# Lyapunov Function and Exponential Trichotomy on Time Scales

Authors: Ji Zhang
Journal: Discrete Dynamics in Nature and Society
 (2011)

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## Abstract

We study the relation between Lyapunov function and
exponential trichotomy for the linear equationxΔ=A(t)x on time scales. Furthermore,
as an application of these results, we give the roughness of exponential trichotomy on
time scales.


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## Body

## 1. Introduction

Exponential trichotomy is important for center manifolds theorems and bifurcation theorems. When people analyze the asymptotic behavior of dynamical systems, exponential trichotomy is a powerful tool. The conception of trichotomy was first introduced by Sacker and Sell [1]. They described SS-trichotomy for linear differential systems by linear skew-product flows. Furthermore, Elaydi and Hájek [2, 3] gave the notions of exponential trichotomy for differential systems and for nonlinear differential systems, respectively. These notions are stronger notions than SS-trichotomy. In 1991, Papaschinopoulos [4] discussed the exponential trichotomy for linear difference equations. And in 1999 Hong and his partners [5, 6] studied the relationship between exponential trichotomy and the ergodic solutions of linear differential and difference equations with ergodic perturbations. Recently, Barreira and Valls [7, 8] gave the conception of nonuniform exponential trichotomy. From their papers, we can see that the exponential trichotomy studied before is just a special case of the nonuniform exponential trichotomy. For more information about exponential trichotomy we refer the reader to papers [9–14].Many phenomena in nature cannot be entirely described by discrete system, or by continuous system, such as insect population model, the large population in the summer, the number increases, a continuous function can be shown. And in the winter the insects freeze to death or all sleep, their number reduces to zero, until the eggs hatch in the next spring, the number increases again, this process is a jump process. Therefore, this population model is a discontinuous jump function, it cannot be expressed by a single differential equation, or by a single difference equations. Is it possible to use a unified framework to represent the above population model? In 1988, Hilger [15] first introduced the theory of time scales. From then on, there are numerous works on this area (see [16–23]). Time scales provide a method to unify and generalize theories of continuous and discrete dynamical systems.In this paper, motivated by [8], we study the exponential trichotomy on time scales. We firstly introduce (λ,μ)-Lyapunov function on time scales. Then we study the relationship between exponential trichotomy and (λ,μ)-Lyapunov function on time scales. We obtain that the linear equation xΔ=A(t)x admits exponential trichotomy, if it has two (λ,μ)-Lyapunov functions with some property; conversely, the linear equation has two (λ,μ)-Lyapunov functions, if it admits strict exponential trichotomy. At last, by using these results we investigate the roughness of exponential trichotomy on time scales. Above all, our paper gives a way to unify the analysis of continuous and discrete exponential trichotomy.This paper is organized as follows. In Section2, we review some useful notions and basic properties on time scales. Our main results will be stated and proved in Section 3. Finally, in Section 4, we study the roughness of exponential trichotomy on time scales.


## 2. Preliminaries on Time Scales

In order to make our paper independent, some preliminary definitions and theories on time scales are listed below.Definition 2.1.
Let𝕋 be a time scale which is an arbitrary nonempty closed subset of the real numbers. The forward jump operator is defined by
(2.1)σ(t)∶=inf⁡⁡{s∈T:s>t},
while the backward jump operator is defined by
(2.2)ρ(t)∶=sup⁡⁡{s∈T:s<t},
for every t∈𝕋. If σ(t)=t, then t is called right-dense. And if ρ(t)=t, then t is called left-dense. Let μ(t)∶=σ(t)-t be the graininess function.For example, the set of real numbersℝ is a time scale with σ(t)=t and μ(t)=0 for t∈𝕋=ℝ and the set of integers ℤ is a time scale with σ(t)=t+1 and μ(t)=1 for t∈𝕋=ℤ.Definition 2.2.
A functionf:𝕋→ℝn is called rd-continuous if it is continuous at right-dense points in 𝕋 and left-sided limits exist at left dense points in 𝕋.We denote the set of rd-continuous functions byCrd(𝕋,ℝn).Definition 2.3.
We say that a functionf:𝕋→ℝn is differentiable at t∈𝕋, if for any ɛ>0, there exist 𝕋-neighborhood U of t and fΔ(t)∈ℝn such that for any s∈U one has
(2.3)‖f(σ(t))-f(s)-fΔ(t)(σ(t)-s)‖≤ɛ(σ(t)-s),
and fΔ(t) is called the derivative of f at t.Relate to the differential properties, iff and g is differentiable at t, then we have the following equalities:(2.4)f(σ(t))=f(t)+μ(t)fΔ(t),(2.5)(fg)Δ(t)=fΔ(t)g(t)+f(σ(t))gΔ(t).Definition 2.4.
We say that a functionp:𝕋→ℝ is regressive if 1+μ(t)p(t)≠0 for all t∈𝕋. Furthermore, if 1+μ(t)p(t)>0, p is called positive regressive. An n×n matrix valued function A(t) on time scale 𝕋 is called regressive if Id+μ(t)A(t) is invertible for all t∈𝕋.Leta,b be regressive. Define (a⊕b)(t)=a(t)+b(t)+μ(t)a(t)b(t) and (⊖a)(t)=-a(t)/(1+μ(t)a(t)) for all t∈𝕋. Then the regressive set is a Abelian group and it is not hard to verify that the following properties hold:(1)
a⊖a=0;(2)
⊖(⊖a)=a;(3)
a⊖b=(a-b)/(1+μ(t)b);(4)
⊖(a⊕b)=(⊖a)⊕(⊖b), where a,b are regressive.In order to make our paper intelligible, the exponential function on time scales which we will use in our paper is defined as following. For the more general conception of exponential function on time scales, please refer to [17].Definition 2.5.
