ARTICLE TEMPLATE Kernel Sections and global dynamics of nonautonomous Euler-Bernoulli beam equations Huatao Chen a , Juan Luis Garc´ ıa Guirao b , Jingfei Jiang a , Dengqing Cao c , Xiaoming Fan d a Division of Dynamics and Control, School of Mathematics and Statistics, Shandong University of Technology, 255000, Zibo, China b Departamento de Matem´ atica Aplicaday Estad´ ıstica, Universidad Polit´ ecnica de Cartagena, 30203-Cartagena, Spain c Division of Dynamics and Control, School of Astronautics, Harbin Institute of Technology, 150001, Harbin, China d School of Mathematics, Southwest Jiaotong University, 610031, Chengdu, China ARTICLE HISTORY Compiled April 5, 2019 ABSTRACT This paper concerns with dynamical behavior for nonautonomous Euler-Bernoulli beam equations with either weakly damping or strongly damping. Issues relevant to existence and Hausdorff dimension estimation of Kernel sections are investigated. It is shown that there exist Kernel sections for the beams, in the case of strongly damping, the techniques rely on splitting method, when the damping is weakly, the proof depends on the stabilization estimations of the system. Moreover, the Hausdorff dimension of Kernel sections are proved to be finite. Eventually, the global dynamics of the beams are studied by numerical simulation on the Kernel Sections and Kernel. KEYWORDS Nonautonomous Euler-Bernoulli beam equations; Kernel sections; Kernel; Hausdorff dimension; Global dynamics 1. Introduction Let D = (0,L) be a bounded domain in R, the following dimensionless nonautonomous Euler-Bernoulli beam equation was proposed as a model for the transversal deflection of an extensible beam of length L, having one end (x = 0) clamped support and the other end (x = L) simply support, under an time-varying longitudinal force Q = f 1 (t) and external excitation P = f (t, x), see Figure 1, u tt + α(u t )+Δ 2 u + f 1 (t) - β Z L 0 |∇u| 2 dx Δu = f (t, x), Email address: [email protected](HC);[email protected](JLG);[email protected](JJ) [email protected](DC);[email protected](XF)
Transcript
ARTICLE TEMPLATE
Kernel Sections and global dynamics of nonautonomous
Euler-Bernoulli beam equations
Huatao Chena , Juan Luis Garcıa Guiraob, Jingfei Jiang a, Dengqing Caoc, XiaomingFand
a Division of Dynamics and Control, School of Mathematics and Statistics,Shandong University of Technology, 255000, Zibo, China
b Departamento de Matematica Aplicaday Estadıstica, Universidad Politecnica de Cartagena,30203-Cartagena, Spain
c Division of Dynamics and Control, School of Astronautics, Harbin Institute of Technology,150001, Harbin, China
d School of Mathematics, Southwest Jiaotong University, 610031, Chengdu, China
ARTICLE HISTORY
Compiled April 5, 2019
ABSTRACTThis paper concerns with dynamical behavior for nonautonomous Euler-Bernoullibeam equations with either weakly damping or strongly damping. Issues relevant toexistence and Hausdorff dimension estimation of Kernel sections are investigated.It is shown that there exist Kernel sections for the beams, in the case of stronglydamping, the techniques rely on splitting method, when the damping is weakly,the proof depends on the stabilization estimations of the system. Moreover, theHausdorff dimension of Kernel sections are proved to be finite. Eventually, the globaldynamics of the beams are studied by numerical simulation on the Kernel Sectionsand Kernel.
