Abstract and Applied Analysis

Volume 2013 (2013), Article ID 246343, 18 pages

http://dx.doi.org/10.1155/2013/246343

Research Article

Boundedness of Solutions for a Class of Sublinear Reversible Oscillators with Periodic Forcing

Tingting Zhang and Jianguo Si

School of Mathematics, Shandong University, Jinan, Shandong 250100, China

Received 3 February 2013; Accepted 30 April 2013

Academic Editor: Wenchang Sun

Copyright © 2013 Tingting Zhang and Jianguo Si. 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.

Abstract.

We study the boundedness of all solutions for the following differential equation where are odd functions, is an even function, are smooth -periodic functions, is a nonzero constant, and is a small parameter. A sufficient and necessary condition for the boundedness of all solutions of the above equation is established. Moreover, the existence of Aubry-Mather sets is obtained as well.

1. Introduction

It is well known that the longtime behavior for periodically forced planar systems can be very intricate. For example, there are equations having unbounded solutions but with infinitely many zeros and with nearby unbounded solutions having randomly prescribed number of zeros and also periodic solutions; see [1]. In contrast to such unbounded phenomenon Littlewood [2] suggested to study the boundedness of all the solutions of the following differential equation:

in the following two cases:

(i)superlinear case: as ;(ii)sublinear case: and as . Later, one calls this subject as Littlewood boundedness problem.

The first result in superlinear case is obtained by Morris [3], who showed that all solutions of

are bounded, where . Later, a series results in superlinear case were obtained by several authors, see [4–13] and references therein. However, in general, it is harder to study the Lagrange stability of sublinear systems since smoothness of sublinear term is insufficient. There are only a few works in sublinear case so far. In 1999, Küpper and You [14] proved the first result in the study of the equation

where and . Later, Liu [15] proved the same result for more general equation

where satisfying the sublinear condition (ii) and some inequalities, and . In 2004, Ortega and Verzini [16] studied the boundedness of (4) in a special case with the variational method. In 2009, Wang [17] gave a sufficient and necessary condition for the boundedness of all solutions for sublinear equation

where .

As is widely known, there is a deep similarity between reversible and Hamiltonian dynamics. Many fundamental results of the Hamiltonian systems possess reversible counterparts. On boundedness problem for sublinear reversible systems, the first results were obtained by Li [18], later, Yang [19], in the study of a sublinear reversible systems

Recently, Wang [20] gave a sufficient and necessary condition for the boundedness of all solutions of the differential equation

with , .

By the discussions about the sublinear Hamiltonian equation (1.3) in [17] motivations, we will study the boundedness of all solutions for a sublinear reversible system like

where and . Furthermore, we also show that (8) has solutions of Mather type. The results obtained in [18–20] can be regarded as corollary of result of this paper.

Remark 1. Using the method of this paper we also can consider the more general equation

provided of adding suitable conditions for . For convenience, we only consider the case .

Remark 2. Adding the perturbation term will lead to a new difficulty for estimating appeared in (86). Fortunately, we can easily verify that is bounded by a constant (see in the proof of Lemma 12).

Throughout this paper, we denote two universal positive constants without regarding their values by and , and suppose that the following conditions hold:

(A1), and , and are odd, is even, and , are both -periodic functions, ;(A2) there is some positive constant such that the inequalities

 are satisfied for and all , where .

We decompose as , where is the average of and has zero mean value. That is and . If we write that , then it is easy to see that and have the same sign when with .

Now we state the main results of this paper.

Theorem 3. Assume that and - hold. Then there exists an such that for any , every solution of (8) is bounded if and only if .

Theorem 4. Under the conditions of Theorem 3, there is an such that, for any , (8) has a solution of Mather type with rotation number . More precisely:

(i)if is rational, the solutions , , are periodic solutions of period q; moreover, in this case

(ii)if is irrational, the solution is either a usual quasi-periodic solution or a generalized one.

We recall that a solution is called generalized quasi-periodic if the closed set

is a Denjoys minimal set.

