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Abstract and Applied Analysis
Volume 2010, Article ID 286040, 12 pages
http://dx.doi.org/10.1155/2010/286040
Research Article

The Lyapunov Stability for the Linear and Nonlinear Damped Oscillator with Time-Periodic Parameters

Department of Mathematics, College of Sciences, Hohai University, Nanjing 210098, China

Received 27 May 2010; Accepted 11 August 2010

Academic Editor: Stefan Siegmund

Copyright © 2010 Jifeng Chu and Ting Xia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let be continuous -periodic functions with . We establish one stability criterion for the linear damped oscillator . Moreover, based on the computation of the corresponding Birkhoff normal forms, we present a sufficient condition for the stability of the equilibrium of the nonlinear damped oscillator , where is a continuous -periodic function, is continuous -periodic in and dominated by the power in a neighborhood of .

1. Introduction

This paper is motivated by the study of the Lyapunov stability of the equilibrium of the linear and nonlinear time-periodic differential equations, which is an interesting problem in the theory of ordinary differential equations and dynamical systems. Let be continuous -periodic functions, that is, , . Recall that the linear oscillator is stable (in the sense of Lyapunov) if any solution of (1.1) satisfies

The research on the stability of second-order linear differential equations goes back to the time of Lyapunov. Many theoretical results concerning this problem can be found in textbooks such as [1]. One famous stability criterion was given by Magnus and Winkler [2] for the Hill equation That is, (1.3) is stable if which can be shown using a Poincaré inequality. Such a stability criterion has been generalized and improved by Zhang and Li in [3], which now is the so-called -criterion. Recently, Zhang in [4] has extended such a criterion to the linear planar Hamiltonian system where are continuous and -periodic functions. See Lemma 2.1 in next section.

Based on Lemma 2.1, in Section 2, we establish one new stability criterion for (1.1) under the following condition: (A) and

It is interesting to compare our new stability criterion with those in the literature. In fact, there have been many stability results for (1.1) in the literature [511]. We refer to [7, 8] for the discussions for (1.1) when is positive and is constant. On the other hand, if is a continuously differential function, (1.1) can be transformed into where Therefore, every stability criterion for the Hill equation (1.6) can produce a stability criterion for (1.1). As far as the authors know, in the literature there are few stability results for the case that is continuous (may not be differential) and can change sign, which is just our case. Here we refer the reader to [12, 13] for some recent stability criterion of planar Hamiltonian systems.

In Section 3, we present a sufficient condition for the stability of the equilibrium of the nonlinear damped oscillator where satisfy (A) and is a continuous, -periodic functions, is a continuous function with continuous derivatives of all orders with respect to the second variable, -periodic in , and

It is well-known that (1.8) can be unstable when linear oscillator (1.1) is stable. For the case Liu proved in [14] that the equilibrium of (1.8) is stable if and only if Recently, Liu et al. [15] has extended such a result to the case and with In this paper, we will deal with the case that with .

During the last decade, an analytical method for studying time-periodic Lagrangian equations has been developed recently by Ortega in a series of papers [1619]. A nice illustrating example for Ortega's approach is the so-called swing [19, 20] That is, the equilibrium of swing (1.10) is stable if and only if its cubic approximation is stable. After that, some researchers have extended the applications of such an analytical method, and some important stability results for several types of Lagrangian equations have been established, for example, see [2123] and the references therein. We also refer the reader to [24] for its extension to the planar nonlinear system.

We will extend such an analytical method to the nonlinear damped oscillator (1.8). The proof is based on a careful computation of certain Birkhoff normal forms together with some stability results of fixed points of area-preserving maps in the plane. As an application, when condition (A) is satisfied, we present a quite complete Lyapunov’s stability result for the equilibrium position of the pendulum of variable length with damping term That is, the stability of its linearized equation implies the stability of the equilibrium of (1.12).

2. Stability Criteria for Linear Damped Oscillator

Let be a fundamental matrix solution for the linear damped oscillator (1.1), where and are real-valued solutions of (1.1) satisfying It is easy to verify that =det satisfies the following equation: which implies that in which we have used the fact The Poincaré matrix of (1.1) is Using the condition (A) and (2.4), we have The eigenvalues of are called the Floquet multipliers of (1.1). Obviously . Therefore we can distinguish (1.1) in the following three cases:(i)elliptic: ,;(ii)hyperbolic: ;(iii)parabolic: .

It is well-known that (1.1) is stable in the sense of Lyapunov if and only if (1.1) is elliptic or is parabolic with further property that all solutions of (1.1) are -periodic solutions in case or -periodic solutions in case . See, for example, [1, Theorem ].

In order to state our new stability criteria for (1.1), we recall the following stability results for the planar Hamiltonian system (1.5).

Lemma 2.1 (see [4]). Let and . Then system (1.5) is stable if one of the following two conditions is satisfied: (a)criterion (b)criterionwhere is the usual norm for is the conjugate exponent of , and is given by where is the Gamma function of Euler. See [4].

Now we are in a position to state the new stability criteria for the linear damped oscillator.