KEYWORDSNonautonomous Euler-Bernoulli beam equations; Kernel sections; Kernel;Hausdorff dimension; Global dynamics
1. Introduction
Let D = (0, L) be a bounded domain in R, the following dimensionless nonautonomousEuler-Bernoulli beam equation was proposed as a model for the transversal deflectionof an extensible beam of length L, having one end (x = 0) clamped support and theother end (x = L) simply support, under an time-varying longitudinal force Q = f1(t)and external excitation P = f(t, x), see Figure 1,
here α(ut) is the damping. For the physical background, the literature [37] may be
Q
P
o x
y
Figure 1. The model for vibration of Euler-Bernoulli beam
consulted.In some circumstances, the nonautonomous Euler-Bernoulli beam equations can
generate nonautonomous infinite dynamical systems (NDS) which can be describedby Process introduced in [18]. The long-time behavior which can be captured byglobal attractor is a fascinating problem in the investigations on infinite dynamicalsystems [8, 9, 38]. There are several types of global attractor for NDS [14, 17, 36], oneof which is called Kernel and Kernel sections proposed by Chepyzhov and Visik [14],which are natural generalizations of maximal invariant set for NDS. In many cases,the Kernel Sections may be very complicated, from the analytical standpoint, veryfew tools can be employed to describe the structure of Kernel Sections. Essentially,the Hausdorff dimension is one of the few pieces of information which is associatedwith these complicated sets, the approach to estimate the Hausdorff dimension wasalso due to Chepyzhov and Visik [14]. Many researchers have studied the NDS asso-ciated with mathematical physics problems by these methods, various and wonderfulphenomenon about long-time behavior are obtained, such as those in [12, 22] and thereferences therein. There exists another definition of attractors associated with NDS,be referred to as pullback attractors [17, 36], the method to derive dimension estima-tion of pullback attractors was proposed by Caraballo et.al [11]. It is very useful toinvestigate the long time dynamical behavior of NDS in which there exists unboundedtime dependent quantity as time tends to infinite. Whence physically relevant, thissituation hardly occurs in nonlinear beam vibration. Thus, we do not choose pullbackattractors as the tool to study the long-time behavior for the vibration of beam.
Euler-Bernoulli beam is very famous in the study of nonlinear elastic dynamic-s. There are abundant literatures related to it and the brief list given below is by nomeans exhaustive. For practical aspect, we refer to [37, 25]. The investigations focus onexistence and uniqueness of solution can refer to [4, 10, 35] and the references therein.Ball [5] have studied the stability for an extensible Euler-Bernoulli beam. The resultson boundary control problems dues to Morgu [33] and Guo et.al [24]. With respect tothe long time behavior of Euler-Bernoulli beam, there also exist many achievements.Such as, Eden and Milani [21] addressed exponential attractors for autonomous Euler-Bernoulli beam. The study on attractors for autonomous Euler-Bernoulli beam withnonlinear damping and nonlinear source can be found in Ref [30], Ref [41] and thereferences therein. Ref [27] was dedicated to investigate the global attractor for anextensible beam equation with localized nonlinear damping and linear memory. Kang[28] have studied the uniform attractors for nonautonomous Euler-Bernoulli beam ex-
2
cited by solely time-vary and independent of displacement external loadings. To ourbest knowledge, there exist few results about the Kernel Sections and their Hausdorffdimension for Euler-Bernoulli beam equations subjected to longitudinal time-varyingloading. In addition, besides the aforementioned investigations on the long time au-tonomous or nonautonomous Euler-Bernoulli beam, there always exists a nonlinearsource which arises in the control theory for beams, from a practical standpoint, thisterm is seldom exists in the nonlinear oscillations of beams. Thus, such nonlinear termis not taken into account in Euler-Bernoulli beam equations concerned in this paper.
Since the elastic dynamics problem mostly described by partial differential equa-tion (PDE), from the view of engineering applications and numerical computation,the investigations on dynamics of these problems are accomplished by studying theassociated molal equations which are also known as equations of motion. The studyon dynamics of nonlinear oscillations arose in engineering comprise two aspects: glob-al dynamics and local dynamics. There exist plentiful results on local dynamics forEuler-Bernoulli beam, such as, characteristics of Non-linear vibration [6], dynamicalresponse [7], chaotic dynamics [3], to name but a few. The global dynamics of nonlinearsystems which can reveal more dynamical informations than local dynamics are im-portant in engineering applications. According to the Melnikov method, Wiggins [39]have studied the global dynamics of four-dimensional perturbed Hamiltonian systems.The global dynamics for parametrically forced mechanical systems were addressed byFeng and Wiggins [23]. Zhang [42] studied the global bifurcation and chaos for para-metrically excited Euler-Bernoulli beam. Due to the lack of analytical tools, numericalmethod is the predominant approach to study the global dynamics of nonlinear sys-tems, Cell to cell mapping method [26] and its derivatives are the powerful tools.
