Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2011, Article ID 958381, 22 pages
doi:10.1155/2011/958381
Research Article
Lyapunov Function and Exponential Trichotomy on
Time Scales
Ji Zhang
College of Mathematics, Jilin University, Changchun 130012, China
Correspondence should be addressed to Ji Zhang, zhangjiccf@gmail.com
Received 20 February 2011; Revised 7 May 2011; Accepted 6 July 2011
Academic Editor: Jinde Cao
Copyright q 2011 Ji Zhang. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly cited.
We study the relation between Lyapunov function and exponential trichotomy for the linear
equation xΔ Atx on time scales. Furthermore, as an application of these results, we give the
roughness of exponential trichotomy on time scales.
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
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Discrete Dynamics in Nature and Society
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 Atx admits exponential trichotomy, if it has two λ, μLyapunov functions with some property; conversely, the linear equation has two λ, μLyapunov 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 Section 2, 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. Let T be a time scale which is an arbitrary nonempty closed subset of the real
numbers. The forward jump operator is defined by
σt : inf{s ∈ T : s > t},
2.1
while the backward jump operator is defined by
ρt : sup{s ∈ T : s < t},
2.2
for every t ∈ T. If σt t, then t is called right-dense. And if ρt t, then t is called leftdense. Let μt : σt − t be the graininess function.
For example, the set of real numbers R is a time scale with σt t and μt 0 for
t ∈ T R and the set of integers Z is a time scale with σt t 1 and μt 1 for t ∈ T Z.
Definition 2.2. A function f : 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 by Crd T, Rn .
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Definition 2.3. We say that a function f : T → Rn is differentiable at t ∈ T, if for any ε > 0,
there exist T-neighborhood U of t and f Δ t ∈ Rn such that for any s ∈ U one has
fσt − fs − f Δ tσt − s ≤ εσt − s,
2.3
and f Δ t is called the derivative of f at t.
Relate to the differential properties, if f and g is differentiable at t, then we have the
following equalities:
fσt ft μtf Δ t,
fg
Δ
t f Δ tgt fσtg Δ t.
2.4
2.5
Definition 2.4. We say that a function p : T → R is regressive if 1 μtpt /
0 for all t ∈ T.
Furthermore, if 1μtpt > 0, p is called positive regressive. An n×n matrix valued function
At on time scale T is called regressive if Id μtAt is invertible for all t ∈ T.
Let a, b be regressive. Define a⊕bt atbtμtatbt and at −at/1
μtat 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 μtb;
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. Let p be positive regressive. We define the exponential function by
t
ep t, s exp
s
1
ln 1 μτpτ Δτ ,
μτ
2.6
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
σt
fsΔs μtft.
2.7
t
From the definition of exponential function we have that if a, b are positive regressive,
then
1 e0 t, s 1 and ea t, t 1;
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2 ea t, s ea s, t;
3 ea t, sea s, r ea t, r;
4 ea t, seb t, s ea⊕b t, s;
Δ
5 ea c, tΔ
t −atea c, σt, where ea c, tt stands for the delta derivative of
ea c, t with respect to t.
Lemma 2.6 L’Hôpital’s Rule. Suppose that f and g are differentiable on T. Then if gt satisfies
gt > 0, g Δ > 0 and limt → ∞ gt ∞, then the existence of limt → ∞ f Δ t/g Δ t implies that
limt → ∞ ft/gt exists and limt → ∞ ft/gt limt → ∞ f Δ t/g Δ t.
3. Exponential Trichotomy and Lyapunov Function on Time Scales
From now on, we always suppose that T is a two-sides infinite time scale and the graininess
function μt is bounded, which means that there exists a M > 0 such that 0 ≤ μt ≤ M. We
consider the following linear equation
xΔ Atx,
3.1
where At is an n × n matrix valued function on time scale T, satisfying that At is rdcontinuous and regressive. To make it simple, in this paper we always require that At is
regressive. If At 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, τT τ, s T t, s;
3 the mapping t, τ → T t, τx 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, Qt, Rt : Rn → Rn for each t ∈ T such that P t Qt Rt Id,
T t, τP τ P tT t, τ,
T t, τQτ QtT t, τ,
T t, τRτ RtT t, τ 3.2
for t, τ ∈ T and there are some constants a > b ≥ 0 and D > 1 such that for t ≥ τ, t, τ ∈ T
T t, τP τ ≤ Dea τ, t,
T t, τRτ ≤ Deb t, τ,
3.3
T t, τRτ ≤ Deb τ, t.