Letp be positive regressive. We define the exponential function by
(2.6)ep(t,s)=exp⁡⁡{∫st1μ(τ)ln⁡⁡(1+μ(τ)p(τ))Δτ},
for all s,t∈𝕋. Here the integral is always understood in Lebesgue's sense and if f∈Crd(𝕋,ℝn), for any t∈𝕋 one has
(2.7)∫tσ(t)f(s)Δs=μ(t)f(t).From the definition of exponential function we have that ifa,b are positive regressive, then(1)
e0(t,s)=1 and ea(t,t)=1;(2)
ea(t,s)=e⊖a(s,t);(3)
ea(t,s)ea(s,r)=ea(t,r);(4)
ea(t,s)eb(t,s)=ea⊕b(t,s);(5)
[ea(c,t)]tΔ=-a(t)ea(c,σ(t)), where [ea(c,t)]tΔ stands for the delta derivative of ea(c,t) with respect to t.Lemma 2.6 (L'Hôpital's Rule).
Suppose thatf and g are differentiable on 𝕋. Then if g(t) satisfies g(t)>0, gΔ>0 and limt→+∞g(t)=∞, then the existence of limt→+∞fΔ(t)/gΔ(t) implies that limt→+∞f(t)/g(t) exists and limt→+∞f(t)/g(t)=limt→+∞fΔ(t)/gΔ(t).


## 3. Exponential Trichotomy and Lyapunov Function on Time Scales

From now on, we always suppose that𝕋 is a two-sides infinite time scale and the graininess function μ(t) is bounded, which means that there exists a M>0 such that 0≤μ(t)≤M. We consider the following linear equation(3.1)xΔ=A(t)x,
where A(t) is an n×n matrix valued function on time scale 𝕋, satisfying that A(t) is rd-continuous and regressive. To make it simple, in this paper we always require that A(t) is regressive. If A(t) is not regressive, from [20] we can know that the linear evolution operator T(t,τ) associated to (3.1) is not invertible and exists only for τ≤t which will make our paper complicated. Here, T(t,τ) is called the linear evolution operator if T(t,τ) satisfies the following conditions:  (1)
T(τ,τ)=Id;(2)
T(t,τ)T(τ,s)=T(t,s);(3)
the mapping(t,τ)→T(t,τ)x is continuous for any fixed x∈ℝn.Equation (3.1) is said to admit an exponential  trichotomy on time scale 𝕋 if there exist projections P(t), Q(t), R(t):ℝn→ℝn for each t∈𝕋 such that P(t)+Q(t)+R(t)=Id,(3.2)T(t,τ)P(τ)=P(t)T(t,τ),T(t,τ)Q(τ)=Q(t)T(t,τ),T(t,τ)R(τ)=R(t)T(t,τ)
for t,τ∈𝕋 and there are some constants a>b≥0 and D>1 such that for t≥τ,t,τ∈𝕋(3.3)‖T(t,τ)P(τ)‖≤Dea(τ,t),‖T(t,τ)R(τ)‖≤Deb(t,τ),
and for t≤τ,t,τ∈𝕋(3.4)‖T(t,τ)Q(τ)‖≤Dea(t,τ),‖T(t,τ)R(τ)‖≤Deb(τ,t).
We say that (3.1) admits a strong exponential trichotomy on time scale 𝕋 if there exist P(t), Q(t), R(t):ℝn→ℝn for each t∈𝕋 and constants a>b≥0, D>1 satisfying (3.2), (3.3), (3.4), as well as a constant c with c≥a and c>1 such that(3.5)‖T(t,τ)P(τ)‖≤Dec(τ,t),t≤τ,t,τ∈T,‖T(t,τ)Q(τ)‖≤Dec(t,τ),t≥τ,t,τ∈T.
Obviously, if for any t∈𝕋 we have μ(t)≡0 or μ(t)≡1, then the notion of exponential trichotomy on time scale 𝕋 becomes the exponential trichotomy for differential equation or difference equation, respectively.Next, we give the notion of Lyapunov function. Consider a functionV:𝕋×ℝn→ℝn. If the following two conditions are satisfied (H1)
for eachτ∈𝕋 set Vτ∶=V(τ,·) and(3.6)Cs(Vτ)∶={0}∪Vτ-1(-∞,0),Cu(Vτ)∶={0}∪Vτ-1(0,+∞).letrs and ru be, respectively, the maximal dimensions of linear subspaces inside Cs(Vτ) and Cu(Vτ), then we have rs+ru=n;(H2)
for everyt≥τ, t,τ∈𝕋 and x∈ℝn, we have T(τ,t)Cs(Vt)⊂Cs(Vτ) and T(t,τ)Cu(Vτ)⊂Cu(Vt),then we say that the functionV is a Lyapunov function.Let(3.7)Hτs∶=⋂r∈TT(τ,r)Cs(Vr)¯⊂Cs(Vτ)¯,Hτu∶=⋂r∈TT(τ,r)Cu(Vr)¯⊂Cu(Vτ)¯.
By the notion of Lyapunov function, we can see that there are subspaces Dτs(V) and Dτu(V) such that Dτs(V)⊕Dτu(V)=ℝn. The function V is called a  (λ,μ)-Lyapunov function if V is a Lyapunov function and for t≥τ,t,τ∈𝕋, V satisfies the following conditions:  (L1)
|V(t,x)|≤N∥x∥,(L2)
forx∈Dτs(V), V2(t,T(t,τ)x)≤eλ(t,τ)V2(τ,x),(L3)
forx∈Dτu(V), V2(t,T(t,τ)x)≥eμ(t,τ)V2(τ,x),(L4)
forx∈Dτs(V)⋃Dτu(V), |V(t,x)|≥∥x∥/N,whereλ<μ and N is a constant with N>1.Furthermore, letS(t) for each t∈𝕋 be a symmetric invertible n×n matrix. Set H(t,x)∶=〈S(t)x,x〉. If V(t,x)∶=-signH(t,x)|H(t,x)| is a Lyapunov function ((λ,μ)-Lyapunov function), then V(t,x) is called a quadric Lyapunov function ((λ,μ)-quadric Lyapunov function). Let ∥S(t)∥∶=supx∈ℝn|H(t,x)|/∥x∥2. Now, we state and prove our main results.Theorem 3.1.
