Abstract and Applied Analysis

Volume 2013 (2013), Article ID 140173, 9 pages

http://dx.doi.org/10.1155/2013/140173

## Nontrivial Periodic Solutions of an -Dimensional Differential System and Its Application

School of Mathematical Science, Yangzhou University, Yangzhou 225002, China

Received 29 March 2013; Accepted 3 August 2013

Academic Editor: Pei Yu

Copyright © 2013 F. B. Gao. 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

Two criteria are constructed to guarantee the existence of periodic solutions for a second-order -dimensional differential system by using continuation theorem. It is noticed that the criteria established are found to be associated with the system’s damping coefficient, natural frequency, parametrical excitation, and the coefficient of the nonlinear term. Based on the criteria obtained, we investigate the periodic motions of the simply supported at the four-edge rectangular thin plate system subjected to the parametrical excitation. The effectiveness of the criteria is validated by corresponding numerical simulation. It is found that the existent range of periodic solutions for the thin plate system increases along with the increase of the ratio of the modulus of nonlinear term’s coefficient and parametric excitation term, which generalize and improve the corresponding achievements given in the known literature.

#### 1. Introduction

In recent years, thin plates have been widely applied to the fields of automobile, marine, space station, shutter and modern aircraft, and so forth. Therefore, the nonlinear dynamic behavior of the thin plate received very considerable attention within many articles available in the technical and scientific literature. See [1–6], for example, and the references therein.

In the aforementioned works, based on the Schauder second fixed point theorem, Dizaji et al. [2] predicted the existence of periodic solutions of the following governing equations of motion: which could be derived from the nonlinear simply supported rectangular thin plate system under the influence of a relatively moving mass.

Zhang [5] and Zhang et al. [6] studied the periodic and chaotic motions of the parametrically excited rectangular thin plates on the basis of multiple scales method and continuation theorem, respectively.

As far as we know, there were few researchers who focused on the existence of periodic solutions of the thin plate system subjected to the parametrical excitation with rigorous theoretical proof. In contrast, the existence of periodic solutions is often shown only by numerical simulation. However, it is the rigorously proved theorem that can throw more light than thousands of beautiful pictures on the basic nature of periodicity.

Recently, there are lots of mathematical researchers devoted to the investigation of the periodic solutions for -Laplacian-like systems, for example, [7–12], which can be reduced to the general second-order systems of ordinary differential equations while . Nevertheless, the results obtained cannot be applied to the general nonlinear equations, for example, see [13–16]. The challenge lies in the growth degree with respect to the nonlinear term which often needs to be less than or equal to for -Laplacian-like systems. For instance, the one-sided growth condition imposed on the nonlinear terms in [7, 8, 10, 17] was given as follows:

In this paper, two criteria are established to guarantee the existence of periodic solutions for a second-order -dimensional differential system. It is noticed that the existence of nontrivial periodic solutions is found to be influenced by the system’s damping coefficient, natural frequency, parametric excitation, and the coefficient of the nonlinear term. Moreover, the parametrical excitation in this paper is not limited to be periodic.

As an application of the criteria obtained, the existence of periodic solutions for the simply supported at the four-edged rectangular thin plate system subjected to parametrical excitation is investigated in Section 4 of this paper. Furthermore, corresponding numerical simulations are carried out to validate the feasibility of the criteria achieved. From the several numerical results, it is noticed that the existent range of periodic solutions for the thin plate system becomes larger with the increase of the ratio of the modulus of nonlinear term’s coefficient and parametric excitation term. By means of analytical arguments and numerical simulation runs, it is easy to find that the proposals given in this study are seldom obtained in the known literature, for example, [2, 5, 6].

#### 2. Preliminaries and Notations

Consider a second-order -dimensional system where , , , and is an symmetric matrix of constants. with , , and ; is a positive constant.

Next, we recall an important lemma which will help us to start the corresponding research.

