the stable manifold theorem.Suppose that x(t,τ,x0)is a solution of equation xΔ=A(t)x+f(t,x)satisfying x(τ,τ,x0)=x0 for t,τ∈Τ. For givenτ∈Τ,set Mτs:=｛x0∈Rn｜supt≥τ,t∈Τ‖x(t,τ,x0)‖＜∞｝and Mτu:=｛x0∈Rn｜supt≤τ∈Τ‖x(t,τ,x0)‖<∞｝.We get the following conclusion.Theorem 9 Suppose that the linear equation xΔ= A(t)x has an exponential di-chotomy on time scaleΤ. If DL/b -DL/a＜1,then｛Mτs｝τ∈Τis a stable manifold and｛Mτu｝τ∈Τis an unstable manifold for the equation xΔ=A(t)x+f(t,x).Next,we prove that the stable(unstable)manifolds of equation xΔ=A(t)x+ f(t,x)on time scale T coincides with that on nearby time scales under some assump-tions. In the remaining of this section we suppose the following hypotheses hold. (M1)There is a constant c with 00 there exists a constant NT>0 such that(M3)The solutions of equation xΔ=A(t)x+f(t,x)are uniformly continuous in initial data and the time scale at time scaleΤ,i.e.for anyε>0,R>0 and L'>0, there existsδε,R,L',>0 such that when(τ',t')∈Τ'2∩[T1,T2]2,(τ,t)∈Τ2∩[T1,T2]2 satisfying and for any x0∈Rn,‖x0‖≤R,we have where [T1,T2]is any interval with length L' We get the following theorem.Theorem 10 Suppose that equation xΔ=A(t)x has an exponential dichotomy on time scaleΤand that(M1)-(M3) hold. IfΤandΤ' are sufficiently close and max｛DL/c-a+ DL/b-c,DL/b-DL/a｝<1,then｛Mτs｝τ∈Τ,｛Mτu｝τ∈Τand｛Mτ's｝τ'∈Τ',｛Mτ'u｝τ'∈Τ' are stable and unstable manifolds for the equation xΔ=A(t)x+f(t,x)on time scaleΤand its nearby time scaleΤ',respectively. Furthermore Mτ's=Mτs and Mτ'u=Mτu forτ'∈Τ',τ∈Τsatisfying that｜τ-τ'｜is suffciently small.In Chapter 5,we discuss the roughness of exponential dichotomy on measure chain. Let T be a measure chain,X be a Banach space and L(X)be the set of all the linear mappings from X to X.The following theorem holds. Theorem 11 Assume that equation xΔ=A(t)x has exponential dichotomy on mea-sure chainΤ.That means that there exist projection mappings P:Τ→L(X)and constant K≥1,a,b∈Crd+R(Τ,R):=｛a∈Crd(Τ,R):1+μ*(t)a(t)>0,t∈Τ｝satisfying aΔb,i.e. inft∈Τ[b(t)-a(t)]>0 such that P(t)T(t,s)=T(t,s)P(s), and hold.Then when supt∈Τ｜a(t)-b(t)｜<∞and B(t) in equation xΔ=A(t)x+B(t)x satisfies‖B(t)‖<δfor all t∈Τandδis sufficiently small, equation xΔ=A(t)x+ B(t)x also has an exponential dichotomy.These complete our paper.