Suppose that (3.1) has two (λ,μ)-quadratic Lyapunov functions, V(t,x) with (r1,r2) and W(t,x) with (l1,l2) satisfying r1<r2<0<l1<l2. If the symmetric invertible n×n matrixes S(t) and T(t) for V(t,x) and W(t,x), respectively, satisfy supt∈𝕋∥S(t)∥<∞ and supt∈𝕋∥T(t)∥<∞, then (3.1) has an exponential trichotomy.In order to prove Theorem3.1, we need some lemmas.Lemma 3.2.
The subspacesDτs(V), Dτu(V), Dτs(W), and Dτu(W) according to Lyapunov functions V and W have the following relations
(3.8)Dτs(V)⊂Dτs(W),Dτu(W)⊂Dτu(V).Proof.
By the knowledge of mathematical analysis, we can easily get thatln(1+lx)/x is a decreasing function for any l∈ℝ. Thus, when 0≤x≤M, we have ln(1+lM)/M≤ln(1+lx)/x≤l. Now, we prove this lemma by contradiction. Suppose that there is x∈Dτs(V)∖Dτs(W). Then there are y∈Dτs(W) and z∈Dτu(W)∖{0} such that x=y+z. Since r1<l1<l2, then by Lemma 2.6 we get
(3.9)lim⁡t→+∞1tln⁡⁡|V(t,T(t,τ)x)|≥lim⁡t→+∞1tln⁡⁡‖T(t,τ)x‖N≥lim⁡t→+∞1tln⁡⁡(‖T(t,τ)z‖-‖T(t,τ)y‖)≥lim⁡t→+∞1tln⁡⁡(V(t,T(t,τ)z)N-N|V(t,T(t,τ)y)|)≥lim⁡t→+∞1tln⁡⁡(el2(t,τ)V(τ,z)N-Nel1(t,τ)|V(τ,y)|)≥lim⁡t→+∞1tln⁡⁡el1(t,τ)=lim⁡t→+∞12t∫τt1μ(s)ln⁡⁡(1+μ(s)l1)Δs≥lim⁡t→+∞t-τ2tln⁡⁡(1+l1M)M=ln⁡⁡(1+l1M)2M>0.
But for x∈Dτs(V), one has
(3.10)lim⁡t→+∞1tln⁡⁡|V(t,T(t,τ)x)|≤lim⁡t→+∞1tln⁡⁡(er1(t,τ)|V(τ,x)|)≤r12<0.
It is a contradiction. So we have Dτs(V)⊂Dτs(W).
By the concept of Lyapunov function, we get that fort≤τ,
(3.11)|V′(τ,x)|≤eλ(τ,t)|V′(t,T(t,τ)x)|,x∈Dτs,V′(τ,x)≥eμ(τ,t)V′(t,T(t,τ)x),x∈Dτu,
where V′=V or W, λ=r1 or l1 and μ=r2 or l2. Using above inequalities and the similar way by which we get Dτs(V)⊂Dτs(W), we can directly obtain that Dτu(W)⊂Dτu(V). Here we omit the detailed proofing. Thus, the proof is completed.Obviously, from Lemma3.2 we have (Dτu(V)⋂Dτs(W))⋂Dτu(W)={0}, (Dτu(V)∩Dτs(W))⋂Dτs(V)={0} and Dτu(W)⋂Dτs(V)={0}. Furthermore, (3.12)dim⁡⁡(Dτu(V)⋂Dτs(W))=dim⁡⁡Dτu(V)+dim⁡⁡Dτs(W)-dim⁡⁡(Dτu(V)+Dτs(W))=n-dim⁡⁡Dτs(V)+n-dim⁡⁡Dτu(W)-dim⁡⁡(Dτu(V)+Dτs(W)).
Thus, (3.13)dim⁡⁡(Dτu(V)⋂Dτs(W))+dim⁡⁡Dτs(V)+dim⁡⁡Dτu(W)≥2n-dim⁡⁡(Dτu(V)+Dτs(W))≥n.
So we obtain (3.14)Dτu(V)⋂Dτs(W)⊕Dτs(V)⊕Dτu(W)=Rn.
Then from the conditions in Theorem 3.1 we have the following results.Whent≥τ, for x∈Dτs(V), we get(3.15)‖T(t,τ)x‖≤N|V(t,T(t,τ)x)|≤Ner1(t,τ)|V(τ,x)|≤N2er1(t,τ)‖x‖,
and for x∈Dτu(V)⋂Dτs(W), we get(3.16)‖T(t,τ)x‖≤N|V(t,T(t,τ)x)|≤Nel1(t,τ)|V(τ,x)|≤N2el1(t,τ)‖x‖.Whent≤τ, for x∈Dτu(W), we get(3.17)‖T(t,τ)x‖≤N|V(t,T(t,τ)x)|≤Nel2(t,τ)|V(τ,x)|≤N2el2(t,τ)‖x‖,
and for x∈Dτu(V)⋂Dτs(W), we get(3.18)‖T(t,τ)x‖≤N|V(t,T(t,τ)x)|≤Ner2(t,τ)|V(τ,x)|≤N2er2(t,τ)‖x‖.Let(3.19)PV(τ):Rn⟶Dτs(V),QV(τ):Rn⟶Dτu(V),PW(τ):Rn⟶Dτs(W),QW(τ):Rn⟶Dτu(W)
be projections. Set P(τ)∶=PV(τ), Q(τ)∶=QW(τ) and R(τ)∶=PW(τ)+QV(τ). From Lemma 3.2, we have QV(τ)PW(τ)=PW(τ)QV(τ)=0. Thus, by simply deducing, we get that R(τ) is also a projection. Set L1(τ)=Id-P(τ). Next, we will give the boundedness of P(τ), Q(τ), and R(τ). By the conditions of Theorem 3.1, we suppose that there is a constant L>0 such that supt∈𝕋∥S(t)∥≤L and supt∈𝕋∥T(t)∥≤L. We have the following lemma.Lemma 3.3.
(3.20)‖P(t)x‖≤2N2‖x‖,‖Q(t)x‖≤2N2‖x‖,‖R(t)x‖≤22N2‖x‖.Proof.
