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

Abstract

Two criteria are constructed to guarantee the existence of periodic solutions for a second-order 𝑛-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: 𝐴̈+𝑃(𝑡)𝐴+𝑄(𝑡)𝐴³=𝐹(𝑡), (1) 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 𝑝-Laplacian-like systems, for example, [7–12], which can be reduced to the general second-order systems of ordinary differential equations while 𝑝≡2. 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 𝑝−1 for 𝑝-Laplacian-like systems. For instance, the one-sided growth condition imposed on the nonlinear terms in [7, 8, 10, 17] was given as follows: lim_|𝑢|→∞𝑔(𝑡,𝑢)|𝑢|^𝑛⩽𝑟∈[0,+∞[,uniformlyfor𝑡∈[0,𝑇],𝑛⩽𝑝−1. (2)

In this paper, two criteria are established to guarantee the existence of periodic solutions for a second-order 𝑛-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 𝑛-dimensional system 𝑥̈+𝜇₀𝑥̇+𝜔²₀+𝛼(𝑡)𝑥+𝜷𝑥³₁,𝑥³₂,…,𝑥³_𝑛^𝑇=𝑒(𝑡), (3) where 𝑥∈ℝ^𝑛, 𝜇₀∈ℝ, 𝜔₀∈ℝ^𝑛, and 𝜷 is an 𝑛×𝑛 symmetric matrix of constants. 𝛼,𝑒∈𝐶(ℝ,ℝ^𝑛) with 𝛼(𝑡+𝑇)≡𝛼(𝑡), 𝑒(𝑡+𝑇)≡𝑒(𝑡), and ∫^𝑇₀𝑒(𝑠)𝑑𝑠=0; 𝑇 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 𝑋 and 𝑌 are two Banach spaces, and let 𝐿∶𝐷(𝐿)⊂𝑋→𝑌 be a Fredholm operator with index zero. Furthermore, Ω⊂𝑋 is an open bounded set and 𝑁∶Ω→𝑌 is L-compact in Ω. If(1)𝐿𝑥≠𝜖𝑁𝑥, ∀(𝑥,𝜖)∈(𝐷(𝐿)∩𝜕Ω)×]0,1],(2)𝑁𝑥∉Im𝐿, ∀𝑥∈Ker𝐿∩𝜕Ω,(3)deg{𝐽𝑄𝑁,Ω∩Ker𝐿,0}≠0, where 𝐽∶Im𝑄→Ker𝐿 is an isomorphism.Then, the equation 𝐿𝑥=𝑁𝑥 has a solution in 𝐷(𝐿)∩Ω.

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 ℝ^𝑛, for ∀𝑎=(𝑎₁,𝑎₂,…,𝑎_𝑛)^𝑇 and |𝑎|=(∑^𝑛_𝑖=1|𝑎_𝑖|²)^1/2. Also, we set 𝐼=[0,𝑇], 𝐶⁰=𝐶⁰(𝐼,ℝ^𝑛), 𝐶¹=𝐶¹(𝐼,ℝ^𝑛) and 𝐶_𝑇={𝑢∈𝐶⁰∣𝑢(0)=𝑢(𝑇)} with the norm |𝑢|_max=max_𝑡∈𝐼|𝑢(𝑡)| and 𝐶¹_𝑇={𝑣∈𝐶¹∣𝑣(0)=𝑣(𝑇),𝑣(̇0)=𝑣(̇𝑇)} with the norm ‖𝑣‖=max{|𝑣|_max,|𝑣|̇_max}. Obviously, 𝐶_𝑇 and 𝐶¹_𝑇 are two Banach spaces. Meanwhile, denote 𝐿∶𝐷(𝐿)⊂𝐶_𝑇⟶𝐶_𝑇,𝐿𝑥=𝑥̈,𝑁∶𝐶_𝑇⟶𝐶_𝑇,[𝑁𝑥](𝑡)=−𝜇₀𝑥̇−𝜔²₀(𝑡𝑥𝑥+𝛼)−𝜷³₁,𝑥³₂,…,𝑥³_𝑛^𝑇+𝑒(𝑡), (4)(5) where 𝐷(𝐿)={𝑥∣𝑥∈𝐶¹_𝑇,𝑥̈∈𝐶(ℝ,ℝ^𝑛)}.

It is easily shown that system (3) can be converted into the equivalent abstract equation 𝐿𝑥=𝑁𝑥. Moreover, from the definition of 𝐿, we see that Ker𝐿=ℝ^𝑛, Im𝐿={𝑦∣𝑦∈𝐶_𝑇,∫^𝑇₀𝑦(𝑠)𝑑𝑠=0}. Therefore, 𝐿 is a Fredholm operator with index zero.

