 Research
 Open Access
AubryMather sets in semilinear asymmetric Duffing equations
 Xiaoming Wang^{1, 2}Email author
 Received: 23 July 2016
 Accepted: 10 November 2016
 Published: 23 November 2016
Abstract
Keywords
 Duffing equations
 asymmetric
 AubryMather sets
 quasiperiodic solutions
MSC
 34C15
 37C55
1 Introduction
As one of the simplest but nontrivial conservative systems, equation (1.1) has been extensively and intensively studied by many researchers as regards its dynamic behavior such as periodic solutions [1–10], and the existence of bounded or unbounded solutions [11–16]. In the existing literature, a lot of work has been done to prove the existence of the periodic solutions, such as [1–10] and the references given therein, but there is little information as regards the dynamics behavior of its periodic solution. On the other hand, AubryMather theory about twist maps heavily indicates that equation (1.1) most likely possesses at least some ‘good’ periodic solutions as well as some generalized quasiperiodic solutions. This is the main objective of this paper and we will prove the existence of AubryMather sets for equation (1.1) under some suitable assumptions.
In the early 1980s, Aubry [17] and Mather [18] have proved independently that invariant curves of integral system will be broken if its perturbation increased gradually and/or the smoothness of integrable system is weakened, when they, respectively, studied a one dimensional liquid crystal model of solid state physics and the qualitative properties of the orbits of an areapreserving twist map of the annulus. They also found that when invariant curves break, they do not disappear completely, some special invariant sets still exist. Today, these invariant sets are called the AubryMather sets. For the planar differential system, AubryMather theory suggests that for its Poincaré mapping if there exist AubryMather sets \(M_{\sigma }\) with a rotation number σ, then the planar differential system possesses AubryMather type solutions \(z_{\sigma }(t)=(x_{\sigma }(t),y_{\sigma }(t))\), such that \(M_{\sigma }\equiv \overline{\{z_{\sigma }(2\pi i),i\in Z\}}\) satisfying the following geometrical properties:
\((1)\) if \(\sigma =\frac{n}{m}\in {\mathbb{Q}}\) with \((n,m)=1\), then \(z_{\sigma }(t)\) is a Birkhoff periodic solution with periodic \(2m\pi \) and \(\arg (z_{\sigma }(t)+m)=\arg (z_{\sigma }(t))+n\), the m solutions \(z_{\sigma }(t+2\pi i), 0\leq i\leq m1\), can be homotopically transformed to m distinct parallel lines;
\((2)\) if \(\sigma \in {{\mathbb{R}}\backslash {\mathbb{Q}}}\), then \(M_{\sigma }\) is either an invariant circle and its orbits are just usual quasiperiodic orbits, or an invariant Cantor set and its orbits become generalized ones. For further interpretations, we refer to the recent work of [19] and [20].
Now a natural and open question is whether equation (1.3) still possesses AubryMather sets when the smoothness of \(\psi (x)\) and \(e(t)\) is further weakened. Our Theorem 1.1 in the following will answer this question and we will deal with a more general case (1.1) than that of (1.3). Owing to the appearance of weak smoothness nonlinearity, the methods in [25–31] are no longer valid. To overcome this difficulty, we first introduce a suitable action and angle variable transformation similar to [29] so that the transformed system of (1.1) is a perturbation of an integral Hamiltonian system, and then apply a new estimate approach developed by the present author (see the recent papers [32–34]) to directly prove the Poincaré map of the transformed system satisfying monotone twist property. Furthermore, the AubryMather theorem on a cylinder with monotone twist assumption by Pei [25] guarantees the existence of AubryMather sets for (1.1), which leads to our desired results. The results of this paper are new and they are natural generalizations and refinements of previously known results obtained in [25, 29].
More exactly, the following theorem is proved.