3.4
and for t ≤ τ, t, τ ∈ T
T t, τQτ ≤ Dea t, τ,
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We say that 3.1 admits a strong exponential trichotomy on time scale T if there exist P t, Qt,
Rt : 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
T t, τP τ ≤ Dec τ, t,
t ≤ τ, t, τ ∈ T,
T t, τQτ ≤ Dec t, τ,
t ≥ τ, t, τ ∈ T.
3.5
Obviously, if for any t ∈ T we have μt ≡ 0 or μt ≡ 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 function V : T × Rn → Rn .
If the following two conditions are satisfied
H1 for each τ ∈ T set Vτ : V τ, · and
Cs Vτ : {0} ∪ Vτ−1 −∞, 0,
Cu Vτ : {0} ∪ Vτ−1 0, ∞.
3.6
let rs and ru be, respectively, the maximal dimensions of linear subspaces inside
Cs Vτ and Cu Vτ , then we have rs ru n;
H2 for every t ≥ τ, t, τ ∈ T and x ∈ Rn , we have T τ, tCs Vt ⊂ Cs Vτ and
T t, τCu Vτ ⊂ Cu Vt ,
then we say that the function V is a Lyapunov function.
Let
Hτs :
T τ, rCs Vr ⊂ Cs Vτ ,
r∈T
Hτu :
T τ, rCu Vr ⊂ Cu Vτ .
3.7
r∈T
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| ≤ Nx,
L2 for x ∈ Dτs V , V 2 t, T t, τx ≤ eλ t, τV 2 τ, x,
L3 for x ∈ Dτu V , V 2 t, T t, τx ≥ eμ t, τV 2 τ, x,
L4 for x ∈ Dτs V Dτu V , |V t, x| ≥ x/N,
where λ < μ and N is a constant with N > 1.
Furthermore, let St for each t ∈ T be a symmetric invertible n × n matrix. Set
Ht, x : Stx, x. If V t, x : − sign Ht, x |Ht, x| is a Lyapunov function λ, μLyapunov function, then V t, x is called a quadric Lyapunov function λ, μ-quadric Lyapunov
function. Let St : supx∈Rn |Ht, x|/x2 . Now, we state and prove our main results.
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Theorem 3.1. Suppose that 3.1 has two λ, μ-quadratic Lyapunov functions, V t, x with r1 , r2 and Wt, x with l1 , l2 satisfying r1 < r2 < 0 < l1 < l2 . If the symmetric invertible n × n matrixes
St and T t for V t, x and Wt, x, respectively, satisfy supt∈T St < ∞ and supt∈T T t < ∞,
then 3.1 has an exponential trichotomy.
In order to prove Theorem 3.1, we need some lemmas.
Lemma 3.2. The subspaces Dτs V , Dτu V , Dτs W, and Dτu W according to Lyapunov functions
V and W have the following relations
Dτs V ⊂ Dτs W,
Dτu W ⊂ Dτu V .
3.8
Proof. By the knowledge of mathematical analysis, we can easily get that ln1 lx/x is a
decreasing function for any l ∈ R. Thus, when 0 ≤ x ≤ M, we have ln1 lM/M ≤ ln1 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
1
1 T t, τx
ln|V t, T t, τx| ≥ lim ln
t → ∞ t
t → ∞ t
N
lim
≥ lim
1
t → ∞ t
ln T t, τz − T t, τy
V t, T t, τz
1
ln
− N V t, T t, τy t → ∞ t
N
el2 t, τV τ, z
1
≥ lim ln
− N el1 t, τ V τ, y
t → ∞ t
N
≥ lim
3.9
1 ln el1 t, τ
t → ∞ t
1 t 1
ln 1 μsl1 Δs
lim
t → ∞ 2t τ μs
≥ lim
≥ lim
t → ∞
t − τ ln1 l1 M ln1 l1 M
> 0.