Let the projections 𝑃∶𝐶¹_𝑇⟶Ker𝐿,[𝑃𝑥](𝑡)=𝑥(0)=𝑥(𝑇),𝑄∶𝐶_𝑇⟶𝐶_𝑇,𝑡)=1Im𝐿𝑄𝑦(𝑇^𝑇₀𝑦(𝑠)𝑑𝑠, (6) then, Im𝑃=Ker𝐿, Ker𝑄=Im𝐿.

Let 𝐿⁻¹ represent the inverse of 𝐿|_{𝐷(𝐿)∩Ker𝑃},𝐿⁻¹∶Im𝐿→𝐷(𝐿)∩Ker𝑃; then 𝐿⁻¹𝑦(𝑡)=^𝑇₀𝐺(𝑡,𝑠)𝑦(𝑠)𝑑𝑠, (7) where 𝐺()=𝑠()𝑡,𝑠𝑡−𝑇𝑇𝑡(),0⩽𝑠<𝑡⩽𝑇,𝑠−𝑇𝑇,0⩽𝑡⩽𝑠⩽𝑇. (8)

From (5) and (7), it is easily verified that 𝑁 is 𝐿-compact on Ω, where Ω is an arbitrary open bounded subset of 𝐶¹_𝑇.

3. Main Results

3.1. Theoretical Proof

Theorem 2. For all 𝑖∈{1,2,…,𝑛}, assume that the following conditions are satisfied:[𝐶₁]0⩽|𝛼_𝑖|_max|𝜆_𝑖|^1/3𝑇/𝜇₀Δ<2, where 𝜆_𝑖 are the eigenvalues of 𝜷 and Δ=|𝜆_𝑖|^1/3−(𝜔²_0𝑖+|𝛼_𝑖|_max)^1/3.[𝐶₂](i) If 𝜆_𝑖≠0, there is a constant 𝛾>0 such that 𝜆_𝑖𝜔>−^20𝑖+𝛼_𝑖,min𝛾²,𝜆_𝑖𝜆>0,_𝑖𝜔<−^20𝑖+𝛼_𝑖,max𝛾²,𝜆_𝑖<0, (9) where 𝛼_𝑖,min=min_𝑡∈𝐼𝛼_𝑖(𝑡) and 𝛼_𝑖,max=max_𝑡∈𝐼𝛼_𝑖(𝑡). (ii) If 𝜆_𝑖≡0, and 𝜔²_0𝑖≠−(1/𝑇)∫^𝑇₀𝛼_𝑖𝑑𝑡,then, system (3) has at least one nontrivial 𝑇-periodic solution if there exist constants 𝑑_𝑖, such that 𝜆_𝑖max<𝜔²_0𝑖+𝛼_𝑖,min𝑑²_𝑖. (10)