Firstly, one has(3.21)V(t,P(t)x)=〈S(t)P(t)x,P(t)x〉≥‖P(t)x‖2N2,V(t,L1(t)x)=-〈S(t)L1(t)x,L1(t)x〉≥‖Q(t)x‖2N2.
Since
(3.22)1N2‖P(t)x-N22S(t)x‖2+1N2‖L1(t)x-N22S(t)x‖2=1N2‖P(t)x‖2+1N2‖L1(t)x‖2+N22‖S(t)x‖2-〈P(t)x,S(t)x〉+〈L1(t)x,S(t)x〉≤〈S(t)P(t)x,P(t)x〉-〈S(t)L1(t)x,L1(t)x〉+N22‖S(t)x‖2-〈P(t)x,S(t)x〉+〈L1(t)x,S(t)x〉=N22‖S(t)x‖2,
then we obtain
(3.23)‖P(t)x‖≤‖P(t)x-N22S(t)x‖+N22‖S(t)x‖≤N22‖S(t)x‖+N22‖S(t)x‖≤2N2‖S(t)x‖≤2N2L‖x‖.
Similarly, we get
(3.24)‖Q(t)x‖≤2N2L‖x‖,‖R(t)x‖≤22N2L‖x‖.Proof of Theorem3.1.
Firstly, we need to show that there are constantsr1′,r2′<0 and l1′,l2′>0 such that when t≥τ, one has
(3.25)er1(t,τ)≤er1′(t,τ),(3.26)el1(t,τ)≤el1′(t,τ),
and when t≤τ, one has
(3.27)el2(t,τ)≤el2′(t,τ),er2(t,τ)≤er2′(t,τ).
For inequality (3.25), by the definition of exponential function, we only need to find a constant r1′<0 such that
(3.28)1+μ(t)r1≤(1+μ(t)r1′)2.
That means
(3.29)r1≤2r1′+μ(t)r1′2.
Thus, if r1/2≤r1′<0, (3.25) always holds. Similarly, we get that if l1′≥l1/2, 0<l2′<(1+Ml2-1)/M and (-1+Mr2-1)/M<r2′<0, then (3.26),  (3.27) always hold, respectively. Here, we notice that since r2 is positive regressive, then 1+Mr2 is significative.
By (3.15)–(3.18) and Lemma 3.3, we get that when t≥τ,
(3.30)‖T(t,τ)P(τ)x‖≤‖T(t,τ)|Dτs(V)‖‖P(τ)x‖≤2N2Ler1(t,τ)‖x‖≤2N2Ler1′(t,τ)‖x‖,‖T(t,τ)R(τ)x‖≤‖T(t,τ)|Dτs(W)⋂Dτu(V)‖‖R(τ)x‖≤22N2Lel1(t,τ)‖x‖≤22N2Lel1′(t,τ)‖x‖,
and when t≤τ,
(3.31)‖T(t,τ)Q(τ)x‖≤‖T(t,τ)|Dτu(W)‖‖Q(τ)x‖≤2N2Lel2(t,τ)‖x‖≤2N2Lel2′(t,τ)‖x‖,‖T(t,τ)R(τ)x‖≤‖T(t,τ)|Dτs(W)⋂Dτu(V)‖‖R(τ)x‖≤22N2Ler2(t,τ)‖x‖≤22N2Ler2′(t,τ)‖x‖.
Set D:=22N2L, a∶=min{|r1′|,l2′}, and b∶=max{|r2′|,l1′}. Obviously, we have a>b and the above inequalities imply that for t≥τ,t,τ∈𝕋 one has
(3.32)‖T(t,τ)P(τ)‖≤Dea(τ,t),‖T(t,τ)R(τ)‖≤Deb(t,τ),
and for t≤τ,t,τ∈𝕋 one has
(3.33)‖T(t,τ)Q(τ)‖≤Dea(t,τ),‖T(t,τ)R(τ)‖≤Deb(τ,t).
Thus, from the definition of exponential trichotomy on time scales, we get that (3.1) has an exponential trichotomy on time scale 𝕋. This completes the proof of Theorem 3.1.Next, we give an approximately converse result of Theorem3.1.Theorem 3.4.
If (3.1) admits a strict exponential trichotomy, then for (3.1) there exist two (λ,μ)-Lyapunov functions, V′(t,x) with (⊖r1′,⊖r2′) and W′(t,x) with (l1′,l2′) satisfying ⊖r1′<⊖r2′<0<l1′<l2′.Proof.
Suppose that (3.1) admits a strict exponential trichotomy with projections P(t), Q(t), R(t), and constants c≥a>b≥0 satisfying (3.2)–(3.5). Set
(3.34)S(t)∶=∫t+∞(T(σ(v),t)P(t))*T(σ(v),t)P(t)ec1⊕b(σ(v),t)Δv-∫-∞t(T(σ(v),t)L1(t))*T(σ(v),t)L1(t)ec2⊕b(σ(v),t)Δv,
where L1(t)=Q(t)+R(t) and b<c2<c1<a. Let H(t,x)∶=〈S(t)x,x〉.
Whent≥τ,t,τ∈𝕋, we get
(3.35)H(t,T(t,τ)x)=∫t+∞‖T(σ(v),τ)P(τ)x‖2ec1⊕b(σ(v),t)Δv-∫-∞t‖T(σ(v),τ)L1(τ)x‖2ec2⊕b(σ(v),t)Δv≤∫τ+∞‖T(σ(v),τ)P(τ)x‖2ec1⊕b(σ(v),τ)Δvec1⊕b(τ,t)-∫-∞τ‖T(σ(v),τ)L1(τ)x‖2ec2⊕b(σ(v),τ)Δvec2⊕b(τ,t)≤ec2⊕b(τ,t)H(τ,x).