2t
M
2M
But for x ∈ Dτs V , one has
1
1
r1
lim ln|V t, T t, τx| ≤ lim ln
< 0.
er1 t, τ|V τ, x| ≤
t → ∞ t
t → ∞ t
2
3.10
It is a contradiction. So we have Dτs V ⊂ Dτs W.
By the concept of Lyapunov function, we get that for t ≤ τ,
V τ, x ≤ eλ τ, tV t, T t, τx,
V τ, x ≥ eμ τ, tV t, T t, τx,
x ∈ Dτs ,
x ∈ Dτu ,
3.11
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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 Lemma 3.2 we have Dτu V Dτs W Dτu W {0}, Dτu V ∩
Dτs W Dτs V {0} and Dτu W Dτs V {0}. Furthermore,
dim Dτu V Dτs W dim Dτu V dim Dτs W − dimDτu V Dτs W
3.12
n − dim Dτs V n − dim Dτu W − dimDτu V Dτs W.
Thus,
dim Dτu V Dτs W dim Dτs V dim Dτu W ≥ 2n − dimDτu V Dτs W ≥ n.
3.13
So we obtain
Dτu V Dτs W ⊕ Dτs V ⊕ Dτu W Rn .
3.14
Then from the conditions in Theorem 3.1 we have the following results.
When t ≥ τ, for x ∈ Dτs V , we get
T t, τx ≤ N|V t, T t, τx| ≤ N er1 t, τ|V τ, x| ≤ N 2 er1 t, τx,
and for x ∈ Dτu V 3.15
Dτs W, we get
T t, τx ≤ N|V t, T t, τx| ≤ N el1 t, τ|V τ, x| ≤ N 2 el1 t, τx.
3.16
When t ≤ τ, for x ∈ Dτu W, we get
T t, τx ≤ N|V t, T t, τx| ≤ N el2 t, τ|V τ, x| ≤ N 2 el2 t, τx,
and for x ∈ Dτu V 3.17
Dτs W, we get
T t, τx ≤ N|V t, T t, τx| ≤ N er2 t, τ|V τ, x| ≤ N 2 er2 t, τx.
3.18
Let
PV τ : Rn −→ Dτs V ,
QV τ : Rn −→ Dτu V ,
PW τ : Rn −→ Dτs W,
QW τ : Rn −→ Dτu W
3.19
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Discrete Dynamics in Nature and Society
be projections. Set P τ : PV τ, Qτ : QW τ and Rτ : PW τ QV τ. From
Lemma 3.2, we have QV τPW τ PW τQV τ 0. Thus, by simply deducing, we get
that Rτ is also a projection. Set L1 τ Id − P τ. Next, we will give the boundedness of
P τ, Qτ, and Rτ. By the conditions of Theorem 3.1, we suppose that there is a constant
L > 0 such that supt∈T St ≤ L and supt∈T T t ≤ L. We have the following lemma.
Lemma 3.3.
P tx ≤
√
2N 2 x,
Qtx ≤
√
2N 2 x,
√
Rtx ≤ 2 2N 2 x.
3.20
Proof. Firstly, one has
V t, P tx StP tx, P tx ≥
P tx2
,
N2
Qtx2
.
V t, L1 tx −StL1 tx, L1 tx ≥
N2
3.21
Since
2
2
N2
1 1 N2
P
−
L
Stx
Stx
−
tx
tx
1
2
2
2
2
N
N
1
1
N2
2
2
tx
tx
P
L
Stx2
1
2
N2
N2
− P tx, Stx L1 tx, Stx
3.22
≤ StP tx, P tx − StL1 tx, L1 tx
N2
Stx2 − P tx, Stx L1 tx, Stx
2
N2
Stx2 ,
2
then we obtain
N2
N2
Stx P tx ≤ P tx −
Stx
2
2
N2
N2
≤ √ Stx Stx
2
2
√
√
≤ 2N 2 Stx ≤ 2N 2 Lx.
3.23
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Similarly, we get
Qtx ≤
√
2N 2 Lx,
√
Rtx ≤ 2 2N 2 Lx.