2. Reversible Systems and Action-Angle Variables

In this section, we will assume that and . Firstly, we consider (8) which is equivalent to the following system:

where . Then we can obtain that (13) is reversible with respect to the transformation by (A1).

Lemma 5. There exists a -invariant diffeomorphism such that (13) is transformed into the following system:

where .

Proof. Introduce a transformation :

where will be determined later. Under this transformation, the system (13) is transformed into a new system as follows:

Now, we define the function by

Since , so we can obtain . Then the new system can be expressed as in (14) by direct computation.

It is easy to know that by (A1), then we can obtain that the transformation is a -invariant diffeomorphism.

Let us consider the auxiliary system

which is a time-independent Hamiltonian system with Hamiltonian

It is easy to see that , , . Note that each level line is a close orbit of system (18), hence, all the solutions of (18) are periodic with period tending to zero as tends to infinity.

Assume that is the solution of (18) with initial conditions , and let be the minimal period. We can find that and satisfy

(i), ;(ii), ;(iii), ;(iv); (v); (vi), ;(vii).

Then we introduce the transformation

which is

where . It is easy to see that by . Since , this transformation is invariant with respect to the involutions and , and we can find that the mapping is a generalized canonical transformation by (iv). In fact,

where .

Under the transformation , the system (18) is transformed into the simpler form

where .

The original system (13) is transformed into the system

where

Let

Clearly, is odd in and is even in by the definitions of and . Thus, by the evenness of and the oddness of and we have

This implies that system (24) is reversible with respect to the involutions .

Lemma 6. For , the following inequalities hold:

(1), (2), (3), (4), (5), (6), (7),

where , , and .

Proof. (1) It is easy to know that is a sum of terms of the form

where . Meanwhile, is a sum terms of the form

Hence, we obtain

by the assumptions on and the definitions of and .

(2) From the expression of , we have

We can find that

where .

(3) From the expression of , we have

(4) From the expression of , we can obtain that

(5) From the definition of , we have

(6) From the definition of , we can obtain

Hence, we can know that

where .

(7) From the expression of , we have

For , we define the domain

Lemma 7. There exists a -invariant diffeomorphism :

such that for some . Under this transformation, (24) is transformed into the system

where

with

Proof. Define a transformation by

By

we get

Let , . The system (24) is transformed into (41).

Lemma 8. For large enough, the following conclusions hold:

(i), (ii).

Proof. In view of

we obtain

By , we have for large enough. Hence, is uniquely determined by the contraction mapping principle. Moreover, , for some , as a consequence of the implicit function theorem and

Above all, if , from (47) and (49), we get

We note that

and the right side hand is sum of the term

where , , . The highest order term in is the one with , namely, . We move the part to the left hand side of (52). Since for large enough, this also provides immediately a bound on . The rest part .

Now, we proceed inductively by assuming that for the estimates

hold and we wish to conclude that the same estimate holds for .

Indeed, if , we have

by

This proves (i) of Lemma 8.

Now we check (ii). In fact, since

we have

From (47), we have for sufficiently large and therefore we obtain .

By the estimates in Lemma 6, we can prove the following inequalities.

Lemma 9. For , the following inequalities hold:

(1), (2), (3), (4), (5), (6), (7).

Proof. (1) From the estimates (1) and (5) of Lemmas 6 and 8, it follows that

(2) Since

we can prove that

Their proofs are similar to the proofs in (1).

Next, we check the last part of   . We get

by the estimate in Lemma 6 and the definition of .

(3) It is clearly by (3) in Lemma 6.

(4) It is clearly by (4) in Lemmas 6 and 8.

(5) We have that

From the last inequalities and (5) in Lemma 6, we obtain

(6) Since

we just have to prove that

In fact,

so we have proved (6).

(7) We have

by (7) in Lemma 6.

3. The Proof of Boundedness

In this section, all the solutions of (8) which are bounded will be proved via the KAM theory for reversible systems developed by Sevryuk [21] or Moser [22, 23] if .