As we all known, once global attractor exists, the important informations on its longterm global behavior are captured by the global attractor, which indicates that theinvestigation on global dynamics can be realized by analysing the structure of globalattractors. Subdivision algorithm proposed by Dellnitz and Hohmann [20] can be usedto efficiently compute global attractors for autonomous dynamical system. Later on,in order to study the structure of random attractors, by employing Pullback technique[1], Keller and Ochs [29] extended this algorithm to random case, it is mentioned herethat this algorithm can be used to compute the Kernel Sections for NDS.
In order to formulate the model in the abstract form, some spaces are intro-duced here. Hs(D) is the usual Sobolev space of order s, s ∈ R, Hs
0 is the closureof C∞0 (D) in Hs(D) and ‖u‖ ≡ ‖u‖L2(D), ‖u‖s ≡ ‖u‖Hs
0 (D), (u, v) ≡ (u, v)L2(D),
((u, v))s = (u, v)Hs0 (D). H
2l (D) denotes space of H2(D) functions subjected to bound-
ary condition (1), define ‖u‖l,2 = ‖u‖2, in term of Section 1.4, Charpter II in Temam[38], it can be verified that the space H2
l (D) equipped with norm ‖u‖l,2 is a Banachspace. Furthermore, set (·, ·)l,2 be the inner product induced by norm ‖u‖l,2, thenH2l (D) is a Hilbert space. Let A := ∆2 : (H4
⋂H2l )(D) ⊂ L2(D)→ L2(D), then A is
self-adjoint, positive, unbounded linear operators and A−1 ∈ L (L2(D)) is compact.So, their eigenvalues {Λi}i∈N satisfy 0 < Λ1 ≤ Λ2 ≤ · · · → ∞ and the correspondingeigenvalues {ei}∞i=1 forms an orthonormal basis in L2(D). So, we can interpret thepower of As, s ∈ R by the method in [38] p54-55. Thus the abstract nonautonomousEuler-Bernoulli beam equation with weakly damping is as follow
utt + αut +Au+[β ‖∇u‖2 − f1(t)
](−∆)u = f(t, x), (2)
3
the following form is the strongly damping beam
utt + αA1
4ut +Au+[β ‖∇u‖2 − f1(t)
](−∆)u = f(t, x). (3)
Actually, the term αA1
4ut which can be used to signify the fractional damping withrespect to space variable [32] indeed exists in the engineering application.
This paper address the dynamical behavior for system (2) and system (3). Firstly,the existence of Kernel Sections for the two systems are derived, moreover, the radiusof Kernel Sections also given. Secondly, following the manner proposed in [14], weshow that the Hausdorff dimensions of Kernel Sections is finite in some cases. Finally,based on the above theoretical results, the global dynamics about the two beams arestudied numerically.
The rest of this paper is organized as follows. The notations and main results aregiven in Section 2. Section 3 is devoted to list some Preliminaries and prove certainLemmas which are crucial to derive the main results. In section 4, the proofs for mainresults are provided.
