Infinity of subharmonics for Duffing equations with convex and oscillatory nonlinearities
- Yinyin Wu^{1} and
- Dingbian Qian^{2}Email author
Received: 27 June 2016
Accepted: 28 November 2016
Published: 23 December 2016
Abstract
The existence of infinity of subharmonics for Duffing equations with convex and oscillatory nonlinearities is shown. This result is a corollary of two theorems. These theorems, one for a weak sub-quadratic potential and another for a geometric case, roughly speaking, are complementary. The approach of this paper is based on the phase-plane analysis for the time map and using the Poincaré-Birkhoff twist theorem.
Keywords
1 Introduction and the main results
Many problems in differential equations and dynamical systems are associated with the perturbations of an autonomous system. One usually concerns the existence of special kind of persistent solutions under perturbation. Recently, there are many researches on this subject using the assumption of ‘oscillatory nonlinearity’ which can be considered as a kind of nondegenerate condition at infinity in view of bifurcation; see [1, 2] and [3]. The approaches of these researches are based on the analysis of the time map for the related autonomous system combining the Poincaré-Birkhoff twist theorem for the planar homeomorphism and topological degree theory.
We will prove that the convexity of the potential and the oscillatory nonlinearity imply that the existence of the infinity of persistent subharmonic solutions, that is, we have the following.
Theorem 1.1
This theorem is a corollary of two theorems. One is in [2] and the other one is the following theorem. It considers the equation with so-called ‘weak sub-quadratic potential’ (see the condition \((G_{1})\) in the following).
Theorem 1.2
If we consider the forced equation (1.1), we can prove that \((\tau_{1}) \Rightarrow(G_{1})\).
Therefore Theorem 1.2 is a generalization of the results both in [6] and in [7].
Now we prove Theorem 1.1. Assume that \(G(x)\) is convex, then \(G(x)\le xg(x)\). \((G_{2})\) implies that \(G(x)\ge \frac{a}{2}x^{2}\), it follows that \(g(x)\ge \frac{a}{2}x\) for \(\vert x\vert \gg1\). Therefore \(\frac{G(x)}{g^{2}(x)}\leq\frac{x}{g(x)}\leq\frac{2}{a}\) for \(\vert x\vert \gg1\). Thus \((G_{0})\) holds. Hence under the assumption \((g_{1})\) and \((g_{2})\) we see that either \((G_{1})\) or \((g_{0})\) and \((G_{0})\) hold. Then Theorem 1.1 is a corollary of Theorem 1.2 and Theorem A in [2].
The key point of our argument is the estimation of the return time when the solution completes j clock-wise turns around the origin in the phase plane for given j under a forced perturbation.
In the rest of the paper we will give the details of the proof for Theorem 1.2.
2 Duffing equation with weak sub-quadratic potential
The key point of applying the Poincaré-Birkhoff twist theorem is constructing an annulus \(\mathcal {A}\) in the \((x,y)\) phase plane bounded by two star-shaped curves \(\gamma^{+}\) and \(\gamma^{-}\), such that the solution starting from \(\gamma^{+}\) and \(\gamma^{-}\) will move more than and less than j clock-wise turns in the time interval \([t_{0}, t_{0}+2k\pi]\), respectively.
When the solution passes through the origin at some time, we cannot compute how many clock-wise turns it moves. To avoid this problem, we introduce a modified equation such that if \(z(t)=(x(t),y(t))\) is a solution of the modified equation, then \(z(t_{0})\neq(0,0)\) for some \(t_{0}\) implies that \(z(t)\neq(0,0)\) for all t. Moreover, the twist property of the fixed point obtained by using the Poinacré-Birkhoff twist theorem will help us to guarantee that the \(2k\pi\)-periodic solution we showed for the modified equation is exactly the \(2k\pi\)-periodic solution of the original equation (2.1) for sufficiently large k.
Let \(z(t;t_{0},z_{0})=(x(t;t_{0},z_{0}),y(t;t_{0},z_{0}))\) be the solution of (2.2) satisfying the initial condition \(z(t_{0};t_{0},z_{0})=z_{0}=(x_{0},y_{0})\). Then we have the following fundamental lemma.
Lemma 2.1
Proof
f is a continuously differentiable function. The existence and uniqueness of the solution associated to the initial condition is ensured by the existence-uniqueness theorem. Moreover, the solution has continuity with respect to initial conditions. It is easy to see that \(z(t)\equiv(0,0)\) is the solution of (2.2) satisfying the initial condition \(z(t_{0};t_{0},z_{0})=(0,0)\), from which it follows that \(z_{0}\neq(0,0) \Rightarrow z(t;t_{0},z_{0})\neq(0,0)\), ∀t.
Moreover, we have the following.
Lemma 2.2
Assume that the conditions of Lemma 2.1 hold. Then there is \(R_{0}>R_{2}>R_{1}>0\), such that \(\theta' (t;t_{0},r_{0},\theta_{0})<0\) whenever \(r(t;t_{0},r_{0},\theta_{0})\ge R_{0}\).