We define the functions , , , , , , and as

Then system (41) is equivalent to the following system:

In addition, one can verify that system (70) is reversible with respect to involution .

Then some estimates on the functions and are given.

Lemma 10. The following inequalities hold:

(1), (2), (3), (4), (5), (6), (7), for .

Proof. (1) It is clear.

(2) Note that , and

it follows that

as .

Moreover, we also have

So

From (72) and (74), it is easy to see that

(3) We have

By (72), and , we have

for .

Let . We find that

so

When , the proof of (3) is similar to the proof of (2).

When , then

(4) The proof of (4) is similar to the proof of (3).

(5) Let . By using the estimates on the functions and , it follows that

when .

When , then

by .

(6) By using the estimates on the functions and , it follows that

(7) By using the estimates on the functions and , it follows that

Let , , and

where is the inverse function of .

Then system (70) is transformed into the following form:

Moreover, one can verify that system (86) is reversible with respect to involution .

It is easy to see that if and only if , and the solutions of system (86) do exist on when .

By using the estimates on and in Lemma 10, the following inequalities can be proved.

Lemma 11. For and , the following inequalities hold:

(1), (2), (3), (4).

Proof. Above all, we know that , so we can get . Then we have

(1) We have that

(2) We have that

(3) We have that

(4) We have that

Lemma 12. The time 1 map of the flow of the system (86) is of the form

where . And there exists a such that, for , sufficiently large and sufficiently small ,

hold. Moreover, the map is reversible with respect to the involution .

Proof. Since

then we get is bounded.

Let . Set with for the flow:

Since

where denotes the vector field of the system (86), we have

which is equivalent to the following equations for and :

Let , . Define , and , where

Next, we will prove that is a contraction map. From the definition of , we have

by Lemma 11 and the boundedness of . Then we have

by the definition of the norm .

Using the contraction principle, one verifies easily that for , (98) has a unique solution in the space . Moreover, and are smooth.

Next, we will estimate and as follows:

In order to prove (93), we just need to prove that

hold for .

(1) When ,

where .

(2) When and , we check the case when firstly

Hence,

Now, we proceed inductively by assuming that for the estimates

hold and we wish to conclude that the same estimate holds for

where . Hence,

(3) We can prove that

similarly to (2) when .

(4) we have that

Hence,

(5) We can prove (103) similarly to (4) for the left .

Proof of Boundedness. From Theorem 1.1 in [21] we can see possesses a sequence of invariant circles tending to infinity. So, in the original system (13), there exists a corresponding sequence of invariant tori in phase space . Then any solution of system (13) is bounded because it must stay within one of those tori.

4. The Proof of Unboundedness

In this section, we will prove that all solutions of (8) are unbounded if . In this case, .

Consider (8) which is equivalent to the following system:

Replacing (18) by an “auxiliary” system

Under the transformation (21), the system (113) is transformed into the form

where

Thus, the system (115) can be written in the form

From the equality

it follows that

Hence, the function is , 1-periodic and change the sign. Since for any , there exists such that

That is, . In view of

we find

Hence, we obtain that or is negative. This proves that there exists a such that and . Therefore, there are and such that for and for , for . Let

Then, if is sufficiently large, on the set , we have

From (117) and (124) we obtain, for ,

Moreover, for and , we have

From (126) and (127), it follows that any solution of (115) with the initial condition always stays in and satisfies with , for all . The proof of Theorem 3 is completed.

5. The Proof of Theorem 4

In this section, we will prove Theorem 4 by using the abstract result on the existence of quasi-periodic solutions proved in [24] in the context Aubry-Mather theory for reversible systems. We only need to show that the Poincaré map (92) has the monotone property; that is,

We can get that

by Lemma 11, and

by Lemma 12. Then we have

where . Therefore, we have

as and . This proves the validity of (128).

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China (Grant nos. 11171185, 10871117) and SDNSF (Grant no. ZR2010AM013).

