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Existence and stability of periodic solutions for a forced pendulum with time-dependent damping
- Fang-Fang Liao^{1, 2}Email author and
- Zaitao Liang^{3}
Received: 27 February 2018
Accepted: 1 July 2018
Published: 5 July 2018
Abstract
In this paper, we study the existence, multiplicity and stability of periodic solutions for a forced pendulum with time-dependent damping. The proof is based on the third order approximation method and a suitable version of the Poincaré–Birkhoff fixed point theorem.
Keywords
- Damped pendulum
- Third order approximation method
- Poincaré–Birkhoff fixed point theorem
MSC
- 34C25
- 34D20
1 Introduction
However, up to now, the problem on the existence and stability of periodic solutions for Eq. (1.2) has not attracted attention in the literature. The purpose of this paper is to fill this gap. In this paper, we study the existence, multiplicity and stability of periodic solutions for Eq. (1.2).
In the third section, we prove that Eq. (1.2) has a twist T-periodic solution if the driving force is not too large. Such twist periodic solution is stable in the sense of Lyapunov [36]. The proof is based on the third order approximation method for nonlinear damped equations, which was developed by Chu et al. [6]. The third order approximation method for general time-periodic Lagrangian equations was developed by Ortega [31] and Zhang [40] and has been applied in [8, 10, 11, 20–22, 24, 38, 39] for different kinds of equations. Recently, in [9], Chu, Liang, and Torres used the Poincaré–Birkhoff fixed point theorem and the third order approximation method to study the existence and stability of periodic oscillations of a rigid dumbbell satellite around its center of mass. We will apply them to study the existence, multiplicity and stability of periodic solutions for a forced pendulum with time-dependent damping.
In the fourth section, we prove that Eq. (1.2) has at least two geometrically distinct T-periodic solutions. Moreover, at least one of them is unstable. Here, we say that Eq. (1.2) has at least two geometrically distinct T-periodic solutions, if such solutions are not differing by a multiple of 2π. Furthermore, we also study the existence of periodic and subharmonic solutions with winding number for Eq. (1.2). The proof is based on a suitable version of the Poincaré–Birkhoff fixed point theorem, which was originally conjectured by Poincaré [33] in 1912 when he studied the restricted three body problems, and was first proved by Birkhoff [3, 4] in 1913. During the last century, different proofs and developments were given. We refer the reader to [13, Sect. 1] for a short review of the Poincaré–Birkhoff fixed point theorem.
2 Preliminaries
2.1 A stability criterion
Given a function \(a(t)\), we denote \(a_{+}(t)=\max\{a(t),0\}\) and \(a_{-}(t)=\max\{-a(t),0\}\) the positive and the negative parts of \(a(t)\).
Based on the method of third order approximation for damped differential equations [6], the following stability criterion was proved in [9].
Theorem 2.1
([9, Theorem 2.2])
2.2 The Poincaré–Birkhoff fixed point theorem
We say that S is isotopic to the inclusion, if there exists a function \(H:A\times[0,1]\rightarrow B\) such that, for every \(\lambda\in[0,1]\), \(H_{\lambda}(x)=H(x,\lambda)\) is a homeomorphism with \(H_{0}(x)=S(x)\) and \(H_{1}(x)=x\). The class of the maps satisfying the above characteristics will be indicated by \(\mathbb{\varepsilon}^{k}(A)\).
The following theorem is a slight modified version of the Poincaré–Birkhoff fixed point theorem proved by Franks in [15, 16] and the statement on the instability was proved by Marò in [25].
Theorem 2.2
3 Stable periodic solutions
Lemma 3.1
([20], Lemma 2.1)
Theorem 3.2
Proof
Finally, by using the Banach contraction mapping theorem, we know that the operator \(\mathbb{T}\) has a unique fixed point u in \(\mathbb {B}\) if the strict inequality in condition (3.5) is satisfied. Note that if the equality in (3.5) holds, we can also obtain the uniqueness from the proof above, although \(\mathbb{T}\) may not be a strict contraction map.
By the uniqueness of the T-periodic solution of Eq. (1.2) in \(\mathbb{B}\), we know that \(\|u\|_{\infty}\) is smaller than other possible T-periodic solutions of Eq. (1.2). Moreover, (3.6) holds. □
The main result of this section reads as follows.
Theorem 3.3
Assume that (3.1) and (3.3) are satisfied. Then there exists a constant \(\rho\in(0,\frac{2\sqrt{2}\omega }{3}]\), such that the T-periodic solution u of Eq. (1.2) obtained in Theorem 3.2 is of twist type if \(\|p\|_{\infty}\leq\rho\).
Proof
Example 3.4
Proof
4 Multiplicity of periodic solutions
The aim of this section is to prove that the Poincaré map associated to Eq. (1.2) fits the hypotheses of Theorem 2.2 exposed in Sect. 2.1.
Theorem 4.1
Proof
In order to apply Theorem 2.2, we need to prove that the Poincaré map S is exact symplectic and satisfies the boundary twist condition (2.7).
In order to apply Theorem 2.2, we take \(A={\mathbb {R}}\times[-\rho,\rho]\). By the solutions of the system (4.2) are globally defined, one can find a larger B such that \(S(A)\subset \operatorname{int}B\). Since the right-hand side of the system (4.2) is analytic with respect to the variables \((u,v)\), the Poincaré map S is also analytic, as follows from the analytic dependence on the initial conditions.
Theorem 4.2
Let u is a periodic solution of (1.2) and take its minimal period \(\tau>0\). Since T is the minimal period of p, we have \(\tau =kT, k\in {\mathbb {N}}\mbox{ with } k\geq1\). Therefore, if \(k=1\), x is a harmonic (periodic) solution of Eq. (1.2); if \(k\in {\mathbb {N}}\mbox{ with } k>1\), u is a subharmonic solution of Eq. (1.2).
Theorem 4.3
Assume that (4.5) and (4.10) hold. Then, for each couple of relatively prime natural numbers N and k, Eq. (1.2) has at least two geometrically distinct k-order subharmonic solutions with winding number N and kT is the minimal period. Moreover, at least one of them is unstable.
Proof
By Theorem 4.2, with T replaced by kT, we see that Eq. (1.2) has at least two geometrically distinct k-order subharmonic solutions with winding number N. Moreover, at least one of them is unstable. We only need to verify that kT is the minimal period of periodic solutions with winding number N for Eq. (1.2).
Example 4.4
- (I)
Equation (4.15) has at least two geometrically distinct 2π-periodic solutions, and at least one of them is unstable.
- (II)
For every integer N, Eq. (4.15) has at least two geometrically distinct 2π-periodic solutions with winding number N, and at least one of them is unstable.
- (III)
For each couple of relatively prime natural numbers N and k, Eq. (4.15) has at least two geometrically distinct k-order subharmonic solutions with winding number N and kT is the minimal period. Moreover, at least one of them is unstable.
Proof
Declarations
Acknowledgements
The authors are very grateful to the referee for constructive suggestions for improving the initial version of the paper.
Availability of data and materials
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
Funding
Fangfang Liao was supported by QingLan project of Jiangsu Province and the National Natural Science Foundation of China (Grant No. 11701375).
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Competing interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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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