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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 140173, 9 pages
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.
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.
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.  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  and Zhang et al.  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 ). 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
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,
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.
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
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.
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.
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.
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.
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