3.24
Proof of Theorem 3.1. Firstly, we need to show that there are constants r1 , r2 < 0 and l1 , l2 > 0
such that when t ≥ τ, one has
er1 t, τ ≤ er1 t, τ,
el1 t, τ ≤ el1 t, τ,
3.25
3.26
and when t ≤ τ, one has
el2 t, τ ≤ el2 t, τ,
er2 t, τ ≤ er2 t, τ.
3.27
For inequality 3.25, by the definition of exponential function, we only need to find a
constant r1 < 0 such that
2
1 μtr1 ≤ 1 μtr1 .
3.28
That means
r1 ≤ 2r1 μtr12 .
3.29
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,
T t, τP τx ≤ T t, τ|Dτs V P τx
√
2N 2 L er1 t, τx ≤ 2N 2 Ler1 t, τx,
T t, τRτx ≤ T t, τ|Dτs W Dτu V Rτx
≤
√
√
√
≤ 2 2N 2 L el1 t, τx ≤ 2 2N 2 Lel1 t, τx,
3.30
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Discrete Dynamics in Nature and Society
and when t ≤ τ,
T t, τQτx ≤ T t, τ|Dτu W Qτx
√
2N 2 L el2 t, τx ≤ 2N 2 Lel2 t, τx,
T t, τRτx ≤ T t, τ|Dτs W Dτu V Rτx
≤
√
3.31
√
√
≤ 2 2N 2 L er2 t, τx ≤ 2 2N 2 Ler2 t, τx.
√
Set D : 2 2N 2 L, 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
T t, τP τ ≤ Dea τ, t,
T t, τRτ ≤ Deb t, τ,
3.32
T t, τRτ ≤ Deb τ, t.
3.33
and for t ≤ τ, t, τ ∈ T one has
T t, τQτ ≤ Dea t, τ,
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 Theorem 3.1.
Theorem 3.4. If 3.1 admits a strict exponential trichotomy, then for 3.1 there exist two λ, μLyapunov 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, Qt,
Rt, and constants c ≥ a > b ≥ 0 satisfying 3.2–3.5. Set
∞
St :
T σv, tP t∗ T σv, tP tec1 ⊕b σv, tΔv
t
−
t
−∞
3.34
∗
T σv, tL1 t T σv, tL1 tec2 ⊕b σv, tΔv,
where L1 t Qt Rt and b < c2 < c1 < a. Let Ht, x : Stx, x.
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When t ≥ τ, t, τ ∈ T, we get
Ht, T t, τx ∞
t
−
≤
T σv, τP τx2 ec1 ⊕b σv, tΔv
t
−∞
∞
T σv, τL1 τx2 ec2 ⊕b σv, tΔv
T σv, τP τx2 ec1 ⊕b σv, τΔvec1 ⊕b τ, t
3.35
τ
−
τ
−∞
T σv, τL1 τx2 ec2 ⊕b σv, τΔvec2 ⊕b τ, t
≤ ec2 ⊕b τ, tHτ, x.
Set V t, x : −signHt, x |Ht, x|. Then if V τ, x > 0, that is, x ∈ Cu Vτ , we have
Hτ, x < 0. By 3.35 we obtain that Ht, T t, τ < 0. Then we get V t, T t, τx > 0, that
is, T t, τx ∈ Cu Vt . That means T t, τCu Vτ ⊂ Cu Vt . Similarly, we have T τ, tCs 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
|Ht, x| ≤
∞
t
≤
≤
−∞
T σv, tL1 tx2 ec2 ⊕b σv, tΔv
D2 ea t, σvea t, σvec1 ⊕b σv, tΔvx2
t
−∞
t
−∞
∞
t
t
∞
t
T σv, tP tx2 ec1 ⊕b σv, tΔv
D2 ea σv, tea σv, tec2 ⊕b σv, tΔvx2
D2 eb t, σveb t, σvec2 ⊕b σv, tΔvx2
D2 eab t, σvΔvx2 t
−∞
D
Δ
2 −eab t, vv
t
−∞
−∞
D2 ea⊕a⊕c2 ⊕b σv, tΔvx2
D2 ec2 b σv, tΔvx2
∞
t
t
D2
ab
Δ
− ea⊕a⊕c2 ⊕b t, v v
Δvx Δvx2
D
a ⊕ a ⊕ c2 ⊕ b
−∞
2
t
−ebc2 t, vΔ
v
Δvx2 .