References

J. Moser, Stable and Random Motions in Dynamical System, Princeton University Press, Princeton, NJ, USA, 1973.

J. E. Littlewood, Some Problems in Real and Complex Analysis, Heath, Lexington, Mass, USA, 1968.

G. R. Morris, “A case of boundedness in Littlewood's problem on oscillatory differential equations,” Bulletin of the Australian Mathematical Society, vol. 14, no. 1, pp. 71–93, 1976.

R. Dieckerhoff and E. Zehnder, “Boundedness of solutions via the twist-theorem,” Annali della Scuola Normale Superiore di Pisa, vol. 14, no. 1, pp. 79–95, 1987.

S. Laederich and M. Levi, “Invariant curves and time-dependent potentials,” Ergodic Theory and Dynamical Systems, vol. 11, no. 2, pp. 365–378, 1991.

M. Levi, “Quasiperiodic motions in superquadratic time-periodic potentials,” Communications in Mathematical Physics, vol. 143, no. 1, pp. 43–83, 1991.

M. Levi, “KAM theory for particles in periodic potentials,” Ergodic Theory and Dynamical Systems, vol. 10, no. 4, pp. 777–785, 1990.

B. Liu, “Boundedness for solutions of nonlinear Hill's equations with periodic forcing terms via Moser's twist theorem,” Journal of Differential Equations, vol. 79, no. 2, pp. 304–315, 1989.

Y. M. Long, “An unbounded solution of a superlinear Duffing's equation,” Acta Mathematica Sinica, vol. 7, no. 4, pp. 360–369, 1991.

Y. Wang, “Unboundedness in a Duffing equation with polynomial potentials,” Journal of Differential Equations, vol. 160, no. 2, pp. 467–479, 2000.

Y. Wang and J. You, “Boundedness of solutions for polynomial potentials with time dependent coefficients,” Zeitschrift für Angewandte Mathematik und Physik, vol. 47, no. 6, pp. 943–952, 1996.

J. You, “Boundedness of solutions of super-linear Duffing's equations,” Scientia Sinica, vol. 8, pp. 805–817, 1991.

X. Yuan, “Boundedness of solutions for Duffing-type equation,” Science in China A, vol. 41, no. 6, pp. 595–605, 1998.

T. Küpper and J. You, “Existence of quasiperiodic solutions and Littlewood's boundedness problem of Duffing equations with subquadratic potentials,” Nonlinear Analysis, vol. 35, no. 5, pp. 549–559, 1999.

B. Liu, “On Littlewood's boundedness problem for sublinear Duffing equations,” Transactions of the American Mathematical Society, vol. 353, no. 4, pp. 1567–1585, 2001.

R. Ortega and G. Verzini, “A variational method for the existence of bounded solutions of a sublinear forced oscillator,” Proceedings of the London Mathematical Society, vol. 88, no. 3, pp. 775–795, 2004.

Y. Wang, “Boundedness for sublinear Duffing equations with time-dependent potentials,” Journal of Differential Equations, vol. 247, no. 1, pp. 104–118, 2009.

X. Li, “Boundedness of solutions for sublinear reversible systems,” Science in China A, vol. 44, no. 2, pp. 137–144, 2001.

X. Yang, “Boundedness of solutions for sublinear reversible systems,” Applied Mathematics and Computation, vol. 158, no. 2, pp. 389–396, 2004.

X. Wang, “Boundedness for sublinear reversible systems with a nonlinear damping and periodic forcing term,” Journal of Mathematical Analysis and Applications, vol. 378, no. 1, pp. 76–88, 2011.

M. B. Sevryuk, Reversible Systems, vol. 1211 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1986.

J. Moser, “Convergent series expansions for quasi-periodic motions,” Mathematische Annalen, vol. 169, pp. 136–176, 1967.

J. Moser, Stable and Random Motions in Dynamical Systems, Princeton University Press, Princeton, NJ, USA, 1973.

S.-N. Chow and M. L. Pei, “Aubry-Mather theorem and quasiperiodic orbits for time dependent reversible systems,” Nonlinear Analysis, vol. 25, no. 9-10, pp. 905–931, 1995.