Lagrangian Stability of a Class of Second-Order Periodic Systems
© Shunjun Jiang et al. 2011
Received: 24 November 2010
Accepted: 5 January 2011
Published: 11 January 2011
We study the following second-order differential equation: , where ( ), and are positive constants, and satisfies . Under some assumptions on the parities of and , by a small twist theorem of reversible mapping we obtain the existence of quasiperiodic solutions and boundedness of all the solutions.
1. Introduction and Main Result
are bounded for all time, that is, whether there are resonances that might cause the amplitude of the oscillations to increase without bound.
The first positive result of boundedness of solutions in the superlinear case (i.e., as ) was due to Morris . By means of KAM theorem, Morris proved that every solution of the differential equation (1.1) is bounded if , where 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 Morris's result and obtained similar result for a large class of superlinear function .
where as and is a -periodic function. After introducing new variables, the differential equation (1.3) can be changed into a Hamiltonian system. Under some suitable assumptions on and , by using a variant of Moser's small twist theorem  to the Pioncaré map, the author obtained the existence of quasi-periodic solutions and the boundedness of all solutions.
where is bounded, is periodic. The idea is also to change the original problem to Hamiltonian system and then use a twist theorem of area-preserving mapping to the Pioncaré map.
The above differential equation essentially possess Hamiltonian structure. It is well known that the Hamiltonian structure and reversible structure have many similar property. Especially, they have similar KAM theorem.
Then, the following conclusions hold true.
(i)There exist and a closed set having positive measure such that for any , there exists a quasi-periodic solution for (1.5) with the basic frequency .
(ii)Every solution of (1.5) is bounded.
where , , and are constants. We want to generalize the result in  to a class of -Laplacian-type differential equations of the form (1.9). The main idea is similar to that in . We will assume that the functions and have some parities such that the differential system (1.9) still has a reversible structure. After some transformations, we change the systems (1.9) to a form of small perturbation of integrable reversible system. Then a KAM Theorem for reversible mapping can be applied to the Poincaré mapping of this nearly integrable reversible system and some desired result can be obtained.
Our main result is the following theorem.
for all , for all , . Then every solution of (1.9) is bounded.
Our main nonlinearity in (1.9) corresponds to in (1.5). Although it is more special than , it makes no essential difference of proof and can simplify our proof greatly. It is easy to see from the proof that this main nonlinearity is used to guarantee the small twist condition.
2. The Proof of Theorem
The proof of Theorem 1.1 is based on Moser's small twist theorem for reversible mapping. It mainly consists of two steps. The first one is to find an equivalent system, which has a nearly integrable form of a reversible system. The second one is to show that Pincaré map of the equivalent system satisfies some twist theorem for reversible mapping.
2.1. Action-Angle Variables
with denoting the Jacobian matrix of .
Below we will see that the symmetric properties (1.10) imply a reversible structure of the system (1.9).
By (1.10) it is easy to see that the system (2.4) is reversible with respect to the involution .
Below we will write the reversible system (2.4) as a form of small perturbation. For this purpose we first introduce action-angle variables .
The function will be extended to the whole real axis as explained below, and the extension will be denoted by . Define on by . Then, we define on such that is an odd function. Finally, we extend to by -periodicity. It is not difficult to verify that has the following properties:
(i) , ;
(iii) is an odd periodic function with period .
It is easily verified that and and so the system (2.11) is reversible with respect to the involution .
2.2. Some Lemmas
To estimate and , we need some definitions and lemmas.
for , .
To describe the estimates in Lemma 2.1, we introduce function space , where is a function of .
Lemma 2.3 (see ).
The following conclusions hold true:
(i)if , then and ;
(ii)if and , then ;
for sufficiently large.
Furthermore, we suppose that among , there are numbers which equal to 0, and among , there are numbers which equal to 0.
In the same way we can consider and we omit the details.
2.3. Some Estimates
The following lemma gives the estimate for and .
, , where .
Since , we first consider and . By Lemma 2.1, . Again , using the conclusion (iii) of Lemma 2.3, we have , where . Note that and , we have . In the same way we can prove . Thus Lemma 2.4 is proved.
It is easy to see that and . Hence, system (2.34) is reversible with respect to the involution .
where , , with and defined in (2.11). It follows , and so (2.35) is also reversible with respect to the involution . Below we prove that and are smaller perturbations.
where , , and and are defined in the same way as and .
Thus, . In the same way, we have .
with , .
2.4. Coordination Transformation
Moreover, the system (2.44) is reversible with respect to the involution G: .
Below we estimate and . We only consider since can be considered similarly or even simpler.
Since , the system (2.44) is reversible in with respect to the involution . Thus Lemma 2.6 is proved.
Now we make average on the nonlinear term in the second equation of (2.44).
Moreover, the system (2.59) is reversible with respect to the involution G: .
Thus Lemma 2.7 is proved.
Below we introduce a small parameter such that the system (2.4) is written as a form of small perturbation of an integrable.
In the same way, . The estimates (2.74) for follow easily from (2.60).
2.5. Poincaré Map and Twist Theorems for Reversible Mapping
We can use a small twist theorem for reversible mapping to prove that the Pioncaré map has an invariant closed curve, if is sufficiently small. The earlier result was due to Moser [11, 12], and Sevryuk . Later, Liu  improved the previous results. Let us first recall the theorem in .
where is a real number, is a small parameter, the functions , , , and are periodic.
Lemma 2.9 (see [14, Theorem 2]).
the mapping has an invariant curve in , the constant and depend on , and . In particular, is independent of .
If satisfy all the conditions of Lemma 2.9, then Lemma 2.9 still holds.
Lemma 2.11 (see [14, Theorem 1]).
The constants and depend on only.
We use Lemmas 2.9 and 2.11 to prove our Theorem 1.1. For the reversible mapping (2.86), , .
2.6. Invariant Curves
Case 1 ( is rational).
Since only depends on , and , all conditions in Lemma 2.9 hold.
Case 2 ( is irrational).
all the assumptions in Lemma 2.11 hold.
Thus, in the both cases, the Poincare mapping always have invariant curves for being sufficient small. Since , we know that for any , there is an invariant curve of the Poincare mapping, which guarantees the boundedness of solutions of the system (2.11). Hence, all the solutions of (1.9) are bounded.
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