Set V(t,x)∶=-signH(t,x)|H(t,x)|. Then if V(τ,x)>0, that is, x∈Cu(Vτ), we have H(τ,x)<0. By (3.35) we obtain that H(t,T(t,τ))<0. Then we get V(t,T(t,τ)x)>0, that is, T(t,τ)x∈Cu(Vt). That means T(t,τ)Cu(Vτ)⊂Cu(Vt). Similarly, we have T(τ,t)Cs(Vt)⊂Cs(Vτ). Thus, V(t,x) is a Lyapunov function. By (3.3), (3.4) and the characters of exponential function on time scales, one has
(3.36)|H(t,x)|≤∫t+∞‖T(σ(v),t)P(t)x‖2ec1⊕b(σ(v),t)Δv+∫-∞t‖T(σ(v),t)L1(t)x‖2ec2⊕b(σ(v),t)Δv≤∫t+∞D2ea(t,σ(v))ea(t,σ(v))ec1⊕b(σ(v),t)Δv‖x‖2+∫-∞tD2ea(σ(v),t)ea(σ(v),t)ec2⊕b(σ(v),t)Δv‖x‖2+∫-∞tD2eb(t,σ(v))eb(t,σ(v))ec2⊕b(σ(v),t)Δv‖x‖2≤∫t+∞D2ea⊖b(t,σ(v))Δv‖x‖2+∫-∞tD2ea⊕a⊕c2⊕b(σ(v),t)Δv‖x‖2+∫-∞tD2ec2⊖b(σ(v),t)Δv‖x‖2=∫t+∞D2-[ea⊖b(t,v)]vΔa⊖bΔv‖x‖2+∫-∞tD2-[e⊖(a⊕a⊕c2⊕b)(t,v)]vΔ⊖(a⊕a⊕c2⊕b)Δv‖x‖2+∫-∞tD2-[eb⊖c2(t,v)]vΔb⊖c2Δv‖x‖2.
Since μ(t)≤M, then
(3.37)1a⊖b=1+μ(t)ba-b≤1+Mba-b,-1⊖(a⊕a⊕c2⊕b)=1+μ(t)(a⊕a⊕c2⊕b)a⊕a⊕c2⊕b≤1a⊕a⊕c2⊕b+M≤12a+c2+b+M,-1b⊖c2=1+μ(t)c2c2-b≤1+Mc2c2-b.
Thus,
(3.38)|H(t,x)|≤[D2(1+Mb)a-b+D2(12a+c2+b+M)+D2(1+Mc2c2-b)]‖x‖2.
Set K∶=D2(1+Mb)/(a-b)+D2(1/(2a+c2+b)+M)+D2((1+Mc2)/(c2-b)).
LetHτs(V)∶=P(τ)ℝn and Hτu(V)∶=L1(τ)ℝn=R(τ)ℝn⊕Q(τ)ℝn. For x∈Hτs(V), one has
(3.39)H(t,x)=∫t+∞‖T(σ(v),t)P(t)x‖2ec1⊕b(σ(v),t)Δv=∫t+∞‖T(σ(v),t)x‖2ec1⊕b(σ(v),t)Δv.
Obviously, one has
(3.40)‖T(t,τ)‖≤‖T(t,τ)P(τ)‖+‖T(t,τ)Q(τ)‖+‖T(t,τ)R(τ)‖.
Then by (3.3)–(3.5), we obtain that
(3.41)‖T(t,τ)‖≤Dea(τ,t)+Deb(t,τ)+Dec(t,τ)fort≥τ,(3.42)‖T(t,τ)‖≤Dea(t,τ)+Deb(τ,t)+Dec(τ,t)fort≤τ.
Thus,
(3.43)|H(t,x)|≥∫t+∞‖x‖2‖T(t,σ(v))‖2ec1⊕b(σ(v),t)Δv≥∫t+∞‖x‖2‖Dea(t,σ(v))+Deb(σ(v),t)+Dec(σ(v),t)‖2ec1⊕b(σ(v),t)Δv≥‖x‖29D2∫t+∞ec1⊕b(σ(v),t)ec⊕c(t,σ(v))Δv=‖x‖29D2∫t+∞-(ec⊕c⊖(c1⊕b)(t,v))vΔc⊕c⊖(c1⊕b)Δv.
Since c⊕c⊖(c1⊕b)=(2c+μ(s)c2-(c1+b+μ(s)c1b))/(1+μ(s)(c1+b+μ(s)b))≤2c-c1-b+Mc2-Mc1b, then we get
(3.44)|H(t,x)|≥‖x‖29D2(2c-c1-b+Mc2-Mc1b).
Similarly, for x∈Hτu(V), we get
(3.45)H(t,x)=-∫-∞t‖T(σ(v),t)x‖2ec2⊕b(σ(v),t)Δv,|H(t,x)|≥‖x‖29D2(c2+b+2c+M(2c2c+2bc+c2)+M2(bc+c2c2+bc2)+M3bc2).
LetL′∶=max{c2+b+2c+M(2c2c+2bc+c2)+M2(bc+c2c2+bc2)+M3bc2,2c-c1-b+Mc2-Mc1b}. Then when x∈Hτs(V)⋃Hτu(V), we get |H(t,x)|≥∥x∥2/9D2L′. Let N2∶=max{9D2L′,K}. We get |H(t,x)|≤K≤N2∥x∥2 for any t∈𝕋, x∈ℝn, and when x∈Hτs(V)⋃Hτu(V), we get |H(t,x)|≥∥x∥2/N2.
Sincec2<c1, then we have c2⊕b<c1⊕b and there are constants r1′ and r2′ with c2⊕b≤r2′<r1′≤c1⊕b. Thus, when x∈Hτs(V), for any t≥τ we get
(3.46)H(t,T(t,τ)x)=∫t+∞‖T(σ(v),τ)P(τ)x‖2ec1⊕b(σ(v),t)Δv≤ec1⊕b(τ,t)H(τ,x),
that is,
(3.47)V2(t,T(t,τ)x)≤ec1⊕b(τ,t)V2(τ,x)≤er1′(τ,t)V2(τ,x),
and when x∈Hτu(V), for any t≥τ we get
(3.48)H(t,T(t,τ)x)=-∫-∞t‖T(σ(v),τ)L1(τ)x‖2ec2⊕b(σ(v),t)Δv≤ec2⊕b(τ,t)H(τ,x),
that is,
(3.49)V2(t,T(t,τ)x)≥ec2⊕b(τ,t)V2(τ,x)≥er2′(τ,t)V2(τ,x).