Lemma 1 (see [18]). *Suppose that and are two Banach spaces, and let be a Fredholm operator with index zero. Furthermore, is an open bounded set and is L-compact in . If*(1)*, ,*(2)*, ,*(3)*, where is an isomorphism.**Then, the equation has a solution in .*

In what follows, for convenience and without loss of generality, some notations are introduced throughout the paper: denotes absolute value and the Euclidean norm on , for and . Also, we set , , and with the norm and with the norm . Obviously, and are two Banach spaces. Meanwhile, denote where .

It is easily shown that system (3) can be converted into the equivalent abstract equation . Moreover, from the definition of , we see that , . Therefore, is a Fredholm operator with index zero.

Let the projections then, , .

Let represent the inverse of ; then where

From (5) and (7), it is easily verified that is -compact on , where is an arbitrary open bounded subset of .

#### 3. Main Results

##### 3.1. Theoretical Proof

Theorem 2. *For all , assume that the following conditions are satisfied:**, where are the eigenvalues of and .**(i) If , there is a constant such that
* *where and .* *(ii) If , and ,**then, system (3) has at least one nontrivial -periodic solution if there exist constants , such that
*

*Proof. *Let us embed system (3) into one parameter family of the systems as follows:

Since is the symmetrical matrix, there is an orthogonal matrix , such that

Integrating both sides of (11) from to gives

Applying integral mean value theorem, there exists a constant , such that

Now, we claim that
where .*Case **1.* If , then, (15) holds clearly.*Case **2.* If , define

By (14) and simple calculation, we obtain
Thus, it can be easily seen that (15) holds.

According to (15), we have

Combining inequalities (18) and (19) yields

Therefore, it follows from condition that . Then, we obtain

As , multiplying both sides of the th component of (11) by and integrating on the interval lead to

Using Hölder’s inequality and (22), we have

It is noticed that and are bounded on the interval . From (21), we obtain

According to the condition , it is easily seen that there exists a constant , such that

Combining (21) and (25), we obtain

For , there exists a , such that . Then, it follows from (11) that

Therefore, we have

Let . For , there is not any solutions of (11) on with for all . Based on the condition , there exist the appropriate constants and , such that the following relationship holds
Thus, the first two conditions of Lemma 1 are satisfied.

Next, we claim that the third condition of Lemma 1 is also satisfied. To verify this, we define the isomorphism , . For all , , we denote

By using the condition again, the following relationships hold, when: ,
It follows from (31) that is homotopic and that
Thus, the last condition of Lemma 1 is also satisfied.

Applying Lemma 1, it can be concluded that the equation has at least one -periodic solution on with . Furthermore, it is obvious that the -periodic solution is nontrivial. Otherwise, there is a constant vector satisfying (11), that is,

By simple computation, we obtain which contradicts the condition in Theorem 2. Then, system (3) has at least one non-trivial -periodic solution. Therefore, we complete the proof of Theorem 2.

Theorem 3. *Assume that and hold for all , then, system (3) has at least one -periodic solution.*

*Proof. *The same proof also works for this theorem. We only need to show that is bounded.

As , multiplying both sides of the th component of (11) by and integrating from to yield

It can be easily found that when using the condition . Combining (21) and (35), we obtain

Noticing that , there is a constant , such that . Therefore, the proof of the boundedness is completed. The rest proof of the theorem is almost identical to that of Theorem 2.

Corollary 4. *If , let . Then, we have . System (11) can be reduced to the following form
**
Thus, one only needs to study system (37) by using the aforementioned results.*

##### 3.2. Application

In this section, we apply some of the main results obtained in the previous section to a well-known model for practical engineering.

We now investigate periodic motions of the simply supported at the four-edged rectangular thin plate system subjected to parametrical excitation (see Figure 1).

According to [5, 19], we have the following partial differential governing equations: where represents the density of the plate, is the bending rigidity, is Young’s modulus, is the Poission ratio, is the stress function, and the damping coefficient.

