R E S E A R C H
Open Access
Existence and exponential stability of
periodic solutions for a class of Hamiltonian
systems on time scales
Li Yang
1, Yongzhi Liao
2and Yongkun Li
1**Correspondence: [email protected] 1Department of Mathematics,
Yunnan University, Kunming, Yunnan 650091, People’s Republic of China
Full list of author information is available at the end of the article
Abstract
In this paper, by using a fixed point theorem and the theory of calculus on time scales, we obtain some sufficient conditions for the existence and exponential stability of periodic solutions for a class of Hamiltonian systems on time scales. We also present numerical examples to show the feasibility of our results. The results of this paper are completely new and complementary to the previously known results even if the time scaleT=RorZ.
Keywords: periodic solution; Hamiltonian system; exponential stability; time scale
1 Introduction
Hamiltonian system, which was introduced by the Irish mathematician SWR Hamilton, is widely used in mathematical sciences, life sciences and so on. Many models in celestial mechanics, plasma physics, space science and bio-engineering are in the form of a Hamil-ton system. Therefore, the study of a HamilHamil-ton system is useful and meaningful in theory and practice. Recently, various Hamiltonian systems have been extensively studied (see [–] and references cited therein).
Most existing results on the study of Hamiltonian systems are for continuous systems. However, both discrete and continuous systems are important in applications, and it is troublesome to study the dynamical properties for continuous and discrete systems, re-spectively. Therefore, it is meaningful to study dynamical systems on time scales (see [– ] and references cited therein), which helps avoid proving results twice, once for differ-ential equations and once for difference equations. There have been some results devoted to Hamiltonian systems on time scales [, , –]. For example, authors in [] intro-duced Hamiltonian systems on time scales and authors in [, ] studied the following Hamiltonian system on time scales:
x(t) =α(t)x(σ(t)) +β(t)y(t),
y(t) = –γ(t)x(σ(t)) –α(t)y(t), (.)
wheret∈T. In [], authors obtained inequalities of Lyapunov for (.), and authors in []
studied the stability of (.) by using Floquet theory. However, to the best of our knowl-edge, up to now, there have been no papers published on the existence and exponential stability of a periodic solution to (.).
Motivated by the above mentioned works, in this paper, we study the existence and ex-ponential stability of periodic solutions to (.), in whichTis a periodic time scale. The main aim of this paper is to study the existence of periodic solutions to (.) by using a fixed point theorem. Moreover, we also study the exponential stability of the periodic so-lution to (.). Our results are new and complementary to the previously known results even if the time scaleT=RorZ.
Remark . It is obvious that whenT=R, (.) reduces to the following continuous time Hamiltonian system:
x(t) =α(t)x(t) +β(t)y(t), y(t) = –γ(t)x(t) –α(t)y(t), t∈R.
WhenT=Z, (.) reduces to the following discrete time Hamiltonian system:
x(n) =α(n)x(n+ ) +β(n)y(n), y(n) = –γ(n)x(n+ ) –α(n)y(n), n∈Z.
For convenience, we denote [a,b]T={t|t∈[a,b]∩T} andϑ=supt∈Tμ(t). For anω -periodic functionf :T→R, we denotef+=maxt∈[,ω]T|f(t)|,f–=mint∈[,ω]T|f(t)|and
¯
f =maxt∈[,ω]Tf(t).
Throughout this paper, we assume that
(H) β(t),γ(t)∈Crd(T,R),α(t)∈Crd(T, (, +∞))are allω-periodic functions ande–α(ω,
)= ,–α∈R+, whereR+=R+(T,R) ={r: +μ(t)r(t) > ,∀t∈T}.
2 Preliminaries
In this section, we introduce some definitions and state some preliminary results.
Definition .[] LetTbe a nonempty closed subset (time scale) ofR. The forward
and backward jump operatorsσ,ρ:T→Tand the graininessμ:T→R+ are defined,
respectively, by
σ(t) =inf{s∈T:s>t}, ρ(t) =sup{s∈T:s<t} and μ(t) =σ(t) –t.
Definition .[] A pointt∈Tis called left-dense ift>infTandρ(t) =t, left-scattered ifρ(t) <t, right-dense ift<supTandσ(t) =t, and right-scattered ifσ(t) >t. IfThas a left-scattered maximumm, thenTk=T\ {m}; otherwiseTk=T. IfThas a right-scattered minimumm, thenTk=T\ {m}; otherwiseTk=T.
Definition .[] A functionr:T→Ris called regressive if
+μ(t)r(t)=
for allt∈Tk. Ifris a regressive function, then the generalized exponential functione ris
defined by
er(t,s) =exp
t
s
ξμ(τ)
r(τ)τ
with the cylinder transformation
ξh(z) =
Log(+hz)
h ifh= ,
z ifh= .