Therefore, V(t,x) is a (⊖r1′,⊖r2′)-Lyapunov function.
Set(3.50)T(t)∶=∫t+∞(T(σ(v),t)L2(t))*T(σ(v),t)L2(t)ed1⊕b(t,σ(v))Δv-∫-∞t(T(σ(v),t)Q(t))*T(σ(v),t)Q(t)ed2⊕b(t,σ(v))Δv,
where L2(t)=P(t)+R(t) and b<d1<d2<a. Let G(t,x)∶=〈T(t)x,x〉 and W(t,x)∶=-signG(t,x)|G(t,x)|. Then
(3.51)G(t,x)=∫t+∞‖(T(σ(v),t)L2(t))x‖2ed1⊕b(t,σ(v))Δv-∫-∞t‖(T(σ(v),t)Q(t))x‖2ed2⊕b(t,σ(v))Δv.
Let Hτs(W)∶=L2(τ)ℝn and Hτu(W)∶=Q(τ)ℝn. Similar to the consideration for V(t,x), we can see that G(t,x) is also a Lyapunov function satisfying
(3.52)G(t,T(t,τ)x)≤ed2⊕b(t,τ)G(τ,x),|G(t,x)|≤D2(1+Maa-b+12a+d1+b+M+1+Mbd1-b)‖x‖2,|G(t,x)|≥‖x‖29D2L′′,forx∈Hτs(W)⋃Hτu(W),
where L′′=max{d1+b+2c+M(2d1c+2bc+c2)+M2(bc+d1c2+bc2)+M3bc2, 2c-d2-b+Mc2-Md2b}.
Sinced1<d2, then we have d1⊕b<d2⊕b and there are constants l1′ and l2′ with d1⊕b≤l1′<l2′≤d2⊕b. Thus, for any t≥τ, we obtain that when x∈Hτs(W),
(3.53)G(t,T(t,τ)x)=∫t+∞‖T(σ(v),τ)L2(τ)x‖2ed1⊕b(t,σ(v))Δv≤ed1⊕b(t,τ)G(τ,x),
that is,
(3.54)W2(t,T(t,τ)x)≤ed1⊕b(t,τ)W2(τ,x)≤el1′(t,τ)W2(τ,x),
and when x∈Hτu(W),
(3.55)G(t,T(t,τ)x)=-∫-∞t‖T(σ(v),τ)L1(τ)x‖2ed2⊕b(t,σ(v))Δv≤ed2⊕b(t,τ)G(τ,x),
that is,
(3.56)W2(t,T(t,τ)x)≥ed2⊕b(t,τ)W2(τ,x)≥el2′(t,τ)V2(τ,x).
Thus, W(t,x) is a (l1′,l2′)-Lyapunov function.
According to all the discussions above we get that when (3.1) has a strict exponential trichotomy, there exists two (λ,μ)-Lyapunov functions, V(t,x) with (⊖r1′,⊖r2′) and W(t,x) with (l1′,l2′) satisfying ⊖r1′<⊖r2′<0<l1′<l2′. This completes the proof.


## 4. Roughness of Exponential Trichotomy on Time Scales

The roughness of exponential dichotomy on time scales had been studied by Zhang and his cooperators in their paper [24] in 2010. In this section, we go further to study the roughness of exponential trichotomy on time scales, using the results which we get from Section 3. We have known that the notion of exponential trichotomy plays a central role when we study center manifolds, so it is important to understand how exponential trichotomy vary under perturbations. Here, we discuss the following linear perturbed equation(4.1)xΔ=A(t)x+B(t)x,
where B(t) is an n×n matrix valued function on time scale 𝕋 with ∥B(t)∥≤δ. For this linear perturbed (4.1) we get the following theorem.Theorem 4.1.
Suppose that (3.1) has a strict exponential trichotomy. If δ>0 is sufficiently small, then the linear perturbed (4.1) also has an exponential trichotomy.Proof.
Since (3.1) has a strict exponential trichotomy, then by Theorem 3.4 there are two (λ,μ)-Lyapunov functions, V(t,x) with (r1,r2) and W(t,x) with (l1,l2) satisfying r1<r2<0<l1<l2. Here, V(t,x) and W(t,x) are defined in Theorem 3.4. Let x(t) be a solution of (3.1) satisfying x(τ)=x. Differentiating on both sides of H(t,x(t))=〈S(t)x(t),x(t)〉, one has
(4.2)HΔ(t,x(t))=〈S(t)x(t),x(t)〉Δ=〈SΔ(t)x(t),x(t)〉+〈S(σ(t))xΔ(t),x(t)〉+〈S(σ(t))x(σ(t)),xΔ(t)〉=〈[SΔ(t)+S(σ(t))A(t)+A*(t)S(σ(t))+μ(t)A*(t)S(σ(t))A(t)SΔ]x(t),x(t)〉.
Let T(t,τ) be the linear evolution operator related to (3.1). Then we get x(t)=T(t,τ)x. And we can see that P(t)x(t)=P(t)T(t,τ)x=T(t,τ)P(τ)x is also a solution of (3.1). By (4.2) we get
(4.3)HΔ(t,P(t)x(t))=〈[SΔ(t)+S(σ(t))A(t)+A*(t)S(σ(t))+μ(t)A*(t)S(σ(t))A(t)SΔ]P(t)x(t),P(t)x(t)〉.
At the same time, by (2.4), (2.7), and (3.46) one has
(4.4)μ(t)HΔ(t,P(t)x(t))=H(σ(t),P(σ(t))x(σ(t)))-H(t,P(t)x(t))≤H(t,P(t)x(t))ec1⊕b(t,σ(t))-H(t,P(t)x(t))=H(t,P(t)x(t))(exp⁡{∫tσ(t)1μ(s)ln⁡(1+⊖(c1⊕b)μ(s))Δs}-1)=⊖(c1⊕b)μ(t)H(t,P(t)x(t)).