Proof. Let us embed system (3) into one parameter family of the systems as follows: 𝑥̈+𝜖𝜇₀𝑥̇+𝜖𝜔²₀+𝛼(𝑡)𝑥+𝜖𝜷𝑥³₁,𝑥³₂,…,𝑥³_𝑛^𝑇=𝜖𝑒(𝑡),𝜖∈]0,1]. (11)
Since 𝜷 is the symmetrical matrix, there is an orthogonal matrix 𝑈, such that 𝑈𝜷𝑈^𝑇=diag𝜆₁,𝜆₂,…,𝜆_𝑛. (12)
Integrating both sides of (11) from 0 to 𝑇 gives ^𝑇₀𝜆_𝑖𝑥³_𝑖𝑑𝑡⩽^𝑇₀𝑥_𝑖𝜔²_0𝑖+𝛼_𝑖(𝑡)𝑑𝑡+𝑇|𝑒|_max,∀𝑖∈{1,2,…,𝑛}. (13)
Applying integral mean value theorem, there exists a constant 𝜉∈]0,𝑇[, such that 𝜆_𝑖𝑥_𝑖(𝜉)³⩽1𝑇^𝑇₀𝑥_𝑖𝜔²_0𝑖+𝛼_𝑖(𝑡)𝑑𝑡+𝑇|𝑒|_max. (14)
Now, we claim that 𝜆_𝑖^1/3𝑥_𝑖(𝜉)⩽𝜔²_0𝑖+𝛼_𝑖max^1/3𝑥_𝑖max+𝐶, (15) where 𝐶=(𝜔²_0𝑖+|𝛼_𝑖|_max)^1/3+𝑇|𝑒|_max+|𝜆_𝑖|^1/3.
Case 1. If |𝑥_𝑖(𝜉)|⩽1, then, (15) holds clearly.
Case 2. If |𝑥_𝑖(𝜉)|>1, define 𝐸₁=𝑡∶𝑡∈𝐼,𝑥_𝑖>1,𝐸₂=𝑡∶𝑡∈𝐼,𝑥_𝑖⩽1. (16)
By (14) and simple calculation, we obtain 𝜆_𝑖^1/3𝑥_𝑖(𝜉⩽1)𝑇_𝐸1𝑥_𝑖𝜔²_0𝑖+𝛼_𝑖(𝑡)𝑑𝑡^1/3+1𝑇_𝐸2𝑥_𝑖𝜔²_0𝑖+𝛼_𝑖(𝑡)𝑑𝑡^1/3⩽𝜔²_0𝑖+𝛼_𝑖max^1/3𝑥_𝑖max+𝜔²_0𝑖+𝛼_𝑖max^1/3. (17) Thus, it can be easily seen that (15) holds.
According to (15), we have 𝜆_𝑖^1/3𝑥_𝑖(𝑡)⩽𝜔²_0𝑖+𝛼_𝑖max^1/3𝑥_𝑖max+𝜆_𝑖^1/3𝑡_𝑡∗𝑥̇_𝑖(𝑠𝑡)𝑑𝑠+𝐶,𝑡∈^∗,𝑡^∗+𝑇,𝜆_𝑖^1/3𝑥_𝑖(𝑡)=𝜆_𝑖^1/3𝑥_𝑖(⩽𝜔𝑡−𝑇)_20𝑖+𝛼_𝑖max^1/3𝑥_𝑖max+𝜆_𝑖^1/3𝑡∗_𝑡−𝑇𝑥̇_𝑖(𝑠𝑡)𝑑𝑠+𝐶,𝑡∈^∗,𝑡^∗+𝑇. (18)(19)
Combining inequalities (18) and (19) yields 𝜆_𝑖^1/3𝑥_𝑖max⩽𝜔²_0𝑖+𝛼_𝑖max^1/3𝑥_𝑖max+𝜆_𝑖^1/32^𝑇₀𝑥̇_𝑖(𝑡)𝑑𝑡+𝐶. (20)
Therefore, it follows from condition [𝐶₁] that Δ>0. Then, we obtain 𝑥_𝑖max⩽𝜆_𝑖^1/32Δ^𝑇₀𝑥̇_𝑖(𝑡𝐶)𝑑𝑡+Δ. (21)
As 𝑥(𝑡)∈𝐶¹_𝑇, multiplying both sides of the 𝑖th component of (11) by 𝑥̇_𝑖(𝑡) and integrating on the interval 𝐼 lead to 𝜇₀^𝑇₀𝑥̇_𝑖²𝑑𝑡+^𝑇₀𝛼_𝑖𝑥_𝑖𝑥̇_𝑖𝑑𝑡=^𝑇₀𝑒_𝑖𝑥̇_𝑖𝑑𝑡. (22)
Using Hölder’s inequality and (22), we have 𝜇₀^𝑇₀𝑥̇_𝑖²𝑑𝑡⩽𝛼_𝑖max𝑥_𝑖max+𝑒_𝑖max𝑇^1/2×^𝑇₀𝑥̇_𝑖²𝑑𝑡^1/2. (23)
It is noticed that 𝛼_𝑖(𝑡) and 𝑒_𝑖(𝑡) are bounded on the interval 𝐼. From (21), we obtain 𝜇₀^𝑇₀𝑥̇_𝑖²𝛼𝑑𝑡⩽^𝑖max𝜆_𝑖^1/3𝑇2Δ^𝑇₀𝑥̇_𝑖²+𝛼𝑑𝑡_𝑖max𝐶Δ+𝑒_𝑖max𝑇^1/2×^𝑇₀𝑥̇_𝑖²𝑑𝑡^1/2. (24)