2. Notations and Main results
2.1. Notations
Invoking the eigenvalues of A, the Hilbert space D(As), s ∈ R with norm ‖ · ‖D(As)
and (·, ·)D(As) can be defined by the mechanism in [38] p55, especially, inner prod-
uct D(A1
2 ) = H2l (D). Moreover, for all s1, s2 ∈ R, s1 > s2, D(As1) can be compact
imbedding in D(As2) and the following holds
‖u‖s1 ≥ Λs1−s2
2
1 ‖u‖s2 , ∀u ∈ D(As1). (4)
Let E1+s = D(A1
2+s)×D(As), 0 ≤ s ≤ 1
2 equipped with Graph norm and the inducedinner product, then they are all Hilbert spaces. It must be kept in mind that although‖A
1
2u‖ = ‖(−∆)u‖, ‖A1
4u‖ = ‖∇u‖, ∀u ∈ D(A) and A1
2 6= −∆, A1
4 6= ∇.Essentially, the operator A1 := −∆ with boundary condition (1) is self-adjoint,
positive, unbounded linear operators from L2(D) to L2(D) and the inverse operatorsare compact. Thus, the power of As1, s ∈ R can be defined, moreover, ‖As1u‖ ≤ ‖A
s
2u‖,∀s ∈ [0, 1], u ∈ D(A).
From the practical view, suppose
f1(t) ∈ C1b(R,R), f(t, x) ∈ C1
b(R, H2l (D)). (5)
although by no means always, it will be convenient to reduce the equation (2) and (3)to an evolution equation of the first order in time in the following manner.
Let v = ut and
Tε : [u, v]T → [u, v + εu] ≡ [u, v]T , (6)
4
where [·, ·]T means the transpose of [·, ·], ε is a positive constant, moreover, the trans-formation Tε is invertible. Then equation (2) can be casted in the following form
dU
dt= A1U + F1(t,U), (7)
where U = [U1, U2]T = [u, v]T ,
A1 =
(−εI, I
−A+ ε(α− ε), −(α− ε)I
),
F1(t,U) =
(F11(t,U)F12(t,U)
)=
(0
−N(t, U1) + f(t, x)
),
N(t, u) =[β ‖∇u‖2 − f1(t)
](−∆)u,
moreover
Nu(t, u) =[β ‖∇u‖2 − f1(t)
](−∆) ·+2β(∇u,∇·)(−∆)u.
Then system (2) and system (7) are equivalent.In order to achieve the stabilization estimation of the solution for system (7) which
is significant to verify the uniformly asymptotical compact of the system, the followingequation should be considered.
dU(1) −U(2)
dt= A1
(U(1) −U(2)
)+ F1(t,U(1))− F1(t,U(2)), (8)
where U(1) = [U(1)1 , U
(1)2 ] = [u1, v1] and U(2) = [U
(2)1 , U
(2)2 ] = [u2, v2] are two solutions
of system (7).Analogously, implementing Tε, the equation (3) is transformed to the following form
dU
dt= A2(U) + F2(t,U) (9)
where U = [U1, U2]T = [u, v]T ,
A2 =
(−εI, I
−A+ ε(αA1
4 − ε)I, −(αA1
4 − ε)I
),
F2(t,U) =
(F21(t,U)F22(t,U)
)=
(F11(t,U)F12(t,U)
),
system (9) is equivalent to system (3).In the treatment to derive the uniformly qusidifferentiable for system (7) and system
(9) in Kernel, the following two variation equations are needed.The variation equation corresponding to (7) is
dU
dt= A1U + F1(U)U, (10)
5
here U = [U1, U2]T = [U , Ut + εU′]T ,
F1(U) =
(F11(U), F13(U)
F12(U), F14(U)
)=
(0, 0
−Nu(t, U1), 0
).
The variation equation associated with (9) is
dU
dt= A2U + F2(U)U, (11)
where U = [U1, U2]T = [U , Ut + εU′]T ,
F2(U) =
(F21(U), F23(U)
F22(U), F24(U)
)=
(F11(U), F13(U)
F12(U), F14(U)
)= F1(U).
Since the equivalent of system (2) and system (7), system (3) and system (9), it isenough to investigate the system (7) and system (9).
The following turns to give some basic theory of NDS. Let (X, d) be complete metricspace and {U(t, τ)}t≥τ , t, τ ∈ R be a family mappings satisfying
(i) U(τ, τ) = id,(ii) U(t, τ) = U(t, s) ◦ U(s, τ),(iii) u 7→ U(t, τ)u is continuous.
