Infinitely Many Periodic Solutions of Duffing Equations with Singularities via Time Map

Abstract

We study the periodic solutions of Duffing equations with singularities . By using Poincaré-Birkhoff twist theorem, we prove that the given equation possesses infinitely many positive periodic solutions provided that satisfies the singular condition and the time map related to autonomous system tends to zero.

1. Introduction

In this paper, we are concerned with the periodic solutions of singular Duffing equations: where is locally Lipschitz continuous and has a singularity at the origin and is continuous and periodic, whose least period is .

The periodic problem of equations with singularities has been widely studied lately because of their background in applied sciences [1–15]. For example, the oscillation problem of a spherical thick shell made of an elastic material can also be modeled by this kind of equations [1].

The opening work on the existence of periodic solutions of ordinary differential equations with singularities was done by Lazer and Solimini [2], in which the equations were studied. It was proved in [2] that if , then (2) has at least one positive -periodic solution if and only if Meanwhile, if , then they constructed a periodic function with negative mean value such that (2) does not have any -periodic solution.

It is well known that time map plays an important role in studying the existence and multiplicity of periodic solutions of Duffing equations without singularities. In case when has a singularity, we can also use time map to deal with the periodic solutions of (1) (see [4] and the related references therein).

Assume that satisfies and the primitive function of satisfies Moreover, the following condition holds:

Let us define The map is usually called time map, which is continuous for large enough. We shall deal with the multiplicity of periodic solutions of (1) by means of asymptotic property of the time map . Assume that the time map satisfies It is easy to check that if satisfies superlinear condition then condition is satisfied. However, the converse is not true. In fact, we can find functions , which do not satisfy (5). But the corresponding time maps satisfy the condition . For example, let us define Obviously, conditions hold and condition (5) does not hold. Next, we will show that condition is satisfied. In case when , we have Therefore, we have Since we get

When the conditions , , and (5) hold, it was proved in [6] that (1) has infinitely many periodic solutions. In the present paper, we will deal with the multiplicity of periodic solutions of (1) under the conditions , , , and . Obviously, the conditions and generalize the condition (5). Since (5) does not hold, the estimating method in [6] is invalid. By taking some new estimating skills, we obtain the following results.

Theorem 1. Assume that conditions and hold. Then (1) has infinitely many positive harmonic solutions satisfying

Theorem 2. Assume that conditions and hold. Then for any integer , (1) has infinitely many positive -order subharmonic solutions satisfying

Remark 3. In the following, for convenience and brevity, we move the singular point to the point . In fact, we can take a transformation to achieve this aim. We will consider singular equations as follows: where is continuous and has a singularity at . We now assume that the following conditions hold: Next, we will deal with the existence and multiplicity of periodic solutions of under conditions , , , and .

2. Basic Lemmas

In this section, we will perform some phase-plane analyses for when conditions , , and hold. Consider the equivalent system of : Let be the solution of (13) through the initial point:

Lemma 4. Assume that conditions and hold. Then every solution of system (13) exists uniquely on the whole -axis.

Proof. Define a potential function Set Then we have where . From and we know that there exists a constant such that From (17) and (18) we get where . Then, for any finite , we have Therefore, is bounded for . Furthermore, exists on the interval . Similarly, we can prove that exists on the interval . The uniqueness of the solution follows directly from the local Lipschitzian condition on .

On the basis of Lemma 4, we can define the Poincaré map as follows: We know that fixed points of the Poincaré map correspond to -periodic solutions of (13).

To show the position of orbit of (13), we introduce a function ,

Lemma 5. There exists a constant such that, for any , is a closed star-shaped curve around the origin.

Proof. Consider autonomous system: Obviously, is one orbit of the autonomous system above. From the expression of we know that there exists such that, for , is a closed curve around the origin. Applying the polar coordinate transformation , to this system, we get In the case when , we have . In the case when and , we have , which implies . Therefore, there exists such that, for , is decreasing strictly. Take . Then for , is a closed star-shaped curve around the origin.

Lemma 6 (see [1]). Assume that conditions , , and hold. Then, for any and , there exists sufficiently large such that, for , where is the solution of (13) through the initial point .

From Lemma 6 we know that if is large enough, then , . Therefore, we can take the polar coordinate transformation Under this transformation, system (13) becomes Let be the solution of (27) satisfying condition with , .

Then we can rewrite the Poincaré map as follows: with , .

Lemma 7. Assume that conditions , , and hold. Then, for any , there exist and such that, for ,

Proof. From we know that there exist constants and such that Moreover, we know from that there exist and such that If , , then If , , then On the other hand, we know from Lemma 6 that there exists large enough such that if and , , then Hence, Consequently, the conclusion of Lemma 7 holds.

Lemma 8. Assume that conditions , , , and hold. Let be a given constant. Then we have

Proof. We now prove the first estimation. From condition we know that there exists a constant such that, for , Then, for , we have Write where From condition we can derive easily that . From (39) we get According to condition , we have that . Hence, we get Next, we prove the second estimation. Let be a sufficiently small constant. In the case when , we write where If , then we have Set Obviously, . From condition we know According to (46), we get that, for , Hence, which, together with (48), means that . On the other hand, we can infer easily from that . Consequently, we have Thus, the proof is completed.