b c2
2
3.36
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Since μt ≤ M, then
1 μtb 1 Mb
1
≤
,
ab
a−b
a−b
1 μta ⊕ a ⊕ c2 ⊕ b
−1
1
1
≤
M ≤
M, 3.37
a ⊕ a ⊕ c2 ⊕ b
a ⊕ a ⊕ c2 ⊕ b
a ⊕ a ⊕ c2 ⊕ b
2a c2 b
1 μtc2 1 Mc2
−1
≤
.
b c2
c2 − b
c2 − b
Thus,
1
D2 1 Mb
2
2 1 Mc2
D
M D
|Ht, x| ≤
x2 .
a−b
2a c2 b
c2 − b
3.38
Set K : D2 1 Mb/a − b D2 1/2a c2 b M D2 1 Mc2 /c2 − b.
Let Hτs V : P τRn and Hτu V : L1 τRn RτRn ⊕ QτRn . For x ∈ Hτs V , one
has
Ht, x ∞
T σv, tP tx2 ec1 ⊕b σv, tΔv
t
∞
3.39
2
T σv, tx ec1 ⊕b σv, tΔv.
t
Obviously, one has
T t, τ ≤ T t, τP τ T t, τQτ T t, τRτ.
3.40
Then by 3.3–3.5, we obtain that
T t, τ ≤ Dea τ, t Deb t, τ Dec t, τ
for t ≥ τ,
3.41
T t, τ ≤ Dea t, τ Deb τ, t Dec τ, t
for t ≤ τ.
3.42
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Thus,
|Ht, x| ≥
∞
T t, σv2
t
≥
x2
ec1 ⊕b σv, tΔv
∞
x2
Dea t, σv Deb σv, t Dec σv, t2
x2 ∞
ec1 ⊕b σv, tec⊕c t, σvΔv
≥
9D2 t
Δ
x2 ∞ − ec⊕cc1 ⊕b t, v v
Δv.
c ⊕ c c1 ⊕ b
9D2 t
ec1 ⊕b σv, tΔv
t
3.43
Since c ⊕ c c1 ⊕ b 2c μsc2 − c1 b μsc1 b/1 μsc1 b μsb ≤ 2c − c1 −
b Mc2 − Mc1 b, then we get
|Ht, x| ≥
x2
.
9D2 2c − c1 − b Mc2 − Mc1 b
3.44
Similarly, for x ∈ Hτu V , we get
Ht, x −
t
−∞
T σv, tx2 ec2 ⊕b σv, tΔv,
x2
.
|Ht, x| ≥
9D2 c2 b 2c M2c2 c 2bc c2 M2 bc c2 c2 bc2 M3 bc2 3.45
Let L : max{c2 b 2c M2c2 c 2bc c2 M2 bc c2 c2 bc2 M3 bc2 , 2c −
c1 − b Mc2 − Mc1 b}. Then when x ∈ Hτs V Hτu V , we get |Ht, x| ≥ x2 /9D2 L . Let
N 2 : max{9D2 L , K}. We get |Ht, x| ≤ K ≤ N 2 x2 for any t ∈ T, x ∈ Rn , and when
x ∈ Hτs V Hτu V , we get |Ht, x| ≥ x2 /N 2 .
Since c2 < 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
Ht, T t, τx ∞
T σv, τP τx2 ec1 ⊕b σv, tΔv ≤ ec1 ⊕b τ, tHτ, x,
3.46
t
that is,
V 2 t, T t, τx ≤ ec1 ⊕b τ, tV 2 τ, x ≤ er1 τ, tV 2 τ, x,
3.47
and when x ∈ Hτu V , for any t ≥ τ we get
Ht, T t, τx −
t
−∞
T σv, τL1 τx2 ec2 ⊕b σv, tΔv ≤ ec2 ⊕b τ, tHτ, x,
3.48
14
Discrete Dynamics in Nature and Society
that is,
V 2 t, T t, τx ≥ ec2 ⊕b τ, tV 2 τ, x ≥ er2 τ, tV 2 τ, x.