Letp,q:T→Rbe two regressive functions, we define
p⊕q:=p+q+μpq, p:= – p
+μp, pq:=p⊕(q).
Then the generalized exponential function has the following properties.
Lemma .[] Assume that p,q:T→Rare two regressive functions,then
(i) e(t,s)≡andep(t,t)≡;
(ii) ep(σ(t),s) = ( +μ(t)p(t))ep(t,s); (iii) ep(t,s) =ep(s,t)=ep(s,t); (iv) ep(t,s)ep(s,r) =ep(t,r).
Lemma .[] Assume that f,g:T→Rare delta differentiable at t∈Tk,then (i) (νf +νg)=νf+νgfor any constantsν,ν;
(ii) (fg)(t) =f(t)g(t) +f(σ(t))g(t) =f(t)g(t) +f(t)g(σ(t)).
Lemma .[] Assume that p(t)≥for t≥s,then ep(t,s)≥.
Definition .[] A functionf :T→Ris positively regressive if +μ(t)f(t) > for all
t∈T.
Lemma .[] Suppose that p∈R+,then
(i) ep(t,s) > for allt,s∈T;
(ii) ifp(t)≤q(t)thenep(t,s)≤eq(t,s)fort,s∈T.
Lemma .[] If p∈R+and p(t) < for all t∈T,then for all s∈Twith s≤t,we have
<ep(t,s)≤exp
t
s
p(u)u < .
Lemma .[] Let x∈ {x∈C(T,R)|x(t+ω) =x(t)}.Then xσexists andxσ=x,
wherex=maxt∈[,ω]T|x(t)|.
Lemma .[] If p∈Rand a,b,c∈T,then
ep(c,·)= –pep(c,·)σ
and
b
a
p(t)ep
Definition . Letz∗(t) = (x∗(t),y∗(t))Tbe a solution of (.) andz(t) = (x(t),y(t))Tbe an arbitrary solution of (.). If there exist positive constantsM> andλwithλ∈R+such that fort∈T,
z(t) –z∗(t)≤Mz–z∗eλ(t,t), t∈T,t≥t,
where|z(t)|=max{|x(t) –x∗(t)|,|y(t) –y∗(t)|},z–z∗=max{|z–z*|,|z–z*|},|zi–
z*
i|=maxs∈(–∞,t]T|zi(s) –z*i(s)|,i= , , then the solutionz∗(t) is said to be exponentially stable.
3 Existence and uniqueness
Lemma . Let(H)hold.Then everyω-periodic solution(x(t),y(t))Tof (.)is of the form:
x(t) =
e–α(ω,)–
t+ω
t β(s)y(s)e–α(s,t)s,
y(t) =–e
(–α)(ω,) t+ω
t
e(–α)(s,t)
–μ(s)α(s)γ(s)x(σ(s))s.
Proof Let (x(t),y(t))Tbe anω-periodic solution of (.). We can rewrite the first equation of (.) as follows:
x(t) –α(t)xσ(t)=β(t)y(t).
Multiplying both sides of the above equation bye–α(t, ), we have
e–α(t, )x(t)
=e–α(t, )β(t)y(t).
Integrating both sides of this equation fromttot+ωand noticing thatx(t+ω) =x(t), we have
x(t) =
e–α(ω, ) –
t+ω
t
β(s)y(s)e–α(s,t)s.
On the other hand, we rewrite the second equation of (.) as follows:
–α(t)μ(t)y(t) +α(t)yσ(t)= –γ(t)xσ(t), which is equivalent to
y(t) +–α(t)yσ(t)= – γ(t)
–α(t)μ(t)x
σ(t).
Multiplying both sides of the above equation bye(–α)(t, ), we have
e(–α)(t, )y(t)
= –γ(t)xσ(t)e(–α)
σ(t), .
Integrating both sides of this equation fromttot+ωand noticing thaty(t+ω) =y(t), we have
y(t) = –e(–α)(ω, )
t+ω
t
e(–α)(s,t)
–μ(s)α(s)γ(s)x
σ(s)s,
Theorem . Assume that(H)and
hold.Then(.)has a unique periodic solution.
Proof LetX={z(t) = (x(t),y(t))T∈C(T,R)|x(t+ω) =x(t),y(t+ω) =y(t)}with the norm
z=max{|x|,|y|}, where|x|=maxt∈[,ω]T|x(t)|and|y|=maxt∈[,ω]T|y(t)|. ThenXis a Banach space. Forz∈X, define the following operator:
:X→X, z= (x,y)T→z= (z,z)T,
Next, we prove thatis a contraction mapping. Together with
for anyz(t) = (x(t),y(t))T,z(t) = (x(t),y(t))T∈X, we have
It follows thatis a contraction. Thereforehas a fixed point inX, that is, (.) has a
unique periodic solution inX. This completes the proof.