By means of Galerkin’s method, (38) can be reduced to the following dimensionless form: where

Obviously, system (11) can be reduced to the system (39), provided that , , , , , and .

In what follows, in order to illustrate the aforementioned theoretical results, several numerical simulations are carried out. For system (39), by Theorem 2, we take a set of initial values , let and and choose appropriate such that .

By using ode45 in MATLAB 7, three families of dynamic characteristics of the thin plate system are illustrated for different values of and , respectively.

For , a centrosymmetric periodic solution in the plane is shown in Figure 2, keeping fixed at 15. Moreover, time history curves with respect to the displacement and velocity of the plate are also shown in Figures 3 and 4 for the same condition. It is straightforward to see that the curves are also periodic.

If we set and , a group of dynamics behavior of the thin plate system in the planes , , and is obtained in Figures 5, 6, and 7, respectively. It can be observed from these figures that the phase portrait is also centrosymmetric and the time history curves are periodic too. Furthermore, according to the corresponding power spectrum, there is a dominant peak at the frequency that is approximately equal to 2.3 with symmetric sidebands surrounding it.

For , when is gradually increased to 20, a set of dynamics characteristic of the thin plate system is addressed in Figures 8, 9 and 10, respectively.

In Figure 8, phase portrait in the plane is also illustrated to be centrosymmetric, though it seems to be more complex than the previous ones. The corresponding periodic time history curves with respect to the displacement and velocity of the plate are depicted in Figures 9 and 10. In addition, the power spectrum in this case associated with the periodic solution admits a distinctive broadband character.

#### 4. Conclusions

This paper primarily deals with the existence of nontrivial periodic solutions for a second-order -dimensional differential system. Moreover, the simply supported at the four-edged rectangular thin plate system subjected to parametrical excitation is investigated as an application. Theoretical analysis and numerical validation produce several important results as follows.(i)From the conditions of the proved theorems, it is easy to find that the nontrivial periodic solutions of the system are mainly influenced by the system’s damping coefficient, natural frequency, parametrical excitation, and the coefficient of the nonlinear term.(ii)By substituting the variables , , , and into the condition of Theorem 2, and combining with the phase diagrams and time history curves displayed above, one can see that there exist a set of -periodic solutions at least for system (39) with , , and under the three sets of different parameter values, respectively.(iii)It is significant that the existent range of periodic solutions for system (39) increases along with the increase of the ratio of and through simple calculation.(iv)In addition, the parametrical excitation term need not be periodic in accordance with the proof of Theorem 2, though it finds expression in periodic form for the above illustrated model.

#### Acknowledgments

The author gratefully acknowledges the support of the National Natural Science Foundation of China (NNSFC) through Grant no. 11302187, and is also very grateful to the referee for his/her careful reading of the original paper.

