Research Article | Open Access
Lagrangian Stability of a Class of Second-Order Periodic Systems
We study the following second-order periodic system: where has a singularity and . Under some assumptions on the and , by Moser's twist theorem we obtain the existence of quasiperiodic solutions and boundedness of all the solutions.
1. Introduction and Main Result
In the early 1960s, Littlewood  asked whether or not the solutions of the Duffing-type equations are bounded for all time, that is, whether there are resonances that might cause the amplitude of the oscillations to increase without bound. Littlewood suggested studying the following two cases: (i)superlinear case: as ,(ii)sublinear case: and as .
The first positive result of boundedness of solutions in the superlinear case (i) was due to Morris . By means of KAM theorem, Morris proved that every solution of the second-order system (1.1) is bounded if and is piecewise continuous and periodic. This result relies on the fact that the nonlinearity can guarantee the twist condition of KAM theorem. Later, several authors (see [3–5]) improved the Morris’s result and obtained similar results for a large class of superlinear function .
In 1999, the first result in the sublinear case was proved by Küpper and You  in the study of where and . The authors transform (1.2) into a perturbation of an integrable Hamiltonian system and then prove that the Poincaré map of the transformed system is close to a so-called twist map. So, the Moser’s twist theorem guarantees the boundedness of all solutions of (1.2). The general sublinear case was considered by Liu  under certain reasonable conditions.
The Littlewood problem for singular potentials is known to be challenging, and there are only very a few results. Recently, Capietto et al.  studied with is a -periodic function and , where and is a positive integer. Under the Lazer-Leach assumption that they prove the boundedness of solutions and the existence of quasiperiodic solution by Moser’ twist theorem. It is the first time that the equation of the boundedness of all solution is treated in case of a singular potential.
In this paper, We consider the following sublinearly growing potential: where .
Our main result is the following theorem.
Theorem 1.1. If is 1-periodic continuous, then all the solutions of (1.5) are bounded.
The idea for proving the boundedness of solutions of (1.5) is as follows. By means of transformation theory, (1.5) is, outside of a large disc in the -plane, transformed into a perturbation of an integrable Hamiltonian system. Then, Poincaré map of the transformed system is close to a so-called twist map in . The Moser’s twist theorem  guarantees the existence of arbitrarily large invariant curves diffeomorphic to circles and surrounding the origin in the -plane. Every such curves is the base of a time-periodic and flow-invariant cylinder in the extended phase space , which confines the solutions in the interior and which leads to a bound of these solutions.
The paper is organized as follows. In Sections 2.1 and 2.2, we give action-angle variables and some estimates which is useful for our proof. In Section 2.3, we will give an asymptotic expression of the Poincaré map and prove the main result by Moser’s twist theorem .
2. Proof of Theorem
2.1. Action-Angle Variables and Some Estimates
Without loss of generality and for brevity of arguments, we assume that the average value of vanishes; that is, . Hence the function is also 1-periodic in and is in .
System (1.5) is equivalent to the planar Hamiltonian system where Hamiltonian is .
In order to introduce action and angle variables, we first consider the auxiliary autonomous system which is integrable with the Hamiltonian The closed curves are just the integral curves of (2.2). Denote by the time period of the integral curve : and by the area enclosed by the closed curve . Let Then, .
It is easy to see that Denote Then,
The following estimates on the functions , , and and , , and are crucial for this paper. We first estimate and . Since is the area enclosed by the closed curve and axis when , we can easily prove that
Let , then we get Since , we have
We now give the estimates on the function and .
Lemma 2.1. One has where , . Note that here and below, one always uses , , or to indicate some constants.
Proof. Now, we estimate the first inequality. We choose as the new variable of integration, then we have
Since and , we have . By direct computation, we have
then we get
When and sufficient large, there exits such that , so we have
Since , we have
Observing that there is such that when and , we have
By (2.15)–(2.18), we have , .
The proof of the second inequality is similar to the first one, so we only give the brief proof.
We choose as the new variable of integration, so we have By direct computation, we have By (2.20), we can easily get where .
By the similar way in estimating , we get which means that
Thus, we complete the proof of Lemma 2.1.
Remark 2.3. It also follows from the definition of , , and Remark 2.2 that In particular,
Remark 2.4. Note that is the inverse function of . By Remark 2.3, we have
We now carry out the standard reduction to the action-angle variables. For this purpose, we define the generating function , where is the part of the the closed curve connecting the point on the -axis and point .
We define the well-known map by It is well-known that the map is symplectic, since From the above discussion, we can easily get In the new variables , the system (2.1) becomes where where is the inverse function of and .
In order to estimate , we need the estimate on the functions and . For this purpose, we first give some definitions which are very similar to those in .
Define a function in terms of and by and a linear differential operator acting on functions of , according to where is a smooth function, and we denote .
The following equality (its proof can be found in ) is crucial for the proof of the following lemmas:
Before giving the estimates on and , we now prove some lemmas which will be used frequently in the following proof.
Lemma 2.5. Suppose that there is a constant such that , then one can find a constant and such that, for ,
Proof. We now prove (2.37). Let
By direct calculation,
By , there is such that
which means that
Thus, we complete the proof (2.37).
Now, we prove (2.38). Let Then, we have By direct computation, we have Since and , it follows that for so for , we have which means that By the definition of , we have Thus, we complete the proof of (2.38) and Lemma 2.5.
By Lemma 2.5, we have the following Lemma which is important to our estimation.
Lemma 2.6. One can find a constant such that, for , where .
Proof. When , we have
By the definitions of and , we have and . By (2.37), we obtain
Suppose that , we have
We now proven that when ,
By direct computation, we have
where denotes linear combination of functions with integer coefficients and .
Since , we have . By (2.37), we obtain By direct computation, we get By assumption (2.59), we have So, we have By (2.62) and (2.65), we have Thus, we have proved (2.55).
The inequality (2.56) can be proved by (2.38), and the process of proof is similar to that of (2.55), so we omit it.
Thus, we have proved Lemma 2.6.
Now, we give the estimates of and .
Lemma 2.7. For sufficient large and , the following estimates hold:
Proof. We now prove the first inequality. It is sufficient to prove that
Case . Differentiating (2.30) by and noting , we have
Now, we choose as the new variable of integration, so
and simplifying, we have
By (2.69)–(2.72), we have
Since , by Lemma 2.6, we have
We observe that
where , so , which means that
By (2.74) and (2.76), we have .
We suppose that where . We will prove .
For this purpose, we firstly estimate We differentiate in (2.73) and using (2.73), then we obtain Since , we have By (2.37), we have By (2.78)–(2.80), we have We suppose that when, We will prove that when, By direct computation, we have where and .
By the assume (2.77), we have By (2.55), we get By (2.85), (2.86) and noting the fact that we obtain By assumption (2.82), we have By (2.89) and the fact that , we have So, by (2.88) and (2.90), we have proved (2.83).
By (2.56), we have By (2.83) and (2.91), we have By the assumption (2.77), the fact that , and noting that where , we have By (2.92) and (2.94), we have which means
We now prove Since we have we have proved (2.83), so we have which means that The proof of Lemma 2.7 is complete.
Now, we give the estimate of .
Proof. This lemma can be proved easily form the definition of and .