Hindawi Publishing Corporation
International Journal of Differential Equations
Volume 2011, Article ID 871574, 11 pages
doi:10.1155/2011/871574
Research Article
Topological Conjugacy between Two Kinds of
Nonlinear Differential Equations via Generalized
Exponential Dichotomy
Xiaodan Chen and Yonghui Xia
Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China
Correspondence should be addressed to Yonghui Xia, yhxia@zjnu.cn
Received 3 July 2011; Accepted 15 August 2011
Academic Editor: Yuri V. Rogovchenko
Copyright q 2011 X. Chen and Y. Xia. 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.
Based on the notion of generalized exponential dichotomy, this paper considers the topological
decoupling problem between two kinds of nonlinear differential equations. The topological
equivalent function is given.
1. Introduction and Motivation
Well-known Hartman’s linearization theorem for differential equations states that a 1:1
cor-respondence exists between solutions of a linear autonomous system ẋ Ax and
those of the perturbed system ẋ Ax fx, as long as f fulfills some goodness
conditions, like smallness, continuity, or being Hartman 1. Based on the exponential
dichotomy, Palmer 2 extended this result to the nonautonomous system. Some other
improvements of Palmer’s linearization theorem are reported in the literature. For examples,
one can refer to Shi 3, Jiang 4, and Reinfelds 5, 6. Recently, Xia et al. 7 generalized
Palmer’s linearization theorem to the dynamic systems on time scales. Consider the linear
system
ẋ Atx,
where x ∈ Rn and At is a n × n matrix function.
1.1
2
International Journal of Differential Equations
Definition 1.1. System 1.1 is said to possess an exponential dichotomy 8 if there exists a
projection P and constants K > 0, α > 0 such that
UtP U−1 s ≤ Ke−αt−s ,
UtI − P U−1 s ≤ Keαt−s ,
for s ≤ t, s, t ∈ R,
1.2
for t ≤ s, s, t ∈ R
hold, where Ut is a fundamental matrix of linear system ẋ Atx.
However, Lin 9 argued that the notion of exponential dichotomy considerably
restricts the dynamics. It is thus important to look for more general types of hyperbolic
behavior. Lin 9 proposed the notion of generalized exponential dichotomy which is more
general than the classical notion of exponential dichotomy.
Definition 1.2. System 1.1 is said to have a generalized exponential dichotomy if there exists
a projection P and K ≥ 0 such that
t
−1
UtP U s ≤ K exp − ατdτ ,
t ≥ s,
s
t
−1
ατdτ ,
UtI − P U s ≤ K exp
1.3
t ≤ s,
s
where αt is a continuous function with αt
0
∞, limt → −∞ t αξdξ ∞.
≥
0, satisfying limt → ∞
t
0
αξdξ
Example 1.3. Consider the system
ẋ1
ẋ2
⎛
⎞
1
−
0
⎜ 3 |t| 1
⎟ x1
⎜
⎟
⎝
.
⎠
1
x2
0
3
|t| 1
1.4
Then, system 1.4 has a generalized exponential dichotomy, but the classical exponential
dichotomy cannot be satisfied.
For this reason, basing on generalized exponential dichotomy, we consider the
topological decoupling problem between two kinds of nonlinear differential equations. We
prove that there is a 1 : 1 correspondence existing between solutions of topological decoupling
systems, namely, ẋt Atxt ft, x and ẋt Atxt gt, x.
International Journal of Differential Equations
3
2. Existence of Equivalent Function
Consider the following two nonlinear nonautonomous systems:
ẋ Atx ft, x,
2.1
ẋ Atx gt, x,
2.2
where x ∈ Rn , At, Bt are n × n matrices.
Definition 2.1. Suppose that there exists a function H : R × Rn → Rn such that
i for each fixed t, Ht, · is a homeomorphism of Rn into Rn ;
ii Ht, x → ∞ as |x| → ∞, uniformly with respect to t;
iii assume that Gt, · H −1 t, · has property ii too;
iv if xt is a solution of system 2.1, then Ht, xt is a solution of system 2.2.
If such a map H exists, then 2.1 is topologically conjugated to 2.2. H is called an equivalent
function.