According to the condition [𝐶₁], it is easily seen that there exists a constant 𝑀>0, such that 𝑥̇^𝑇_0𝑖²𝑑𝑡⩽𝑀. (25)
Combining (21) and (25), we obtain |𝑥|_max⩽𝜆_𝑖^1/3√𝑛𝑀𝑇+√2Δ𝑛𝐶Δ≜𝑀₁. (26)
For 𝑥(0)=𝑥(𝑇), there exists a 𝑡₀∈]0,𝑇[, such that 𝑥̇(𝑡₀)=0. Then, it follows from (11) that 𝑥̇_𝑖=^𝑡_𝑡0𝑥̈_𝑖(𝑠)𝑑𝑠⩽𝜇₀^𝑇₀𝑥̇_𝑖𝑑𝑡+𝜔²_0𝑖+𝛼_𝑖max𝑥_𝑖max𝑇+𝜆_𝑖𝑥_𝑖³_max𝑇+𝑒_𝑖max𝑇. (27)
Therefore, we have |𝑥̇|_max⩽𝜇₀√𝑛𝑀𝑇+𝜔²_0𝑖+|𝛼|_max√𝑛𝑥_𝑖max𝑇+√𝑛𝜆_𝑖𝑥_𝑖³_max𝑇+√𝑛𝑒_𝑖max𝑇⩽𝜇₀√+𝜔𝑛𝑀𝑇_20𝑖+|𝛼|_max√𝑛𝑇𝜆2Δ^𝑖1/3√+√𝑀𝑇+2𝐶𝑛𝑇8Δ³𝜆_𝑖(𝑀𝑇)^3/2+6𝜆_𝑖^2/3𝜆𝐶𝑀𝑇+12_𝑖^1/3𝐶²(𝑀𝑇)^1/2+8𝐶³+√𝑛𝑒_𝑖max𝑇≜𝑀₂. (28)
Let Ω∶={𝑥∶|𝑥|_max<𝑀₁+1,|𝑥̇|_max<𝑀₂+1}. For ∀𝜖∈]0,1], there is not any solutions of (11) on 𝜕Ω with for all 𝑥∈Ker𝐿∩𝜕Ω. Based on the condition [𝐶₂], there exist the appropriate constants 𝑀₁ and 𝛾, such that the following relationship holds ^𝑇₀[𝑁𝑥](𝑡)𝑑𝑡=−^𝑇₀𝑥₁𝜔²₀₁+𝛼₁(𝑡)+𝜆₁𝑥²₁𝑑𝑡,…,_𝑇0𝑥_𝑛𝜔²_0𝑛+𝛼_𝑛(𝑡)+𝜆_𝑛𝑥²_𝑛𝑑𝑡^𝑇≠(0,0,…,0)^𝑇,for𝑥_𝑖>𝛾,𝑖={1,2,…,𝑛}. (29) 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 𝐽∶Im𝑄→Ker𝐿, 𝐽(𝑥)=𝑥. For all 𝜇^∗∈[0,1], 𝑖=1,2,…,𝑛, we denote 𝐻𝑥,𝜇^∗=𝜇^∗𝑥+1−𝜇^∗𝑇×sgn𝜆₁^𝑇₀𝑥₁𝜔²₀₁+𝛼₁(𝑡)+𝜆₁𝑥²₁𝜆𝑑𝑡,…,sgn_𝑛^𝑇₀𝑥_𝑛𝜔²_0𝑛+𝛼_𝑛(𝑡)+𝜆_𝑛𝑥²_𝑛𝑑𝑡^𝑇,𝜆_𝑖𝐻≠0,𝑥,𝜇^∗=𝜇^∗𝑥+1−𝜇^∗𝑇×sgn^𝑇₀𝜔²₀₁+𝛼₁(𝑡×)𝑑𝑡_𝑇0𝑥₁𝜔²₀₁+𝛼₁(𝑡)𝑑𝑡,…,sgn^𝑇₀𝜔²_0𝑛+𝛼_𝑛(𝑡)𝑑𝑡_𝑇0𝑥_𝑛𝜔²_0𝑛+𝛼_𝑛(𝑡)𝑑𝑡^𝑇,𝜆_𝑖≡0. (30)
By using the condition [𝐶₂] again, the following relationships hold, when: 𝑥∈𝜕Ω∩ℝ^𝑛, 𝑥_𝑖𝐻𝑥_𝑖,𝜇^∗=𝜇^∗𝑥²_𝑖+1−𝜇^∗𝑇sgn𝜆_𝑖×^𝑇₀𝑥²_𝑖𝜔²_0𝑖+𝛼_𝑖(𝑡)+𝜆_𝑖𝑥²_𝑖𝜆𝑑𝑡>0,_𝑖𝑥≠0,_𝑖𝐻𝑥_𝑖,𝜇^∗=𝜇^∗𝑥²_𝑖+1−𝜇^∗𝑇×sgn^𝑇₀𝜔²_0𝑖+𝛼_𝑖(𝑡×)𝑑𝑡_𝑇0𝑥²_𝑖𝜔²_0𝑖+𝛼_𝑖(𝑡𝜆)𝑑𝑡>0,_𝑖≡0. (31) It follows from (31) that 𝐻(𝑥,𝜇^∗) is homotopic and that {}{𝐻()}{𝐻()}deg𝐽𝑄𝑁,Ω∩Ker𝐿,0=deg⋅,0,Ω∩Ker𝐿,0=deg⋅,1,Ω∩Ker𝐿,0≠0. (32) Thus, the last condition of Lemma 1 is also satisfied.