Theorem 1.1
 \((A_{1})\) :

the limit \(\lim_{\vert x \vert \to +\infty }f_{x}(t,x)=0\), uniformly in \(t\in [0,2\pi ]\);
 \((A_{2})\) :

there exist constants \(d\geq 0, \mu >0\), such that$$ \operatorname{sgn}(x) \bigl[f(t,x)xf_{x}(t,x) \bigr]>\mu ,\quad\textit{for } \vert x \vert \geq d. $$
 (i)
when \(\sigma =\frac{n}{m}\) is rational, and \((n,m)=1\), the solutions \(z_{\sigma }^{i}(t)=z_{\sigma }(t+2\pi i) , 0\leq i\leq m1\), are mutually unlinked periodic solutions of period m;
 (ii)when σ is irrational, the solution \(z_{\sigma }(t)\) is either a usual quasiperiodic solution or a generalized one exhibiting a Denjoy minimal set (see the definition in [35])$$ M_{\sigma }\equiv \overline{ \bigl\{ z_{\sigma }(2\pi i),i\in { \mathbb{Z}} \bigr\} }. $$
Remark 1.1
The rest of the paper is organized as follows: The proof of Theorem 1.1 will be given in Section 4. Section 2 introduces some basic results which are necessary for the proof of Theorem 1.1. In Section 2.1, we introduce a polar coordinate type actionangle variable which transform equation (1.1) into a perturbation of an integral Hamiltonian system, and then in Section 2.2, we will give some estimates on the corresponding action and angle variables functions. Section 3 deals with the proof of monotone twist property of the Poincaré map P of the new system around infinity. Finally, Section 5 presents some examples and remarks on Theorem 1.1.
2 Some basic results
2.1 Actionangle variables
 (i)
\(C'(t)=S(t), S'(t)=\alpha C^{+}(t)\beta C^{}(t)\);
 (ii)
\((S(t))^{2}+\alpha (C^{+}(t))^{2}+\beta (C^{}(t))^{2}\equiv \alpha \);
 (iii)
\(\vert C(t) \vert \leq \max \{1,\sqrt{\frac{\alpha }{\beta }}\}:=C_{\infty }, \vert S(t) \vert \leq \sqrt{\alpha }\).
2.2 Some estimates on action and angle variables functions
According to the assumptions of \((A_{1})\) and \((A_{2})\), it is easy to prove the existence and uniqueness of the solution of the initial value problem associated with (2.4). Moreover, this solution has continuous derivatives with respect to initial data.
For notional convenience, hereinafter, we write x, y, θ, r instead of \(x(\theta (t;\theta _{0},r_{0}),r(t;\theta _{0}, r_{0}))\), \(y(\theta (t;\theta _{0},r_{0}),r(t;\theta _{0},r_{0})),\theta (t;\theta _{0},r_{0}), r(t;\theta _{0},r_{0})\), respectively.
Now in the following, we will give some growth estimation properties with respect to the action and angle variables functions \(r(t;\theta _{0},r_{0})\) and \(\theta (t;\theta _{0},r_{0})\).
Proposition 2.1
Proof
Consequently, by (2.6), \(r(t;\theta _{0},r _{0})\to +\infty \) as \(r_{0}\to +\infty \) uniformly for \(t\in [0,2\pi ]\). □
By (2.6), we can easily verify the following.
Proposition 2.2
Proposition 2.3
Proof
Hence, by (2.3) and Proposition 2.1, there exists a constant \(\bar{r_{2}}>0\), such that \(\frac{d\theta }{dt}\geq \frac{\omega }{2}\) if \(r_{0}\geq \bar{r_{2}}\).
If we choose \(\bar{r}=\max \{\bar{r_{1}},\bar{r_{2}}\}\), then \(r_{0}\geq \bar{r}\) implies \(\frac{d\theta }{dt}\geq \frac{\omega }{2}\).
Using the same arguments as above, one can prove that the inequality on the right side of (2.7) holds. □
3 Monotone twist property
Now in the following, we will start to examine the behavior of \(\frac{\partial \theta (2\pi ;\theta _{0},r_{0}) }{\partial r_{0}} \) when \(r_{0}\gg 1\).
Lemma 3.1
 (i)
\(a_{1}(t)=o(\frac{1}{r_{0}})\);
 (ii)
\(a_{2}(t)=o(1)\);
 (iii)
\(a_{1}(t)\cdot a_{3}(s)=o(1)\).
