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.
Keywordsperiodic solution Hamiltonian system exponential stability time scale
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.
- Ahlbrandt C, Peterson A: Discrete Hamiltonian Systems: Difference Equations, Continued Fractions, and Riccati Equations. Kluwer Academic, Boston; 1996.View ArticleGoogle Scholar
- Tang XH, Zhang MR: Lyapunov inequalities and stability for linear Hamiltonian systems. J. Differ. Equ. 2012, 252: 358-381. 10.1016/j.jde.2011.08.002View ArticleGoogle Scholar
- Liu ZL, Su JB, Wang ZQ: A twist condition and periodic solutions of Hamiltonian systems. Adv. Math. 2008, 218: 1895-1913. 10.1016/j.aim.2008.03.024MathSciNetView ArticleGoogle Scholar
- Han ZQ: Computations of cohomology groups and nontrivial periodic solutions of Hamiltonian systems. J. Math. Anal. Appl. 2007, 330(1):259-275. 10.1016/j.jmaa.2006.07.047MathSciNetView ArticleGoogle Scholar
- Sun JT, Chen HB, Nieto JJ: Homoclinic solutions for a class of subquadratic second-order Hamiltonian systems. J. Math. Anal. Appl. 2011, 373: 20-29. 10.1016/j.jmaa.2010.06.038MathSciNetView ArticleGoogle Scholar
- Hilger S: Analysis on measure chains-a unified approach to continuous and discrete calculus. Results Math. 1990, 18: 18-56. 10.1007/BF03323153MathSciNetView ArticleGoogle Scholar
- Bohner M, Peterson A: Dynamic Equations on Time Scales, an Introduction with Applications. Birkhäuser, Boston; 2001.View ArticleGoogle Scholar
- Kaufmann ER, Raffoul YN: Periodic solutions for a neutral nonlinear dynamical equations on a time scale. J. Math. Anal. Appl. 2006, 319: 315-325. 10.1016/j.jmaa.2006.01.063MathSciNetView ArticleGoogle Scholar
- Zhou JW, Li YK: Sobolev’s spaces on time scales and its applications to a class of second order Hamiltonian systems on time scales. Nonlinear Anal. 2010, 73: 1375-1388. 10.1016/j.na.2010.04.070MathSciNetView ArticleGoogle Scholar
- Zhou JW, Li YK: Variational approach to a class of second order Hamiltonian systems on time scales. Acta Appl. Math. 2012, 117: 47-69. 10.1007/s10440-011-9649-zMathSciNetView ArticleGoogle Scholar
- Zhang HT, Li YK: Existence of positive periodic solutions for functional differential equations with impulse effects on time scales. Commun. Nonlinear Sci. Numer. Simul. 2009, 14: 19-26. 10.1016/j.cnsns.2007.08.006MathSciNetView ArticleGoogle Scholar
- Li YK, Yang L, Wu WQ: Anti-periodic solutions for a class of Cohen-Grossberg neural networks with time-varying delays on time scales. Int. J. Syst. Sci. 2011, 42: 1127-1132. 10.1080/00207720903308371MathSciNetView ArticleGoogle Scholar
- Li YK, Wang C: Uniformly almost periodic functions and almost periodic solutions to dynamic equations on time scales. Abstr. Appl. Anal. 2011., 2011: Article ID 341520Google Scholar
- Ahlbrandt CD, Bohner M: Hamiltonian systems on time scales. J. Math. Anal. Appl. 2000, 250: 561-578. 10.1006/jmaa.2000.6992MathSciNetView ArticleGoogle Scholar
- He XF, Zhang QM, Tang XH: On inequalities of Lyapunov for linear Hamiltonian systems on time scales. J. Math. Anal. Appl. 2011, 381: 695-705. 10.1016/j.jmaa.2011.03.036MathSciNetView ArticleGoogle Scholar
- Zafer A: The stability of linear periodic Hamiltonian systems on time scales. Appl. Math. Lett. 2013, 26: 330-336. 10.1016/j.aml.2012.09.014MathSciNetView ArticleGoogle Scholar
- Bohner M, Zafer A: Lyapunov type inequalities for planar linear dynamic Hamiltonian systems. Appl. Anal. Discrete Math. 2013. 10.2298/AADM130211004BGoogle Scholar
- Adivar M, Raffoul YN: Existence of periodic solutions in totally nonlinear delay dynamic equations. Electron. J. Qual. Theory Differ. Equ. 2009, 1: 1-20.MathSciNetGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.