#### References

- S. I. Chang, A. K. Bajaj, and C. M. Krousgrill, “Nonlinear vibrations and chaos in harmonically
excited rectangular plates with one-to-one internal resonance,”
*Nonlinear Dynamics*, vol. 4, pp. 433–460, 1993. View at Google Scholar - A. F. Dizaji, H. A. Sepiani, F. E. Ebrahimi, A. Allahverdizadeh, and H. A. Sepiani, “Schauder fixed point theorem based existence of periodic solution for the response of Duffing's oscillator,”
*Journal of Mechanical Science and Technology*, vol. 23, pp. 2299–2307, 2009. View at Google Scholar - N. Malhotra and N. S. Namachchivaya, “Chaotic dynamics of shallow arch structures under 1 : 1 internal resonance conditions,”
*Journal of Engineering Mechanics*, vol. 123, pp. 612–619, 1997. View at Google Scholar - W. M. Tian, N. S. Namachchivaya, and N. Malhotra, “Nonlinear dynamics of a shallow arch under periodic excitation-II. 1 : 1 internal resonance,”
*International Journal of Non-Linear Mechanics*, vol. 29, pp. 367–386, 1994. View at Google Scholar - W. Zhang, “Global and chaotic dynamics for a parametrically excited thin plate,”
*Journal of Sound and Vibration*, vol. 239, pp. 1013–1036, 2001. View at Google Scholar - W. Zhang, F. B. Gao, and L. H. Chen, “Periodic solutions for a thin plate with parametrical excitation,” in
*Proceedings of the 12th NCNV and 9th NCNDSM*, Zhenjiang, China, 2009. - P. Amster, P. De Nápoli, and M. C. Mariani, “Periodic solutions for
*p*-Laplacian like systems with delay,”*Dynamics of Continuous, Discrete & Impulsive Systems*, vol. 13, no. 3-4, pp. 311–319, 2006. View at Google Scholar · View at MathSciNet - W. S. Cheung and J. L. Ren, “Periodic solutions for
*p*-Laplacian Rayleigh equations,”*Nonlinear Analysis. Theory, Methods & Applications*, vol. 65, no. 10, pp. 2003–2012, 2006. View at Publisher · View at Google Scholar · View at MathSciNet - F. B. Gao, W. Zhang, S. K. Lai, and S. P. Chen, “Periodic solutions for
*n*-generalized Liénard type*p*-Laplacian functional differential system,”*Nonlinear Analysis. Theory, Methods & Applications*, vol. 71, no. 12, pp. 5906–5914, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - F. B. Gao and W. Zhang, “Periodic solutions for a
*p*-Laplacian-like NFDE system,”*Journal of the Franklin Institute*, vol. 348, no. 6, pp. 1020–1034, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - F. B. Gao, S. P. Lu, and W. Zhang, “Existence and uniqueness of periodic solutions for a
*p*-Laplacian Duffing equation with a deviating argument,”*Nonlinear Analysis. Theory, Methods & Applications*, vol. 70, no. 10, pp. 3567–3574, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - R. Manásevich and J. Mawhin, “Periodic solutions for nonlinear systems with
*p*-Laplacian-like operators,”*Journal of Differential Equations*, vol. 145, no. 2, pp. 367–393, 1998. View at Publisher · View at Google Scholar · View at MathSciNet - M. L. Bertotti and M. Delitala, “On the existence of limit cycles in opinion formation processes under time periodic influence of persuaders,”
*Mathematical Models & Methods in Applied Sciences*, vol. 18, no. 6, pp. 913–934, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - D. Jiang, D. O'Regan, R. P. Agarwal, and X. Xu, “On the number of positive periodic solutions of functional differential equations and population models,”
*Mathematical Models & Methods in Applied Sciences*, vol. 15, no. 4, pp. 555–573, 2005. View at Publisher · View at Google Scholar · View at MathSciNet - L. Nie, Z. Teng, and L. Hu, “Existence and stability of periodic solution of a stage-structured model with state-dependent impulsive effects,”
*Mathematical Methods in the Applied Sciences*, vol. 34, no. 14, pp. 1685–1693, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - R. Ouifki and M. L. Hbid, “Periodic solutions for a class of functional differential equations with state-dependent delay close to zero,”
*Mathematical Models & Methods in Applied Sciences*, vol. 13, no. 6, pp. 807–841, 2003. View at Publisher · View at Google Scholar · View at MathSciNet - S. P. Lu and W. G. Ge, “Sufficient conditions for the existence of periodic solutions to some second order differential equations with a deviating argument,”
*Journal of Mathematical Analysis and Applications*, vol. 308, no. 2, pp. 393–419, 2005. View at Publisher · View at Google Scholar · View at MathSciNet - R. E. Gaines and J. L. Mawhin,
*Coincidence Degree, and Nonlinear Differential Equations*, Springer, Berlin, Germany, 1977. View at MathSciNet - C. Y. Chia,
*Nonlinear Analysis of Plate*, McMraw-Hill, New York, NY, USA, 1980.