Thus, we obtain that
(4.5)HΔ(t,P(t)x(t))≤⊖(c1⊕b)H(t,P(t)x(t)).
Then by (4.3) and (4.5) we can see
(4.6)SΔ(t)+S(σ(t))A(t)+A*(t)S(σ(t))+μ(t)A*(t)S(σ(t))A(t)≤⊖(c1⊕b)S(t).
Now, Let y(t) be a solution of equation xΔ=A(t)x+B(t)x satisfying y(τ)=y. Since y(σ(t))=y(t)+μ(t)yΔ(t)=y(t)+μ(t)(A(t)+B(t))y(t), one has
(4.7)HΔ(t,y(t))=〈S(t)y(t),y(t)〉Δ=〈SΔ(t)y(t),y(t)〉+〈S(σ(t))yΔ(t),y(t)〉+〈S(σ(t))y(σ(t)),yΔ(t)〉=〈SΔ(t)y(t),y(t)〉+〈S(σ(t))A(t)y(t),y(t)〉+〈A*(t)S(σ(t))y(t),y(t)〉+〈μ(t)A*(t)S(σ(t))A(t)y(t),y(t)〉+〈S(σ(t))B(t)y(t),y(t)〉+〈μ(t)S(σ(t))B(t)y(t),A(t)y(t)〉+〈S(σ(t))y(t),B(t)y(t)〉+〈μ(t)S(σ(t))A(t)y(t),B(t)y(t)〉+〈μ(t)S(σ(t))B(t)y(t),B(t)y(t)〉≤⊖(c1⊕b)H(t,y(t))+‖S(σ(t))‖‖B(t)‖‖y(t)‖2+2‖μ(t)A(t)‖‖S(σ(t))‖‖B(t)‖‖y(t)‖2+‖S(σ(t))‖‖B(t)‖‖y(t)‖2+μ(t)‖S(σ(t))‖‖B(t)‖2‖y(t)‖2.
By the variation of constants formula, we haveT(t,s)=Id+∫stA(v)T(v,s)Δv. Thus, by (2.7) we obtain
(4.8)T(σ(t),t)=I+∫tσ(t)A(v)T(v,t)Δv=Id+μ(t)A(t)T(t,t)=Id+μ(t)A(t).
Therefore, by Definition 2.5 and (2.7), (3.41), we obtain
(4.9)‖Id+μ(t)A(t)‖≤Dea(t,σ(t))+Deb(σ(t),t)+Dec(σ(t),t)=D[1+μ(t)(⊖a)]+D(1+μ(t)b)+D(1+μ(t)c)≤3D+DM(b+c).
Thus, one has
(4.10)‖μ(t)A(t)‖≤‖Id+μ(t)A(t)‖+‖Id‖≤3D+DM(b+c)+1.
From Theorem 3.4, we know
(4.11)‖S(t)‖=sup⁡x∈Rn⁡H(t,x)‖x(t)‖2≤N2,‖y(t)‖≤N|V(t,y(t))|.
Thus,
(4.12)HΔ(t,y(t))≤⊖(c1⊕b)H(t,y(t))+[2N2δ+2(3D+1+DMb+DMc)N2δ+MN2δ2]N2|H(t,y(t))|.
Let M(δ)∶=[2N2δ+2(3D+1+DMb+DMc)N2δ+MN2δ2]N2. It is not hard to see that M(δ)→0 when δ→0. Set E(δ)∶=⊖(c1⊕b)+M(δ). Then when δ is sufficiently small, we have E(δ)<0 and
(4.13)1+μ(t)E(δ)=1+μ(t)[-(c1⊕b)1+μ(t)(c1⊕b)+M(δ)]=11+μ(t)(c1⊕b)+μ(t)M(δ)>0.
So E(δ) is positive regressive. Then for H(t,y(t))>0, (4.12) can be written as
(4.14)HΔ(t,y)≤E(δ)H(t,y(t)).
Notice that
(4.15)(H(t,y(t))e⊖E(δ)(t,τ))Δ=HΔ(t,y(t))e⊖E(δ)(σ(t),τ)+⊖E(δ)e⊖E(δ)(t,τ)H(t,y(t))=HΔ(t,y(t))e⊖E(δ)(σ(t),τ)+⊖E(δ)e⊖E(δ)(t,σ(t))e⊖E(δ)(σ(t),τ)H(t,y(t))=HΔ(t,y(t))e⊖E(δ)(σ(t),τ)+⊖E(δ)11+⊖E(δ)μ(t)e⊖E(δ)(σ(t),τ)H(t,y(t))=(HΔ(t,y)-E(δ)H(t,y(t)))e⊖E(δ)(σ(t),τ)≤0.
Thus, we get
(4.16)H(t,y(t))≤eE(δ)(t,τ)H(τ,y(τ)).
Then we can see that if y(t)∈Cs(Vt), we have V(t,y(t))<0 and thus H(t,y(t))>0. By (4.16) we have H(τ,y(τ))>0 and thus V(τ,y(τ))<0, that is, y(τ)∈Cs(Vτ). Let U(t,τ) be the linear evolution operator associated to (4.1). Then y(t)=U(t,τ)y(τ). Thus, we get
(4.17)U(τ,t)Cs(Vt)⊂Cs(Vτ).
Let L1(t)=Q(t)+R(t). We can see that L1(t)x(t) is also a solution of (3.1). By (4.2) one has
(4.18)HΔ(t,L1(t)x(t))=〈[SΔ(t)+S(σ(t))A(t)+A*(t)S(σ(t))+μ(t)A*(t)S(σ(t))A(t)SΔ]L1(t)x(t),L1(t)x(t)〉.
Similar to the proof of (4.5), we have HΔ(t,L1(t)x(t))≤⊖(c2⊕b)H(t,L1(t)x(t)). Then we get
(4.19)SΔ(t)+S(σ(t))A(t)+A*(t)S(σ(t))+μ(t)A*(t)S(σ(t))A(t)≤⊖(c2⊕b)S(t).