3.49
Therefore, V t, x is a r1 , r2 -Lyapunov function.
Set
∞
T t :
T σv, tL2 t∗ T σv, tL2 ted1 ⊕b t, σvΔv
t
−
t
−∞
3.50
∗
T σv, tQt T σv, tQted2 ⊕b t, σvΔv,
P t Rt and b < d1 < d2 < a. Let Gt, x : T tx, x and Wt, x :
where L2 t − sign Gt, x |Gt, x|. Then
Gt, x ∞
t
−
T σv, tL2 tx2 ed1 ⊕b t, σvΔv
3.51
t
2
−∞
T σv, tQtx ed2 ⊕b t, σvΔv.
Let Hτs W : L2 τRn and Hτu W : QτRn . Similar to the consideration for V t, x, we
can see that Gt, x is also a Lyapunov function satisfying
Gt, T t, τx ≤ ed2 ⊕b t, τGτ, x,
1
1 Mb
1 Ma
M
|Gt, x| ≤ D2
x2 ,
a−b
2a d1 b
d1 − b
|Gt, x| ≥
x2
,
9D2 L
for x ∈ Hτs W
3.52
Hτu W,
where L max{d1 b 2c M2d1 c 2bc c2 M2 bc d1 c2 bc2 M3 bc2 , 2c − d2 − b Mc2 − Md2 b}.
Since d1 < 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,
Gt, T t, τx ∞
T σv, τL2 τx2 ed1 ⊕b t, σvΔv ≤ ed1 ⊕b t, τGτ, x,
3.53
t
that is,
W 2 t, T t, τx ≤ ed1 ⊕b t, τW 2 τ, x ≤ el1 t, τW 2 τ, x,
and when x ∈ Hτu W,
3.54
Discrete Dynamics in Nature and Society
Gt, T t, τx −
t
−∞
T σv, τL1 τx2 ed2 ⊕b t, σvΔv ≤ ed2 ⊕b t, τGτ, x,
15
3.55
that is,
W 2 t, T t, τx ≥ ed2 ⊕b t, τW 2 τ, x ≥ el2 t, τV 2 τ, x.
3.56
Thus, Wt, 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 Wt, 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
xΔ Atx Btx,
4.1
where Bt is an n × n matrix valued function on time scale T with Bt ≤ 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, μLyapunov functions, V t, x with r1 , r2 and Wt, x with l1 , l2 satisfying r1 < r2 < 0 <
l1 < l2 . Here, V t, x and Wt, x are defined in Theorem 3.4. Let xt be a solution of 3.1
satisfying xτ x. Differentiating on both sides of Ht, xt Stxt, xt, one has
H Δ t, xt Stxt, xtΔ
SΔ txt, xt Sσt xΔ t, xt Sσtxσt, xΔ t
SΔ t SσtAt A∗ tSσt
μtA∗ tSσtAt xt, xt .
4.2
16
Discrete Dynamics in Nature and Society
Let T t, τ be the linear evolution operator related to 3.1. Then we get xt T t, τx. And
we can see that P txt P tT t, τx T t, τP τx is also a solution of 3.1. By 4.2 we
get
H Δ t, P txt SΔ t SσtAt A∗ tSσt
μtA∗ tSσtAt P txt, P txt .
4.3
At the same time, by 2.4, 2.7, and 3.46 one has
μtH Δ t, P txt Hσt, P σtxσt − Ht, P txt
≤ Ht, P txtec1 ⊕b t, σt − Ht, P txt
σt
4.4
1
ln 1 c1 ⊕ bμs Δs − 1
Ht, P txt exp
μs
t
c1 ⊕ bμtHt, P txt.
Thus, we obtain that
H Δ t, P txt ≤ c1 ⊕ bHt, P txt.
4.5
Then by 4.3 and 4.5 we can see
SΔ t SσtAt A∗ tSσt μtA∗ tSσtAt ≤ c1 ⊕ bSt.