Lemma 9. Assume that conditions , , , and hold. Let be a given positive integer. Then, for any given positive integer , there is a constant such that, for ,

Proof. From Lemmas 6 and 7 we know that, for any sufficiently large , there is a constant such that, for and , Let be a solution of (13) satisfying . Then the solution will move clockwise during the time period . Without loss of generality, we assume that lies in the first quadrant. Then there exist such that Next, we will estimate (), respectively. We first estimate . If , then . Let us define an auxiliary function where . Then we have that, for , which implies that is decreasing on the interval . Therefore, we get that, for , which means Hence, we obtain Similarly, we can obtain We next estimate . If , then . Therefore, we have which implies that is increasing on the interval . Furthermore, we have that, for , which yields Consequently, we get We now estimate . If , then . Define Then we have that, for , which implies that is decreasing on the interval . Therefore, we get that, for , which implies Hence, Similarly, we get According to Lemma 6, if we take , then we have , and . From Lemma 8 and (59)–(70) we know that, for any sufficiently small , there exists such that It follows that Therefore, the motion rotates clockwise a turn in a period less than . Consequently, can rotate a sufficiently large number of turns during the period provided that () is satisfied.
The proof is thus completed.

3. Infinity of Harmonic Solutions

To prove Theorem 1, we first prove the following proposition.

Proposition 10. Assume that conditions , , , and hold. Then has infinitely many harmonic solutions satisfying

Proof. From Lemma 6 we know that there exist ( is given in Lemma 5) and such that, for , and , . For , we consider with , . Obviously, there exists an integer such that On the other hand, it follows from Lemma 9 that there exists such that Meanwhile, there exists an integer such that From (75) and (76) we know that the area-preserving homeomorphism is twisting on the annulus . Obviously, we have provided that . Hence, , where is an open region with boundary . Finally, we know from Lemma 5 that both and are closed star-shaped curves with respect to the origin . Thus, we have proved that all conditions of the generalized Poincaré-Birkhoff theorem [16, 17] are satisfied. Consequently, the Poincaré map has at least two fixed points in annulus and then (13) has two periodic solutions (, ). Therefore, are periodic solutions of . On the other hand, since the period of any periodic solution of must be multiple of the period of , then is the minimal period of . Therefore, are harmonic solutions of .
Similarly, we can find a sequence such that the area-preserving homeomorphism is twisting on the annuli Therefore, the Poincaré map has at least two fixed points in each , . Consequently, (13) has two periodic solutions , () and then are periodic solutions of . Similarly, we know that are harmonic solutions of . Since , we have Furthermore, we know from Lemma 6 that, for , Thus we have proved Proposition 10.

Proof of Theorem 1. Consider the equivalent equation of (1): where . Obviously, satisfies conditions , , and . To use Proposition 10, we only need to prove that condition holds for function . Set Then we have From conditions and we get . Therefore, all conditions of Proposition 10 are satisfied. Accordingly, (82) has infinitely many harmonic solutions satisfying Recalling that (82) is obtained by taking a parallel transformation to (1), we know that the conclusion of Theorem 1 holds.

Remark 11. In [16], the Poincaré-Birkhoff theorem was proved in case that the inner closed curve of the annulus is star shaped. From [17] we know that there is a need for both boundaries of the annulus to be star shaped in the Poincaré-Birkhoff theorem.

4. Infinity of Subharmonic Solutions

To prove Theorem 2, we first prove the following proposition.

Proposition 12. Assume that conditions , , , and hold. Then, for any given integer , has infinitely many -order subharmonic solutions satisfying

Proof. Let be a given integer. From Lemmas 6 and 9 we know that there exists ( is given in Lemma 5) and such that, for , For , we consider with , . Obviously, there exists a positive prime integer such that On the other hand, it follows from Lemma 8 that there exists such that From (90) and (91) we know that the map is twisting on the annulus . Using the generalized Poincaré-Birkhoff twist theorem, we know that has at least two fixed points in , which satisfy It follows that (13) has two -periodic solutions and then has two -periodic solutions .
Next, we will prove that is the minimal period of . Assume by contradiction that is the minimal period of . Then we have (). Since is periodic, we know from (88) that there exists a positive integer such that Furthermore, Hence, we have . Since is a prime integer, we get a contradiction. This proves that is the minimal period of . Consequently, are -order subharmonic solutions of .
In a similar manner, we can find a sequence such that the area-preserving homeomorphism is twisting on the annuli Therefore, the Poincaré map has at least two fixed points in each , . Consequently, (13) has two periodic solutions and then are periodic solutions of  . Similarly, we know that are -order subharmonic solutions of  . Since , we have Furthermore, we know from Lemma 6 that, for , Thus we have proved Proposition 12.

Proof of Theorem 2. Using Proposition 12 and the same method as proving Theorem 1, we can prove Theorem 2.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was supported by the Research Fund for the Doctoral Program of Higher Education of China, no. 11AA0013, and the Grant of Beijing Education Committee Key Project, no. KZ201310028031.