Boundedness of solutions for semilinear Duffing’s equation with asymmetric nonlinear term
© Jiang et al.; licensee Springer. 2013
Received: 3 November 2012
Accepted: 3 September 2013
Published: 7 November 2013
In this paper we study the following second-order periodic system:
where has a singularity. Under some assumptions on the , and , by Ortega’s small twist theorem, we obtain the existence of quasi-periodic solutions and boundedness of all the solutions.
Keywordsboundedness of solutions singularity small twist theorem
1 Introduction and main result
are bounded for all time, i.e., 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 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, 4]) improved the result of (1.1) and obtained a similar result for a large class of superlinear functions .
i.e., differential equation (1.1) is semilinear, similar results also hold. But the proof is more difficult since there may be a resonant case.
where is 2π-periodic in t. They defined a new function , where , has limits and the similar property to in . Then the authors proved the boundedness of solutions for (1.2). We observe that in  is unbounded while in  is bounded and that is the major difference between  and . The idea in [5, 6] is to change the original problem to a Hamiltonian system and then use a twist theorem of area-preserving mapping to the Poincaré map.
they proved the boundedness of solutions and the existence of a quasi-periodic solution by the Moser twist theorem. It was the first time that the equation of the boundedness of all solutions was treated in case of a singular potential.
Motivated by the papers [5–7], we observe that in (1.3) is smooth and bounded, so a natural question is to find sufficient conditions on such that all solutions of (1.3) are bounded when is unbounded. The purpose of this paper is to deal with this problem.
Our main result is the following theorem.
Theorem 1 Under assumptions (1.6)-(1.8), all the solutions of (1.5) are defined for all , and for each solution , we have .
The main idea of our proof is acquired from . The proof of Theorem 1 is based on a small twist theorem due to Ortega . Hypotheses (1.6)-(1.8) of our theorem are used to prove that the Poincaré mapping of (1.5) satisfies the assumptions of Ortega’s theorem.
Moreover, we have the following theorem on solutions of Mather type.
Theorem 2 Assume that satisfies (1.8); then, there is such that for any , equation (1.5) has a solution of Mather type with rotation number ω. More precisely,
Case 2: ω is irrational. The solution is either a usual quasi-periodic solution or a generalized one.
2 Proof of the theorem
2.1 Action-angle variables and some estimates
The closed curves are just the integral curves of (2.2).
We now give the estimates on the functions and .
where , . Note that here and below we always use C, or to indicate some constants.
By (2.3)-(2.5) we have , .
Thus Lemma 1 is proved. □
Thus the time period is dominated by when h is sufficiently large. From the relation between and , we know is dominated by when h is sufficiently large.
We now carry out the standard reduction to the action-angle variables. For this purpose, we define the generating function , where C is the part of the closed curve connecting the point on the y-axis and point .
In order to estimate , we need the estimate on the functions .
The lemma was first proved in , later Capietto et al.  gave a different proof; using the method of induction-hypothesis, Jiang and Fang  also gave another proof. So, for concision, we omit the proof.
2.2 New action and angle variables
is also a Hamiltonian system with the Hamiltonian function I and now the action, angle and time variables are H, t and θ.
Now we give the estimates of R. By a similar way to that in estimating Lemma 2.3 in , we have the following lemma.
By Lemmas 1 and 3, we have the estimates on .
Lemma 4 for .
For the estimate of , we need the estimate on . By Lemma 1 and noticing that , we have the following lemma.
Lemma 5 for .
Obviously, if , the solution of (2.15) with the initial data is defined in the interval and . So, the Poincaré map of (2.15) is well defined in the domain .
Lemma 6 ( Lemma 5.1)
The Poincaré map of (2.15) has the intersection property.
The proof is similar to the corresponding one in .
for some constant which is independent of the arguments t, ρ, θ, ϵ.
uniformly in .
2.3 Poincaré map and twist theorems
We will use Ortega’s small twist theorem to prove that the Poincaré map P has an invariant closed curve if ϵ is sufficiently small. Let us first recall the theorem in .
Lemma 7 (Ortega’s theorem)
the mapping has an invariant curve in , the constant ϵ is independent of δ.
Similar to the way of estimating , by a direct calculation, we have the following lemma.
Proof This lemma was proved in , so we omit the details. □
For estimate and , we need the estimates of x and .
Now we can give the estimates of and .
Thus Lemma 10 is proved. □
2.4 Proof of Theorem 1
The other assumptions of Ortega’s theorem are easily verified. Hence, there is an invariant curve of P in the annulus , which implies the boundedness of our original equation (1.5). Then Theorem 1 is proved.
2.5 Proof of Theorem 2
We apply Aubry-Mather theory. By Theorem B in  and the monotone twist property of the Poincaré map P guaranteed by , it is straightforward to check that Theorem 2 is correct.
Thanks are given to referees whose comments and suggestions were very helpful for revising our paper.
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