4 Exponential stability of periodic solution
In this section, we study the exponential stability of the periodic solution to (.).
Theorem . Assume that(H)and(H)hold.Suppose further thatα–>γ+.Then the
periodic solution of (.)is exponentially stable.
Take a constantλ> with –λ∈R+such thatλ>β+( +ϑ λ) andα–
>γ+, where=
mint∈[,ω]Teλ(t,t). Let
M>max
λ
λ–β+( +ϑ λ),
α–
α– –γ+
,
where=maxt∈[,ω]Teλ(t,t). It is easy to verify thatM> and hence, we have
w(t)≤ w<Meλ(t,t)w, ∀t∈(–∞,t]T.
We claim that
w(t)<Meλ(t,t)w, ∀t∈(t, +∞)T, (.)
which means that
u(t)<Meλ(t,t)w, ∀t∈(t, +∞)T (.)
and
v(t)<Meλ(t,t)w, ∀t∈(t, +∞)T. (.)
By way of contradiction, assume that (.) does not hold, then we have the following three cases.
Case one:(.) is true and (.) is not true. Then there existst∈(t, +∞)Tsuch that
u(t)≥Meλ(t,t)w, u(t)<Meλ(t,t)w, t∈(t,t)T.
Hence, there must be a constantp≥ such that
u(t)=pMeλ(t,t)w, u(t)<Meλ(t,t)w, t∈(t,t)T.
In view of (.), we have
u(t)=
e(–α)(t,t)u(t) +
t
t
e(–α)(t,s)β(s)v(s)s
≤e(–α)(t,t)u(t)+
t
t
e(–α)(t,s)β(s)v(s)s
≤e(–α)(t,t)w+β+pMw
t
t
e(–α)(t,s)eλ(s,t)s
=e(–α)(t,t)w+β+pMweλ(t,t)
t
t
e–αλ(s,t)s
=pMweλ(t,t)
pMe–αλ(t,t) +β
+
t
t
e–αλ(s,t)s
≤pMweλ(t,t)
Meλ(t,t) +β
+
t
t
=pMweλ(t,t)
Case three:Both (.) and (.) are untrue. By case one and case two, we can obtain a contradiction. Therefore, (.) holds. Hence, we have that
z(t) –z∗(t)≤Mz–z∗eλ(t,t), t∈T,t≥t,
which means that the periodic solutionz∗(t) of (.) is exponentially stable. This completes
the proof.
By Theorem . in [], we have the following corollary.
or
ω
α(t)t+
ω
β(t)t
/ ω
γ+(t)t /
< , γ+(t) =max
,γ(t).
Then(.)has a stableω-periodic solution.
5 Examples
In this section, we present two examples to illustrate the feasibility of our results obtained in previous sections.
Example . Consider the following Hamiltonian system onT=R:
x(t) =α(t)x(t) +β(t)y(t),
y(t) = –γ(t)x(t) –α(t)y(t), t∈R, (.)
in which we take the coefficients as follows:
α(t) = +sint, β(t) = .sint, γ(t) = .cost.
SinceT=R, thenμ(t) = . By calculating, we have
α+=α¯= , α–= , β+= ., γ+= .,
exp
ω
–α(s)ds
=e–π
andθ≈. < . All the conditions in Theorem . and Theorem . are satisfied. Hence, (.) has an exponentially stable π-periodic solution.
Example . Consider the following Hamiltonian system onT=Z:
x(n) =α(n)x(n+ ) +β(n)y(n),
y(n) = –γ(n)x(n+ ) –α(n)y(n), n∈Z, (.)
in which we take the coefficients as follows:
α(n) = . + .sinnπ
, β(n) = .cos
nπ
, γ(t) = .sin
nπ
.
SinceT=Z, thenμ(t) = . By calculating, we haveα+=α¯= .,α–= .,β+= .,γ+= . and
| –ωk=–( –α(k))|exp{ω–log| –α(k)|}
|ω–
k=( –α(k))|
≈., θ≈. < .
Remark . Since in (.) and (.),β(t) orβ(n) may be negative, Theorem . in [] is not suitable for our examples. But from our results, we can obtain that both (.) and (.) have exponentially stableω-periodic solutions.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the manuscript and typed, read and approved the final manuscript.
Author details
1Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, People’s Republic of China.2School of
Mathematics and Computer Science, Panzhihua University, Panzhihua, Sichuan 617000, People’s Republic of China.
Acknowledgements
This study was supported by the National Natural Sciences Foundation of People’s Republic of China under Grant 10971183.
Received: 6 January 2013 Accepted: 7 June 2013 Published: 25 June 2013 References
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