Theorem 2.2. Suppose that ẋ Atx has a generalized exponential dichotomy. If ft, x, gt, x
fulfill
ft, x ≤ Ft,
ft, x1 − ft, x2 ≤ rt|x1 − x2 |,
gt, x ≤ Gt,
gt, x1 − gt, x2 ≤ rt|x1 − x2 |,
2.3
Nt, F, G ≤ B,
Nt, r ≤ L < 1,
where
Nt, F, G t
−∞
t
K exp − α ϕ dϕ Fs Gsds,
s
∞
K exp
t
Nt, r t
−∞
t
α ϕ dϕ Fs Gsds,
2.4
s
t
t
∞
K exp − α ϕ dϕ rsds K exp
α ϕ dϕ rsds,
s
t
s
where Ft, Gt, rt ≥ 0 are integrable functions and B, L are positive constants, then the nonlinear
4
International Journal of Differential Equations
nonautonomous system 2.1 is topologically equivalent to the nonlinear nonautonomous system
2.2. Moreover, the equivalent functions Ht, x, Gt, y fulfill
G t, y − y ≤ B.
|Ht, x − x| ≤ B,
2.5
In what follows, we always suppose that the conditions of Theorem 2.2 are satisfied.
Denote that Xt, t0 , x0 is a solution of 2.2 satisfying the initial condition Xt0 x0 and that
Y t, t0 , y0 is a solution of 2.1 satisfying the initial condition Y t0 y0 . To prove the main
results, we first prove some lemmas.
Lemma 2.3. For each τ, ξ, system
z Atz − ft, Xt, τ, ξ gt, Xt, τ, ξ z
2.6
has a unique bounded solution ht, τ, ξ with |ht, τ, ξ| ≤ B.
Proof. Let B be the set of all the continuous bounded functions xt with |xt| ≤ B. For each
τ, ξ and any zt ∈ B, define the mapping T as follows:
T zt t
UtP U−1 s gs, Xs, τ, ξ z − fs, Xs, τ, ξ ds
−∞
∞
−
UtI − P U s gs, Xs, τ, ξ z − fs, Xs, τ, ξ ds.
2.7
−1
t
Simple computation leads to
|T zt| ≤
t UtP U−1 sFs Gsds
−∞
∞ UtI − P U−1 sFs Gsds
t
≤
t
−∞
t
K exp − α ϕ dϕ Fs Gsds
s
t
∞
K exp
t
≤ B,
s
α ϕ dϕ Fs Gsds
2.8
International Journal of Differential Equations
5
which implies that T is a self-map of a sphere with radius B. For any z1 t, z2 t ∈ B,
|T z1 t − T z2 t| ≤
t UtP U−1 srsz1 s − z2 sds
−∞
∞ UtI − P U−1 srsz1 s − z2 sds
t
≤ z1 − z2 t
−∞
t
K exp − α ϕ dϕ rsds
t
∞
K exp
t
2.9
s
α ϕ dϕ rsds
s
≤ Lz1 − z2 .
Due to the fact that L < 1, T has a unique fixed point, namely, z0 t, and
z0 t t
−∞
−
UtP U−1 s gs, Xs, τ, ξ z0 s − fs, Xs, τ, ξ ds
∞
UtI − P U−1 s gs, Xs, τ, ξ z0 s − fs, Xs, τ, ξ ds,
2.10
t
it is easy to show that z0 t is a bounded solution of 2.6. Now, we are going to show that
the bounded solution is unique. For this purpose, we assume that there is another bounded
solution z1 t of 2.6. Thus, z1 t can be written as follows:
z1 t UtU−1 0x0
t
UtU−1 s gs, Xs, τ, ξ z1 s − fs, Xs, τ, ξ ds
0
UtU−1 0x0
t
UtP I − P U−1 s gs, Xs, τ, ξ z1 s − fs, Xs, τ, ξ ds
0
UtU−1 0x0
t
UtP U−1 s gs, Xs, τ, ξ z1 s − fs, Xs, τ, ξ ds
−∞
−
0
−∞
∞
UtP U−1 s gs, Xs, τ, ξ z1 s − fs, Xs, τ, ξ ds
UtI − P U−1 s gs, Xs, τ, ξ z1 s − fs, Xs, τ, ξ ds
0
−
∞
t
UtI − P U−1 s gs, Xs, τ, ξ z1 s − fs, Xs, τ, ξ ds.
2.11
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International Journal of Differential Equations
Note that
0
−∞
UtP U−1 s gs, Xs, τ, ξ z1 s − fs, Xs, τ, ξ ds
−1
UtU 0
0
−∞
U0P U−1 s gs, Xs, τ, ξ z1 s − fs, Xs, τ, ξ ds
0
≤ UtU−1 0
U0P U−1 s gs, Xs, τ, ξ z1 s − fs, Xs, τ, ξ ds
−∞
−1
≤ UtU 0
0
−∞
2.12
0
K exp − α ϕ dϕ Fs Gsds,
s
0
which implies that −∞ U0P U−1 sgs, Xs, τ, ξ z1 s − fs, Xs, τ, ξds is convergent;
denote it by x1 . That is,
0
UtP U−1 s gs, Xs, τ, ξ z1 s − fs, Xs, τ, ξ ds UtU−1 0x1 .