Proof
By using Lemma 3.2, the proof can be obtained immediately. □
Next we are going to give a detailed proof of Lemma 3.2.
Lemma 3.2
 (i)
\(\frac{xf(t,x)}{r}\rightarrow 0\), \(\frac{x^{2}f_{x}(t,x)}{r}\rightarrow 0\);
 (ii)
\(\frac{yf(t,x)}{r}\rightarrow 0\), \(\frac{yxf_{x}(t,x)}{r}\rightarrow 0\), \(\frac{y^{2}xf(t,x)f_{x}(t,x)}{r^{2}}\rightarrow 0\).
Proof
Set \(F_{1}(\varepsilon )=\max \{\vert f(t,x) \vert :t\in [0,2\pi ],\vert x \vert \leq X\}, F_{2}(\varepsilon )=\max \{\vert f_{x}(t,x) \vert :t\in [0,2\pi ],\vert x \vert \leq X\}\).
Since \(\varepsilon >0\) is arbitrary, the proof of (i) is complete.
Since \(\varepsilon >0\) is arbitrary, (ii) is proved. □
Combining the previous lemmas, we have the following.
Lemma 3.3
 (i)
\(\theta _{r_{0}}(t;\theta _{0}, r_{0})\rightarrow 0\);
 (ii)
\(r_{r_{0}}(t;\theta _{0}, r_{0})=1+o(1)\);
 (iii)
\(\theta _{\theta _{0}}(t;\theta _{0}, r_{0})=1+o(1)\).
Proof
Therefore, for all \(t\in (0,2\pi ]\), sending \(r_{0}\rightarrow +\infty \), we have \(r_{r_{0}}(t)=1+o(1)\) and \(\theta _{r_{0}}(t)=(1+o(1))\int _{0}^{t}a_{1}(s)\,ds\rightarrow 0\). This completes the proof of (i) and (ii).
Similar to the proof of (ii), one can see that \(\theta _{\theta _{0}}(t;\theta _{0}, r_{0})=1+o(1)\) for \(\forall t\in (0,2\pi ]\), as \(r_{0}\rightarrow +\infty \). This completes the proof of Lemma 3.3. □
Now, we will give an estimate of upper bound and lower bound for \(a_{1}(t)\).
Lemma 3.4
 (i)
If \(\vert x(t;\theta _{0},r_{0}) \vert \leq d\) for all \(t \in [0,2\pi ]\), then there exists a constant \(K_{d}>0\), such that \(\vert a_{1}(t) \vert \leq \frac{K_{d}}{r^{2}(t)}\).
 (ii)
If \(\vert x(t;\theta _{0},r_{0}) \vert \geq d\) for all \(t \in [0,2\pi ]\), then there exists a constant \(L_{d}>0\), such that \(\vert a_{1}(t) \vert \geq \frac{L_{d}}{r^{2}(t)}\). Moreover, \(\vert x(t;\theta _{0},r_{0}) \vert \geq d\) implies that \(a_{1}(t)<0\).
Proof
(i) If \(\vert x(t;\theta _{0},r_{0}) \vert \leq d\), we take \(f_{d}=\max_{\vert x \vert \leq d, t\in [0,2\pi ]}\vert f(x,t)xf_{x}(x,t) \vert \).
Set \(K_{d}= \frac{df_{d}}{4}\), we get \(\vert a_{1}(t) \vert \leq \frac{K_{d}}{r^{2}(t)}\).
Denote \(a_{1}(t)=a_{1}^{+}(t)a_{1}^{}(t)\), where \(a_{1}^{\pm }(t)=\max \{\pm a_{1}(t),0\}\). To estimate the integral of \(a_{1}^{+}(t)\) on \([0,2\pi ]\) is smaller than the integral of \(a_{1}^{}(t)\) on \([0,2\pi ]\), we need the following lemma. □
Lemma 3.5
Proof
By Proposition 2.3, we know that \(\Delta t\rightarrow 0\) if and only if \(\Delta \theta \rightarrow 0\).
In the following, we will give an estimate of \(\frac{\partial \theta (2\pi ;\theta _{0},r_{0}) }{\partial r_{0}}\) if \(r_{0}\gg 1\).