Applying Lemma 1, it can be concluded that the equation 𝐿𝑥=𝑁𝑥 has at least one 𝑇-periodic solution 𝑥(𝑡) on Ω with |𝑥|_∞⩽𝑀₁. Furthermore, it is obvious that the 𝑇-periodic solution 𝑥(𝑡) is nontrivial. Otherwise, there is a constant vector 𝑑={𝑑₁,𝑑₂,…,𝑑_𝑛} satisfying (11), that is, 𝜔²₀+𝛼(𝑡)𝑑+𝜷𝑑³₁,𝑑³₂,…,𝑑³_𝑛^𝑇=𝑒(𝑡). (33)

By simple computation, we obtain 𝜔²_0𝑖+𝛼_𝑖,min⩽𝜆_𝑖max𝑑²_𝑖, (34) which contradicts the condition in Theorem 2. Then, system (3) has at least one non-trivial 𝑇-periodic solution. Therefore, we complete the proof of Theorem 2.

Theorem 3. Assume that 0⩽((|𝜆_𝑖|^1/3(𝜔²_0𝑖+|𝛼_𝑖|_max)^1/2𝑇)/Δ)<2 and 𝜆_𝑖<0 hold for all 𝑖∈{1,2,…,𝑛}, then, system (3) has at least one 𝑇-periodic solution.

Proof. The same proof also works for this theorem. We only need to show that ∫^𝑇₀|𝑥̇_𝑖|²𝑑𝑡 is bounded.
As 𝑥(𝑡)∈𝐶¹_𝑇, multiplying both sides of the 𝑖th component of (11) by 𝑥_𝑖(𝑡) and integrating from 0 to 𝑇 yield −^𝑇₀𝑥̇_𝑖²𝑑𝑡=−𝜖^𝑇₀𝑥²_𝑖𝜔²_0𝑖+𝛼_𝑖(𝑡)+𝜆_𝑖𝑥²_𝑖𝑑𝑡+𝜖^𝑇₀𝑒_𝑖𝑥_𝑖𝑑𝑡. (35)
It can be easily found that Δ>0 when using the condition 0⩽((|𝜆_𝑖|^1/3(𝜔²_0𝑖+|𝛼_𝑖|_max)^1/2𝑇)/Δ)<2. Combining (21) and (35), we obtain 𝑥̇^𝑇_0𝑖²𝑑𝑡⩽𝑥_𝑖²_max𝜔²_0𝑖+𝛼_𝑖max𝑇+𝑥_𝑖max𝑒_𝑖max𝑇⩽𝜆_𝑖^2/3𝑇²𝜔²_0𝑖+𝛼_𝑖max4Δ^2𝑇₀𝑥̇_𝑖²+𝐶𝜔𝑑𝑡_20𝑖+𝛼_𝑖maxΔ²+𝑒_𝑖maxΔ𝜆_𝑖^1/3𝑇×^𝑇₀𝑥̇_𝑖+𝐶𝑑𝑡₂𝑇𝜔²_0𝑖+𝛼_𝑖maxΔ²+𝐶𝑇𝑒_𝑖maxΔ. (36)
Noticing that 0⩽((|𝜆_𝑖|^1/3(𝜔²_0𝑖+|𝛼_𝑖|_max)^1/2𝑇)/Δ)<2, there is a constant 𝑀^∗>0, such that ∫^𝑇₀|𝑥̇_𝑖|²𝑑𝑡⩽𝑀^∗. 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 ∫^𝑇₀𝑒(𝑡)𝑑𝑡≠0, let 𝑒(𝑡)=𝑒(𝑡)−(1/𝑇)∫^𝑇₀𝑒(𝑠)𝑑𝑠. Then, we have ∫^𝑇₀𝑒(𝑡)𝑑𝑡=0. System (11) can be reduced to the following form 𝑥̈+𝜇₀𝑥̇+𝜔²₀+𝛼(𝑡)𝑥+𝜷𝑥³₁,𝑥³₂,…,𝑥³_𝑛^𝑇−1𝑇^𝑇₀𝑒(𝑠)𝑑𝑠=𝑒(𝑡). (37) 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).