2.13
UtI − P U−1 s gs, Xs, τ, ξ z1 s − fs, Xs, τ, ξ ds UtU−1 0x2 .
2.14
−∞
Similarly,
∞
0
Therefore, it follows from the expression of z1 t that
z1 t UtU−1 0x0 − x1 x2 −
t
−∞
∞
UtP Us gs, Xs, τ, ξ z1 s − fs, Xs, τ, ξ ds
2.15
UtI − P U−1 s gs, Xs, τ, ξ z1 s − fs, Xs, τ, ξ ds.
t
t
Noticing that z1 t is bounded, −∞ UtP Usgs, Xs, τ, ξ z1 s − fs, Xs,
∞
τ, ξds − t UtI − P U−1 sgs, Xs, τ, ξ z1 s − fs, Xs, τ, ξds is also bounded.
So, UtU−1 0x0 − x1 x2 is bounded. But we see that z Atz does not have a nontrivial
bounded solution. Thus, UtU−1 0x0 − x1 x2 0; it follows that
z1 t t
−∞
−
UtP U−1 s gs, Xs, τ, ξ z1 s − fs, Xs, τ, ξ ds
∞
t
UtI − P U−1 s gs, Xs, τ, ξ z1 s − fs, Xs, τ, ξ ds.
2.16
International Journal of Differential Equations
7
Simple calculating shows
|z1 t − z0 t| ≤
t UtP U−1 srs|z1 s − z0 s|ds
−∞
∞ UtI − P U−1 srs|z1 s − z0 s|ds
t
t
≤ z1 − z0 −∞
t
K exp − α ϕ dϕ rs
s
t
∞
K exp
t
2.17
α ϕ dϕ rs
s
≤ Lz1 − z0 .
Therefore, z1 − z0 ≤ Lz1 − z0 , consequently z1 t ≡ z0 t. This implies that the
bounded solution of 2.6 is unique. We may call it htτ, ξ. From the above proof, it is easy
to see that |ht, τ, ξ| ≤ B.
Lemma 2.4. For each τ, ξ, the system
z Atz ft, Xt, τ, ξ z − gt, Xt, τ, ξ
2.18
has a unique bounded solution g t, τ, ξ and |
g t, τ, ξ| ≤ B.
Proof. The proof is similar to that of Lemma 2.3.
Lemma 2.5. Let xt be any solution of the system 2.1, then zt 0 is the unique bounded solution
of system
z Atz ft, xt z − ft, xt.
2.19
Proof. Obviously, z ≡ 0 is a bounded solution of system 2.19. We show that the bounded
solution is unique. If not, then there is another bounded solution z1 t, which can be written
as follows:
−1
z1 t UtU 0z1 0 t
0
UtU−1 s fs, Xs, τ, ξ z1 s − fs, Xs, τ, ξ ds. 2.20
8
International Journal of Differential Equations
By Lemma 2.3, we can get
z1 t t
−∞
UtP U−1 s fs, Xs, τ, ξ z1 s − fs, Xs, τ, ξ ds
∞
−
UtI − P U−1 s fs, Xs, τ, ξ z1 s − fs, Xs, τ, ξ ds.
2.21
t
It follows that
|z1 t| ≤
t UtP U−1 srs|z1 s|ds
−∞
∞ UtI − P U−1 srs|z1 s|ds
t
≤
t
−∞
t
K exp − α ϕ dϕ rs|z1 s|ds
2.22
s
t
∞
K exp
t
α ϕ dϕ rs|z1 s|ds
s
≤ L|z1 t|.
That is, z1 ≤ L|z1 |. Consequently, z1 t ≡ 0. This completes the proof of Lemma 2.5.
Lemma 2.6. Let yt be any solution of the system 2.2, then zt 0 is the unique bounded solution
of system
z Atz g t, yt z − g t, yt .
2.23
Proof. Obviously, z ≡ 0 is a bounded solution of system 2.23. We will show that the bounded
solution is unique. If not, then there is another bounded solution z1 t. Then, z1 t can be
written as follows:
z1 t UtU−1 0z1 0 t
UtU−1 s gs, Y s, τ, ξ z1 s − gs, Y s, τ, ξ ds. 2.24
0
By Lemma 2.3, we can get
z1 t t
−∞
−
UtP U−1 s gs, Y s, τ, ξ z1 s − gs, Y s, τ, ξ ds
∞
t
UtI − P U−1 s gs, Y s, τ, ξ z1 s − gs, Y s, τ, ξ ds.
