Existence and exponential stability of periodic solutions for a class of Hamiltonian systems on time scales
© Yang et al.; licensee Springer 2013
Received: 6 January 2013
Accepted: 7 June 2013
Published: 25 June 2013
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 scale or ℤ.
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 Hamilton system. Therefore, the study of a Hamilton system is useful and meaningful in theory and practice. Recently, various Hamiltonian systems have been extensively studied (see [1–5] and references cited therein).
where . In , authors obtained inequalities of Lyapunov for (1.1), and authors in  studied the stability of (1.1) by using Floquet theory. However, to the best of our knowledge, up to now, there have been no papers published on the existence and exponential stability of a periodic solution to (1.1).
Motivated by the above mentioned works, in this paper, we study the existence and exponential stability of periodic solutions to (1.1), in which is a periodic time scale. The main aim of this paper is to study the existence of periodic solutions to (1.1) by using a fixed point theorem. Moreover, we also study the exponential stability of the periodic solution to (1.1). Our results are new and complementary to the previously known results even if the time scale or ℤ.
For convenience, we denote and . For an ω-periodic function , we denote , and .
Throughout this paper, we assume that
(H1) , are all ω-periodic functions and , , where .
In this section, we introduce some definitions and state some preliminary results.
Definition 2.1 
Definition 2.2 
A point is called left-dense if and , left-scattered if , right-dense if and , and right-scattered if . If has a left-scattered maximum m, then ; otherwise . If has a right-scattered minimum m, then ; otherwise .
Definition 2.3 
Then the generalized exponential function has the following properties.
Lemma 2.1 
Assume that are two regressive functions, then
(i) and ;
Lemma 2.2 
Assume that are delta differentiable at , then
(i) for any constants , ;
Lemma 2.3 
Assume that for , then .
Definition 2.4 
A function is positively regressive if for all .
Lemma 2.4 
Suppose that , then
(i) for all ;
(ii) if then for .
Lemma 2.5 
Lemma 2.6 
Let . Then exists and , where .
Lemma 2.7 
where , , , , then the solution is said to be exponentially stable.
3 Existence and uniqueness
which completes the proof. □
Theorem 3.1 Assume that (H1) and
hold. Then (1.1) has a unique periodic solution.
which means that .
It follows that Φ is a contraction. Therefore Φ has a fixed point in , that is, (1.1) has a unique periodic solution in . This completes the proof. □
4 Exponential stability of periodic solution
In this section, we study the exponential stability of the periodic solution to (1.1).
Theorem 4.1 Assume that (H1) and (H2) hold. Suppose further that . Then the periodic solution of (1.1) is exponentially stable.
By way of contradiction, assume that (4.4) does not hold, then we have the following three cases.
which is a contradiction.
which is also a contradiction.
which means that the periodic solution of (1.1) is exponentially stable. This completes the proof. □
By Theorem 2.1 in , we have the following corollary.
Then (1.1) has a stable ω-periodic solution.
In this section, we present two examples to illustrate the feasibility of our results obtained in previous sections.
and . All the conditions in Theorem 3.1 and Theorem 4.1 are satisfied. Hence, (5.1) has an exponentially stable 2π-periodic solution.
All the conditions in Theorem 3.1 and Theorem 4.1 are satisfied. Hence, (5.2) has an exponentially stable 6-periodic solution.
Remark 5.1 Since in (5.1) and (5.2), or may be negative, Theorem 2.1 in  is not suitable for our examples. But from our results, we can obtain that both (5.1) and (5.2) have exponentially stable ω-periodic solutions.
This study was supported by the National Natural Sciences Foundation of People’s Republic of China under Grant 10971183.
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