Lemma 3.6
For \(r_{0}\gg 1\), we have \(\theta _{r_{0}}(2\pi )<0\).
Proof
Consequently, if \(\sqrt{r_{0}}>\frac{(\beta _{1}^{2} L_{d} +\beta _{2}^{2}K_{d})D}{2\pi L_{d}\beta _{1}^{2}}\), then \(\theta _{r_{0}}(2\pi )<0\). □
4 Proof of Theorem 1.1
Now we are in a position to prove Theorem 1.1.
Proof of Theorem 1.1.
According to Lemma 3.6 and the AubryMather theory framework developed by Pei [25], we know that the Poincaré map P of system (2.4) is a monotone twist map if \(r_{0}\gg 1\). At last, using similar arguments as in [25], one may extend the Poincaré map P to a new map P̂ which is a global monotone twist homeomorphism of the cylinder \(\mathbf{S}^{1}\times \mathbb{R}\) and coincides with P on \(\mathbf{S}^{1}\times [r_{0},+\infty )\) with a fixed constant \(r_{0}\gg 1\). Therefore, the existence of AubryMather sets \(M_{\sigma }\) of P̂ is guaranteed by the AubryMather theorem by Pei [25]. Moreover, for some small \(\varepsilon _{0}>0\), all those AubryMather sets with rotation number \(\sigma \in (2\omega \pi ,2\omega \pi +\varepsilon _{0}) \) lie in the domain \(\mathbf{S}^{1}\times [r_{0},+\infty )\). Hence they are just the AubryMather sets of the Poincaré map of P. From the above discussions, we have showed the existence of AubryMather sets, this implies that equation (1.1) has a solution \(z_{\sigma }(t)=(x_{\sigma }(t),x'_{\sigma }(t))\) with rotation number σ. Thus, the proof of Theorem 1.1 is completed. □
5 Examples and remarks
Example 5.1
Bounded perturbation
Consider \(f(t, x) =\operatorname{sgn}(x)\arctan x\cdot (1+\vert p(t) \vert )\), where \(p(t)\) is a continuous function satisfying \(p(t +2\pi ) = p(t)\). We can easily check \(f(t, x)\in C^{0,1}(\mathbf{S}^{1}\times \mathbb{R})\) is a bounded function satisfying conditions \((A_{1})\) and \((A_{2})\) in Theorem 1.1 when \(d=1\) and \(\mu =\frac{\pi }{4}\frac{1}{2}\) for any \(\alpha \neq \beta \) in (5.2). But \(f(t, x)\) does not satisfy the conditions \((H_{1})\)\((H_{4})\) of Theorem 1 in [29]. Therefore, the results obtained in this paper can be viewed as natural generalizations and refinements of the results in [29].
Example 5.2
Unbounded perturbation
Let \(\alpha =\beta =\lambda \) in (5.2) and \(f(t,x)=\operatorname{sgn}(x)\ln (1+\vert x \vert )\cdot (1+\vert p(t) \vert )\), where \(p(t)\) is a continuous function satisfying \(p(t +2\pi ) = p(t)\). It is obvious that \(f(t, x)\in C^{0,1}(\mathbf{S}^{1}\times \mathbb{R})\) is an unbounded function satisfying conditions \((A_{1})\) and \((A_{2})\) in Theorem 1.1 if \(d=e^{2}\) and \(\mu =1\). It is clear that the conditions \((D_{1})\) and \((D_{2})\) of Theorem B in [25] are not satisfied. Thus our situation is more general than the results obtained in [25].
Remark 5.1
Comparing the estimation method in this paper with [25, 29], it seems that our estimation processes are much simpler than those used in [25, 29] to some degree. Moreover, it is easy to see that the results of this paper generalize and refine the early results published so far, our results are new and not covered by any known work in the literature.
Remark 5.2
Finally, let us note that our method could be useful in more general situations such as equations depending nonlinearly on the derivative, Hamiltonian or more general second order differential equations. These extensions should be developed elsewhere.
Declarations
Acknowledgements
This research was supported by the National Natural Science Foundation of China (11461056).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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