2.25
International Journal of Differential Equations
9
Then, it follows that
|z1 t| ≤
t UtP U−1 srs|z1 s|ds
−∞
∞ UtI − P U−1 srs|z1 s|ds
t
≤
t
−∞
t
K exp − α ϕ dϕ rs|z1 s|ds
2.26
s
t
∞
K exp
t
α ϕ dϕ rs|z1 s|ds
s
≤ L|z1 t|.
That is, z1 ≤ Lz1 . Consequently, z1 t ≡ 0. This completes the proof of Lemma 2.6.
Now, we define two functions as follows:
Ht, x x ht, t, x,
Gt, x y g t, t, y .
2.27
2.28
Lemma 2.7. For any fixed t0 , x0 , Ht, Xt, t0 , x0 is a solution of the system 2.2.
Proof. Replace τ, ξ by t, Xt, τ, ξ in 2.6; system 2.6 is not changed. Due to the
uniqueness of the bounded solution of 2.6, we can get that ht, t, Xt, t0 , x0 ht, t0 , x0 .
Thus,
Ht, Xt, t0 , x0 Xt, t0 , x0 ht, t0 , x0 .
2.29
Differentiating it, noticing that Xt, t0 , x0 , ht, t0 , x0 are the solutions of the 2.1, and 2.6,
respectively; therefore, we can obtain
Ht, Xt, t0 , x0 AtXt, t0 , x0 ft, Xt, t0 , x0 Atht, t0 , x0 − ft, Xt, t0 , x0 gt, Xt, t0 , x0 ht, t0 , x0 AtHt, Xt, t0 , x0 gt, Ht, t0 , x0 .
It indicates that Ht, Xt, t0 , x0 is the solution of the system 2.2.
Lemma 2.8. For any fixed t0 , y0 , Gt, Y t, t0 , y0 is a solution of the system 2.1.
Proof. The proof is similar to Lemma 2.7.
Lemma 2.9. For any t ∈ R, y ∈ Rn , Ht, Gt, y ≡ y.
2.30
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International Journal of Differential Equations
Proof. Let yt be any solution of the system 2.2. From Lemma 2.8, Gt, yt is a solution of
system 2.1. Then, by Lemma 2.7, we see that Ht, Gt, yt is a solution of 2.2, written as
y1 t. Denote Jt y1 t − yt. Differentiating, we have
J t y1 t − y t
Aty1 t g t, y1 t − Atyt − g t, yt
AtJt g t, yt Jt − g t, yt ,
2.31
which implies that Jt is a solution of system 2.23. On the other hand, following the
definition of H and G and Lemmas 2.3 and 2.4, we can obtain
|Jt| H t, G t, yt − yt
≤ H t, G t, yt − G t, yt G t, yt − yt
h t, t, G t, yt g t, t, y 2.32
≤ 2B.
This implies that Jt is a bounded solution of system 2.23. However, by Lemma 2.6,
system 2.23 has only one zero solution. Hence, Jt ≡ 0, consequently y1 t ≡ yt, that is,
Ht, Gt, y yt. Since yt is any solution of the system 2.2, Lemma 2.9 follows.
Lemma 2.10. For any t ∈ R, x ∈ Rn , Gt, Ht, x ≡ x.
Proof. The proof is similar to Lemma 2.10.
Now, we are in a position to prove the main results.
Proof of Theorem 2.2. We are going to show that Ht, · satisfies the four conditions of
Definition 2.1.
Proof of Condition (i)
For any fixed t, it follows from Lemmas 2.9 and 2.10 that Ht, · is homeomorphism and
Gt, · H −1 t, ·.
Proof of Condition (ii)
From 2.27 and Lemma 2.3, we derive |Ht, x − x| |ht, t, x| ≤ B. So, |Ht, x| → ∞ as
|x| → ∞, uniformly with respect to t.
Proof of Condition (iii)
From 2.23 and Lemma 2.4, we derive |Gt, y − y| |
g t, t, y| ≤ B. So, |Gt, y| → ∞ as
|x| → ∞, uniformly with respect to t.
International Journal of Differential Equations
11
Proof of Condition (iv)
Using Lemmas 2.7 and 2.8, we easily prove that Condition iv is true.
Hence, systems 2.1 and 2.2 are topologically conjugated. This completes the proof
of Theorem 2.2.
Acknowledgments
The authors would like to express their gratitude to the editor and anonymous reviewers
for their careful reading which improved the presentation of this paper. This work was
supported by the National Natural Science Foundation of China under Grant no. 10901140
and ZJNSFC under Grant no. Y6100029.
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