4.6
Now, Let yt be a solution of equation xΔ Atx Btx satisfying yτ y. Since yσt yt μtyΔ t yt μtAt Btyt, one has
Δ
H Δ t, yt Styt, yt
SΔ tyt, yt SσtyΔ t, yt Sσtyσt, yΔ t
SΔ tyt, yt SσtAtyt, yt A∗ tSσtyt, yt
μtA∗ tSσtAtyt, yt SσtBtyt, yt
μtSσtBtyt, Atyt Sσtyt, Btyt
μtSσtAtyt, Btyt μtSσtBtyt, Btyt
2
≤ c1 ⊕ bH t, yt SσtBtyt
2
2
2μtAtSσtBtyt SσtBtyt
2
μtSσtBt2 yt .
4.7
Discrete Dynamics in Nature and Society
17
By the variation of constants formula, we have T t, s Id by 2.7 we obtain
T σt, t I σt
t
s
AvT v, sΔv. Thus,
AvT v, tΔv
4.8
t
Id μtAtT t, t Id μtAt.
Therefore, by Definition 2.5 and 2.7, 3.41, we obtain
Id μtAt ≤ Dea t, σt Deb σt, t Dec σt, t
D 1 μta D 1 μtb D 1 μtc
4.9
≤ 3D DMb c.
Thus, one has
μtAt ≤ Id μtAt Id ≤ 3D DMb c 1.
4.10
From Theorem 3.4, we know
St sup
Ht, x
≤ N2,
xt2
yt ≤ N V t, yt .
4.11
x∈Rn
Thus,
H Δ t, yt ≤ c1 ⊕ bH t, yt 2N 2 δ 23D 1 DMb DMcN 2 δ
4.12
MN δ N H t, yt .
2 2
2
Let Mδ : 2N 2 δ 23D 1 DMb DMcN 2 δ MN 2 δ2 N 2 . It is not hard to see that
Mδ → 0 when δ → 0. Set Eδ : c1 ⊕ b Mδ. Then when δ is sufficiently small, we
have Eδ < 0 and
1 μtEδ 1 μt −
c1 ⊕ b
Mδ
1 μtc1 ⊕ b
!
1
μtMδ > 0.
1 μtc1 ⊕ b
4.13
So Eδ is positive regressive. Then for Ht, yt > 0, 4.12 can be written as
H Δ t, y ≤ EδH t, yt .
4.14
18
Discrete Dynamics in Nature and Society
Notice that
Δ
H t, yt eEδ t, τ H Δ t, yt eEδ σt, τ EδeEδ t, τH t, yt
H Δ t, yt eEδ σt, τ
EδeEδ t, σteEδ σt, τH t, yt
H Δ t, yt eEδ σt, τ
4.15
1
eEδ σt, τH t, yt
1 Eδμt
H Δ t, y − EδH t, yt eEδ σt, τ ≤ 0.
Eδ
Thus, we get
H t, yt ≤ eEδ t, τH τ, yτ .
4.16
Then we can see that if yt ∈ Cs Vt , we have V t, yt < 0 and thus Ht, yt > 0. By 4.16
we have Hτ, yτ > 0 and thus V τ, yτ < 0, that is, yτ ∈ Cs Vτ . Let Ut, τ be the
linear evolution operator associated to 4.1. Then yt Ut, τyτ. Thus, we get
Uτ, tCs Vt ⊂ Cs Vτ .
4.17
Let L1 t Qt Rt. We can see that L1 txt is also a solution of 3.1. By 4.2 one has
H Δ t, L1 txt SΔ t SσtAt A∗ tSσt
μtA∗ tSσtAt L1 txt, L1 txt .
4.18
Similar to the proof of 4.5, we have H Δ t, L1 txt ≤ c2 ⊕ bHt, L1 txt. Then we get
SΔ t SσtAt A∗ tSσt μtA∗ tSσtAt ≤ c2 ⊕ bSt.
4.19
Following the process for getting 4.12, one has
H Δ t, yt ≤ c2 ⊕ bH t, yt MδH t, yt .
4.20
Let Fδ : c2 ⊕ b − Mδ. When δ is sufficiently small, we obtain Fδ < 0 and
1 μtFδ 1 μt −
c2 ⊕ b
− Mδ
1 μtc2 ⊕ b
1
− Mδ.
1 μta ⊕ b
!