Proof
Lemma 2.2 concludes that \(z(t;t_{0},z_{0})\) moves clock-wise around the origin O if \(\vert z(t;t_{0}, z_{0})\vert \ge R_{0}\). □
Lemma 2.3
Proof
Note that in equation (2.2) \(yx'=y^{2}>0\) whenever \(x=0\) and \(y\neq0\). Then for any solution \(z(t;t_{0},z_{0})\) of (2.2) with \(z_{0}\neq(0,0)\), we have \(\theta' (t;t_{0},r_{0},\theta _{0})<0\) whenever \(\theta(t;t_{0},r_{0},\theta_{0})=k\pi+1/2\pi\), \(k\in \Bbb {Z}\). Thus Lemma 2.3 in [7] shows that we have \(\theta (t_{2};t_{0},r_{0},\theta_{0})-\theta(t_{1};t_{0},r_{0},\theta_{0})<\pi\) for all \(t_{2}>t_{1}\). This proves the lemma. □
The following lemma is similar to that in [7] with the modified proof as in [7].
Lemma 2.4
Proof
The lemma is thus proved by induction. □
Lemma 2.5
Assume that the conditions of Lemma 2.1 hold. Then there exists a sequence \(\{h_{m}^{-}\}\) with \(\lim_{m\to\infty} h_{m}^{-}=+\infty\) such that \(T_{j}^{+}(h_{m}^{-})<+\infty\). Moreover, if \((G_{1})\) holds, then there exists another sequence \(\{h_{m}^{+}\}\) with \(\lim_{m\to\infty} h_{m}^{+}=+\infty\) such that \(\lim_{m\to\infty} T_{j}^{-}(h_{m}^{+})=+\infty\).
Proof
- (1)If \(r^{(m)}\leq \vert z(t;t_{0},z_{0})\vert \leq \zeta_{j+1}(r^{+}(h^{-}_{m}))\) for \(t\in[t_{0},t_{0}+L_{m}]\), then$$\theta(t_{0}+L_{m};t_{0},z_{0})- \theta_{0}= \int_{t_{0}}^{t_{0}+L_{m}}\theta '(t)\,dt \leq-aL_{m}\leq-2(j+1)\pi. $$
- (2)If there exists a time \(t_{1}'\in [t_{0},t_{0}+L_{m}) \) such that \(\vert z(t_{1}';t_{0},z_{0})\vert < r^{(m)}\) or \(\vert z(t_{1}';t_{0},z_{0})\vert >\zeta _{j+1}(r^{+}(h^{-}_{m}))\), then we have \(t_{1}\in [t_{0},t_{1}') \) such that \(\vert z(t;t_{0},z_{0})\vert \ge R_{0}\) for \(t\in[t_{0},t_{1}]\) andor$$\bigl\vert z(t_{1};t_{0},z_{0})\bigr\vert < r^{(m)},\qquad \vert z_{0}\vert \ge r^{-} \bigl(h^{-}_{m}\bigr)\ge\zeta_{j+1}\bigl(r^{(m)}\bigr), $$$$\bigl\vert z(t_{1};t_{0},z_{0})\bigr\vert > \zeta_{j+1}\bigl(r^{+}\bigl(h^{-}_{m}\bigr)\bigr),\qquad \vert z_{0}\vert \le r^{+}\bigl(h^{-}_{m}\bigr). $$
Hence for both cases, we show that the solution \(z(t;t_{0},z_{0})\), \(z_{0}\in\gamma_{h^{-}_{m}} \) completes at least j clock-wise turns in \([t_{0},t_{0}+L_{m}]\). Moreover, the above argument shows that for \(T\ge L_{m}\), the solution \(z(t;t_{0},z_{0})\), \(z_{0}\in\gamma_{h^{-}_{m}} \) completes at least j clock-wise turns in \([t_{0},t_{0}+T]\), from which it follows that \(T_{j}^{+}(h_{m}^{-})\leq L_{m}<+\infty\).
By the way, using the same method as employed above we can prove that if \(\vert z_{0}\vert \ge\zeta_{j+1}(K_{j+1})\) then \(T_{j}(t_{0},z_{0})\) is well defined.
Now we are in the position to prove Theorem 1.2.
Proof of Theorem 1.2
Moreover, in the above argument we can choose the same \({\mathcal {A}}_{m,l}\) for any \(f\in{\mathcal {B}}_{\mu_{0}}\), \({\mu_{0}}\) sufficiently small. Further, denote the two \(2k\pi\)-periodic solutions obtained above by \(z_{f,1}(t)= z_{1}(t)\) and \(z_{f,2}(t)=z_{2}(t)\). It is easy to find a compact annulus \({\mathcal {C}}_{m,l}\) such that \(z_{f,i}(t)\), \(i=1,2\), are in \({\mathcal {C}}_{m,l}\).
Theorem 1.2 is thus proved. □
Usually, we can use the generalized version of the Poincaré-Birkhoff fixed point theorem by Ding [10] as in [1, 7, 9]. The result in [10], however, requires an extra assumption, i.e. the strictly star-shapedness of the outer boundary of the annular region, as recently pointed out in [11]. Now we refer to [8], Corollary 2, which is a direct reduction to the classical Poincaré-Birkhoff theorem for the standard annulus, already settled in [12]. We also refer to other versions of the Poincaré-Birkhoff theorem due to Franks [13], and Qian-Torres [14], where independent proofs are given.
Declarations
Acknowledgements
This work is supported by the National Natural Science Foundation of China No. 11271277 and the Foundation of Wuxi Institute of Technology No. ZK201501.
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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