Following the process for getting (4.12), one has
(4.20)HΔ(t,y(t))≤⊖(c2⊕b)H(t,y(t))+M(δ)|H(t,y(t))|.
Let F(δ)∶=⊖(c2⊕b)-M(δ). When δ is sufficiently small, we obtain F(δ)<0 and
(4.21)1+μ(t)F(δ)=1+μ(t)[-(c2⊕b)1+μ(t)(c2⊕b)-M(δ)]=11+μ(t)(a⊕b)-M(δ).
So F(δ) is positive regressive. Then for H(t,y(t))<0, (4.20) can be written as
(4.22)HΔ(t,y)≤F(δ)H(t,y(t)).
Following the similar process as (4.15), we get
(4.23)H(t,y(t))≤eF(δ)(t,τ)H(τ,y(τ)).
Then if y(τ)∈Cu(Vτ), we have V(τ,y(τ))>0 and thus H(τ,y(τ))<0. By (4.23) we have H(t,y(t))<0 and thus V(t,y(t))>0, that is, y(t)∈Cu(Vt). So we get
(4.24)U(t,τ)Cu(Vτ)⊂Cu(Vt).
By the notion of Lyapunov function and (4.17), (4.24) we can see that V(t,x) is a Lyapunov function of (4.1).
Let(4.25)Hτ′u(V)∶=⋂r∈TU(τ,r)Cu(Vr)¯⊂Cu(Vτ)¯,Hτ′s(V)∶=⋂r∈TU(τ,r)Cs(Vr)¯⊂Cs(Vτ)¯.
By the notion of Lyapunov function, we know that there exist two subspaces Dτ′u(V) and Dτ′s(V) such that Dτ′u(V)⊂Hτ′u(V), Dτ′s(V)⊂Hτ′s(V), and Dτ′u(V)⊕Dτ′s(V)=ℝn.
Now, we prove thatV(t,x) is a (λ,μ)-Lyapunov function. Since c1>c2, we have c1⊕b>c2⊕b. By simple computation, we obtain ⊖(c1⊕b)<⊖(c2⊕b). Therefore, when δ is sufficiently small, we get E(δ)<F(δ). Then there are constants r1,r2 such that E(δ)≤r1<r2≤F(δ). Thus, by inequality (4.16), for y(τ)∈Dτ′s one has
(4.26)V2(t,y(t))≤eE(δ)(t,τ)V2(τ,y(τ))≤er1(t,τ)V2(τ,y(τ)).
By inequality (4.23), for y(τ)∈Dτ′u, we obtain
(4.27)V2(t,y(t))≥eF(δ)(t,τ)V2(τ,y(τ))≥er2(t,τ)V2(τ,y(τ)).
Then by (4.26) and (4.27) we can see that V(t,y) is a (r1,r2)-Lyapunov function of (4.1).
Next, we will prove thatW(t,y) is also a (λ,μ)-Lyapunov function of (4.1). Similar to the consideration of H(t,y), for G(t,y) we have that if y∈Cs(Wt), then W(t,y(t))<0, that is, G(t,y(t))>0. Thus, we get
(4.28)GΔ(t,y(t))≤(d1⊕b+M(δ))G(t,y(t)).
Let E′(δ):=d1⊕b+M(δ). By the similar deducing as (4.16), one has
(4.29)G(t,y(t))≤eE′(δ)(t,τ)G(τ,y(τ)).
Thus, when G(t,y(t))>0, that is, W(t,y(t))<0, we have G(τ,y(τ))>0, that is, W(τ,y(τ))<0. That means
(4.30)U(τ,t)Cs(Wt)⊂Cs(Wτ).
Wheny∈Cu(Wt), we have W(t,y(t))>0, that is, G(t,y(t))<0. Then
(4.31)GΔ(t,y(t))≤(d2⊕b-M(δ))G(t,y(t)).
Let F′(δ)=d2⊕b-M(δ). By the similar deducing as (4.15), one has
(4.32)G(t,y(t))≤eF′(δ)(t,τ)G(τ,y(τ)).
Then when G(τ,y(τ))<0, that is, W(τ,y(τ))>0, we have G(t,y(t))<0, that is, W(τ,y(τ))>0. That means
(4.33)U(t,τ)Cu(Wτ)⊂Cu(Wt).
By (4.30) and (4.33), we know that W(t,y) is a Lyapunov function of (4.1).
Sinced1<d2, we have d1⊕b<d2⊕b. Then when δ is sufficiently small, we get E′(δ)<F′(δ). Therefore, there are constants l1,l2 such that E′(δ)≤l1<l2≤F′(δ).
Let(4.34)Hτ′u(W)∶=⋂r∈TU(τ,r)Cu(Wr)¯⊂Cu(Wτ)¯,Hτ′s(W)∶=⋂r∈TU(τ,r)Cs(Wr)¯⊂Cs(Wτ)¯.
By the notion of Lyapunov function, we can know that there exist Dτ′u(W) and Dτ′s(W) such that Dτ′u(W)⊂Hτ′u(W), Dτ′s(W)⊂Hτ′s(W), and Dτ′u(W)⊕Dτ′s(W)=ℝn.
Furthermore, by (4.29) and (4.32), for any t≥τ, when y(τ)∈Dτ′s(W), one has
(4.35)W2(t,y(t))≤eE′(δ)(t,τ)W2(τ,y(τ))≤el1(t,τ)W2(τ,y(τ)),
and when y(τ)∈Dτ′u(W), one has
(4.36)W2(t,y(t))≥eF′(δ)(t,τ)W2(τ,y(τ))≥el2(t,τ)W(τ,y(τ)).
Thus, we can see that W(t,y) is a (l1,l2)-Lyapunov function of (4.1). Therefore, we know that (4.1) has two (λ,μ)-quadratic Lyapunov functions, V(t,x) with (r1,r2) and W(t,x) with (l1,l2) satisfying r1<r2<0<l1<l2. Then by Theorem 3.1 we get that (4.1) has an exponential trichotomy. The proof is completed.



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*Source: 958381-2011-08-29.xml*