Figure 1 

The model of a rectangular thin plate subjected to parametrical excitation.

According to [5, 19], we have the following partial differential governing equations: 𝐷∇⁴𝜕𝑤+𝜌ℎ²𝑤𝜕𝑡²−𝜕²𝑤𝜕𝑥²𝜕²𝜙𝜕𝑦²−𝜕²𝑤𝜕𝑦²𝜕²𝜙𝜕𝑥²𝜕+2²𝑤𝜕𝜕𝑥𝜕𝑦²𝜙𝜕𝑥𝜕𝑦+𝜇𝜕𝑤∇𝜕𝑡=0,₄𝜕𝜙=𝐸ℎ²𝑤𝜕𝑥𝜕𝑦²−𝜕²𝑤𝜕𝑥²𝜕²𝑤𝜕𝑦², (38) where 𝜌 represents the density of the plate, 𝐷=𝐸ℎ³/12(1−𝜈²) is the bending rigidity, 𝐸 is Young’s modulus, 𝜈 is the Poission ratio, 𝜙 is the stress function, and 𝜇 the damping coefficient.

By means of Galerkin’s method, (38) can be reduced to the following dimensionless form: 𝑦̈+𝜇_∗𝑦̇+𝜔²_∗+𝛼_∗(𝑡)𝑦+𝛽_∗𝑦³=0, (39) where 𝜇_∗=121−𝜈²𝜌𝐸𝑎𝑏𝜇𝜋²ℎ²,𝛼_∗(𝑡)=121−𝜈²ℎ²𝑎𝑏𝑝₁𝜋²𝐷cosΩ₁𝜔𝑡,_2∗=9𝑎²+𝑏²²𝑏⁴−𝑏²𝑝₀𝜋²𝐷,𝛽_∗=31−𝜈²ℎ²81𝑎²+𝑏²4𝑎𝑏³. (40)

Obviously, system (11) can be reduced to the system (39), provided that 𝑛=1, 𝜇₀=𝜇_∗, 𝜔₀=𝜔_∗, 𝛼=𝛼_∗, 𝛽=𝛽_∗, and 𝑒(𝑡)≡0.

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 (0.03,0.77), let 𝜇_∗=0.02 and Ω₁=2 and choose appropriate 𝑝₀,𝑝₁ such that 𝜔_∗=1.

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

For 𝛼_∗(𝑡)=2cos2𝑡, a centrosymmetric periodic solution in the (𝑦,𝑦)̇ plane is shown in Figure 2, keeping 𝛽_∗ 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.

Figure 2 

Phase portrait for (𝑦,𝑦)̇.

Figure 3 

Time history curve (𝑡,𝑦).

Figure 4 

Time history curve (𝑡,𝑦)̇.

If we set 𝛼_∗(𝑡)=7cos2𝑡 and 𝛽_∗=10, a group of dynamics behavior of the thin plate system in the planes (𝑦,𝑦)̇, (𝑡,𝑦), and (𝑡,𝑦)̇ 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.

Figure 5 

Phase portrait (𝑦,𝑦)̇.

Figure 6 

Time history curve (𝑡,𝑦).

Figure 7 

Time history curve (𝑡,𝑦)̇.

For 𝛼_∗(𝑡)=13.05cos2𝑡, when 𝛽_∗ is gradually increased to 20, a set of dynamics characteristic of the thin plate system is addressed in Figures 8, 9 and 10, respectively.

Figure 8 

Phase portrait (𝑦,𝑦)̇.

Figure 9 

Time history curve (𝑡,𝑦).

Figure 10 

Time history curve (𝑡,𝑦)̇.

In Figure 8, phase portrait in the (𝑦,𝑦)̇ 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 𝑛-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 𝜇_∗, 𝜔_∗, 𝛼_∗, and 𝛽_∗ into the condition [𝐶₁] of Theorem 2, and combining with the phase diagrams and time history curves displayed above, one can see that there exist a set of 𝑇-periodic solutions at least for system (39) with 𝑇<8.3039𝑒−003, 2.1393𝑒−004, and 3.4035𝑒−004 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 |𝛽_∗| and |𝛼_∗|_max 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.