Theorem 2.2. Assume that (A) holds. Then (1.1) is stable if one of the following two conditions is satisfied: (a)criterion (b)criterion

Proof. The linear damped oscillator (1.1) is equivalent to the following planar system: Using the change of variable system (2.13) can be written as System (2.15) is stable if and only if Under the condition (A), if (2.16) is satisfied, one may easily see that which implies that (1.1) is stable. Using Lemma 2.1, system (2.15) is stable when the inequality (2.11) or the inequality is satisfied, where One may easily verify that (2.18) is just inequality (2.12).

The following corollary is a direct result of Theorem 2.2.

Corollary 2.3. Assume that (A) holds and Then there exists a positive constant such that (1.1) is stable when

Theorem 2.2 can be applied to the damped Mathieu's equation where and

Corollary 2.4. Assume that is a continuous -periodic function with Then there exists a constant such that (2.20) is stable when

Remark 2.5. In this paper, the condition is crucial. In fact, for the case , (1.1) and (1.8) will be dissipative, and therefore would be unstable for all (but would be stable for ). For our case , (1.1) becomes an conservative equation at although (1.1) is dissipative at the other time This fact play an important role in our analysis.

3. Stability of the Nonlinear Damped Oscillator

Assume that (1.1) is elliptic and the Floquet multipliers of (1.1) are and with , , , In general, the Poincaré Matrix is conjugate, in the symplectic group to the rigid rotation

Definition 3.1. One says that (1.1) is -elliptic, if (1.1) is elliptic and its Poincaré matrix is a rigid rotation.

Theorem 3.2. Assume that (1.1) is elliptic. Then there always exist some and such that under the transformation the Poincaré matrix of the transformed equation is a rigid rotation where and

Proof. The proof is similar to the proof of [24, Theorem ] and [16, Proposition ], here we omit it.

Remark 3.3. When the linear oscillator is stable, we can assume that (1.1) satisfies the following condition: where is one of the eigenvalues of and . In fact, Theorem 3.2 guarantees that (3.5) holds for the case of ellipticity after a temporal transformation. When (1.1) is parabolic and stable, condition (3.5) is always satisfied, because all solutions of (1.1) are either -periodic or -periodic in this case.

The proof of the main results is based on the theory of stability of fixed points of area-preserving maps in the plane [17, 25]. Let be an area-preserving map defined in an open neighborhood of the origin, and is a fixed point of . It is assumed further that is sufficiently smooth. For convenience, the complex notation is used.

Lemma 3.4 (see [17, Lemma ]). Assume that for some Then there exists a real-valued homogeneous polynomial of degree such that

Now we assume that satisfies the conditions of Lemma 3.4 with The polynomial given by the lemma can be expressed in the form where and

Lemma 3.5 (see [17, Proposition ]). Assume that satisfies (3.7) with for some is given by (3.8), and one of the following conditions hold: for each and for some , and for each where Then is stable with respect to .

Before stating our result, we present the following simple result on the linear inhomogeneous system: which is a consequence of the formula of variation of constants together with (3.5).

Lemma 3.6. Let be the solution of (3.10) with Assume that (3.5) is satisfied. Then one has where is given by (2.4).

Proof. Using the formula of variation of constants, one may easily see that By easy computations, we obtain Therefore,

The main result of this section reads as follows.

Theorem 3.7. Assume that (A) holds and Furthermore, suppose that the following two conditions are satisfied: the linear oscillator (1.1) is stable or Then the trivial solution of the nonlinear oscillator (1.8) is stable.

Proof. We prove the result assuming that condition (3.5) is satisfied. The result holds for the general case because we have Remark 3.3.
Let be the solution of the nonlinear system (1.8) with and , where . The theorem of differentiability with respect to initial conditions implies that These two expansions are uniform in .
We look at the nonlinear oscillator (1.8) as one equation of the kind (3.10) with where Since we have assumed that (3.5) holds, we can apply Lemma 3.6 to (3.10) to obtain where is given by (3.16).
Combining (3.15) and (3.18), we have the expansion Then satisfies (3.7) with given by The coefficient in (3.8) is given by Since and are linearly independent solutions of (1.1), we know that for all
Assume that condition (H) holds and . Then and for all When for each , (C) is satisfied. If for some then the definition of and the negativity of imply that for all and therefore (C) holds. When and , the result holds by similar analysis.

Every stability criterion for the linear oscillator (1.1) together with the assumption (H) produces a stability criterion for the nonlinear system (1.8). For example, we have the following.

Corollary 3.8. Assume that (A), (H) holds and Furthermore suppose that (2.11) or (2.12) holds. Then the trivial solution of (1.8) is stable.

The following stability result for the swing with damping term (1.12) follows directly from Theorem 3.7.

Corollary 3.9. Assume that (A) holds and is a positive function. Then the trivial solution of (1.12) is stable if the linearized oscillator is stable.

Proof. Equation (1.12) can be regarded as one of (1.8) with and Since is a positive -periodic function, the result is now a direct consequence of Theorem 3.7.

Acknowledgments

The research of J. Chu is supported by the National Natural Science Foundation of China (Grant no. 10801044), Jiangsu Natural Science Foundation (Grant No. BK2008356), the Program for New Century Excellent Talents in University (Grant no. NCET-10-0325), and the Fundamental Research Funds for the Central Universities.

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