4.21
Discrete Dynamics in Nature and Society
19
So Fδ is positive regressive. Then for Ht, yt < 0, 4.20 can be written as
H Δ t, y ≤ FδH t, yt .
4.22
Following the similar process as 4.15, we get
H t, yt ≤ eFδ t, τH τ, yτ .
4.23
Then if yτ ∈ Cu Vτ , we have V τ, yτ > 0 and thus Hτ, yτ < 0. By 4.23 we have
Ht, yt < 0 and thus V t, yt > 0, that is, yt ∈ Cu Vt . So we get
Ut, τCu Vτ ⊂ Cu Vt .
4.24
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
Hτu V :
Uτ, rCu Vr ⊂ Cu Vτ ,
r∈T
Hτs V :
4.25
Uτ, rCs Vr ⊂ Cs Vτ .
r∈T
By the notion of Lyapunov function, we know that there exist two subspaces Dτu V and
Dτs V such that Dτu V ⊂ Hτu V , Dτs V ⊂ Hτs V , and Dτu V ⊕ Dτs V Rn .
Now, we prove that V 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 Eδ < Fδ. Then there are constants r1 , r2 such that Eδ ≤ r1 < r2 ≤ Fδ.
Thus, by inequality 4.16, for yτ ∈ Dτs one has
V 2 t, yt ≤ eEδ t, τV 2 τ, yτ
≤ er1 t, τV 2 τ, yτ .
4.26
By inequality 4.23, for yτ ∈ Dτu , we obtain
V 2 t, yt ≥ eFδ t, τV 2 τ, yτ
≥ er2 t, τV 2 τ, yτ .
4.27
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 that Wt, y is also a λ, μ-Lyapunov function of 4.1. Similar to
the consideration of Ht, y, for Gt, y we have that if y ∈ Cs Wt , then Wt, yt < 0, that
is, Gt, yt > 0. Thus, we get
GΔ t, yt ≤ d1 ⊕ b MδG t, yt .
4.28
20
Discrete Dynamics in Nature and Society
Let E δ : d1 ⊕ b Mδ. By the similar deducing as 4.16, one has
G t, yt ≤ eE δ t, τG τ, yτ .
4.29
Thus, when Gt, yt > 0, that is, Wt, yt < 0, we have Gτ, yτ > 0, that is, Wτ, yτ <
0. That means
Uτ, tCs Wt ⊂ Cs Wτ .
4.30
When y ∈ Cu Wt , we have Wt, yt > 0, that is, Gt, yt < 0. Then
GΔ t, yt ≤ d2 ⊕ b − MδG t, yt .
4.31
Let F δ d2 ⊕ b − Mδ. By the similar deducing as 4.15, one has
G t, yt ≤ eF δ t, τG τ, yτ .
4.32
Then when Gτ, yτ < 0, that is, Wτ, yτ > 0, we have Gt, yt < 0, that is, Wτ, yτ >
0. That means
Ut, τCu Wτ ⊂ Cu Wt .
4.33
By 4.30 and 4.33, we know that Wt, y is a Lyapunov function of 4.1.
Since d1 < 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
Hτu W :
Uτ, rCu Wr ⊂ Cu Wτ ,
r∈T
Hτs W :
4.34
Uτ, rCs Wr ⊂ Cs Wτ .
r∈T
By the notion of Lyapunov function, we can know that there exist Dτu W and Dτs W such
that Dτu W ⊂ Hτu W, Dτs W ⊂ Hτs W, and Dτu W ⊕ Dτs W Rn .
Furthermore, by 4.29 and 4.32, for any t ≥ τ, when yτ ∈ Dτs W, one has
W 2 t, yt ≤ eE δ t, τW 2 τ, yτ ≤ el1 t, τW 2 τ, yτ ,
4.35
and when yτ ∈ Dτu W, one has
W 2 t, yt ≥ eF δ t, τW 2 τ, yτ ≥ el2 t, τW τ, yτ .
4.36
Thus, we can see that Wt, 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 Wt, x with
Discrete Dynamics in Nature and Society
21
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.
Acknowledgment
The author expresses her sincere thanks to Professors Yong Li and Zhenxin Liu for their
encouragement and valuable suggestions.
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