Then {U(t, τ)}t≥s, t, s ∈ R is called a Process which was introduced by Dafermos [18].For the sake of brevity, {U(t, τ)} is used to denote {U(t, τ)}t≥s,t, s ∈ R.
A function u(s), s ∈ R is said to be a complete trajactory of Process {U(t, τ)}, if
U(t, τ)u(τ) = u(t), t ≥ τ, t, τ ∈ R. (12)
A complete trajactory u(s), s ∈ R of a Process {U(t, τ)} is bounded if the set{u(s)|s ∈ R} is bounded in the norm of X. For more detail about Process and itstrajactory, the reader can refer to [18].
The following are definitions of Kernel and Kernel Sections and their related propo-sition proposed by Chepyzhov and Visik in [14].
Definition 2.1. The Kernel K of a Process {U(t, τ)} consists of all bounded completetrajactory of {U(t, τ)}:
K = {u(·)|u(·) satisfies (12) and ‖u(s)‖X ≤ Cu,∀s ∈ R}
Definition 2.2. The Kernel section K(s) ⊂ E of a Kernel K at time s ∈ R is definedby
K(s) = {u(s)|u(·) ∈ K}. (13)
Proposition 2.3. Let K be the kernel of a Process {U(t, τ)}, Then
U(t, τ)K(τ) = K(t), t ≥ τ, t, τ ∈ R. (14)
Moreover, according to Schmalfuess [36], if the X is connected, then the Kernel Sec-tions are also connected.
6
The next proposition is very important to study global dynamics by the structureof Kernel Sections.
Proposition 2.4. If X is a convex separable space, then the Kernel SectionsK (τ), τ ∈ R can be decomposed into a Basic Kernel Sections BK (τ), τ ∈ R and aremaind C K (τ), τ ∈ R
K (τ) = BK (τ)⋃
C K (τ), τ ∈ R,
such that basin of BK (τ) is prevalent in X, but
basin C K (τ)\basin BK (τ), τ ∈ R,
is shy.
The definition of shy and prevalent can be refer to Binir [9]. For any τ ∈ R, by thepullback technique [1], the proposition is immediate derived from the proof of Theorem5.2 in Birnir [9].
2.2. Main results
This subsection is intended to list the main results. Let
ε0 =Λ1α
2Λ1 + α2, 0 < ε ≤ ε0, (15)
the following theorem focuses on the existence and uniqueness of the solution forsystem (7).
Theorem 2.5. For any T ∈ R+ and initial value t = τ : U = Uτ ∈ E1, system (7)possesses unique (mild) solution U(t, τ) ∈ C([τ, τ + T ], E1), t ∈ [τ, τ + T ] with thefollowing form
U(t, τ) = eA1(t−τ)Uτ +
∫ t
τeA1(t−s)F1(U)(τ)dτ.
Theorem 2.5 indicates that system (7) can induce a NDS described by Process, rep-resented by {Sw(t, τ), t ≥ τ, τ ∈ R}, in short, Sw(t, τ), then Sw(t, τ) := T−1
ε Sw(t, τ)Tεis the Process generated by system (2).
Similarly, let
ε00 = min
{Λ
1
2
1 ,Λ
3
4
1α
8(Λ1
2
1 + α2)
}, 0 < ε ≤ ε0, (16)
In light of the next theorem, system (9) and system (3) can generate NDS expressedby Process, denoted by {Ss(t, τ)} and Sw(t, τ) correspondingly, moreover, Ss(t, τ) =T−1ε Ss(t, τ)Tε.
Theorem 2.6. For any τ ∈ R, T ∈ R+ and initial value Uτ ∈ E1, system (9) possess-es unique (mild) solution U(t, τ) ∈ C([τ, τ + T ], E1), t ∈ [τ, τ + T ] with the following
7
form
U(t, τ) = eA2(t−τ)Uτ +
∫ t
τeA2(t−s)F2(U)(s)ds.
Let
M0(t) = M0(f1, f) =β
3ε
∣∣∣∣f ′1(t) + 2εf1(t)
2β
∣∣∣∣2 +‖f(t, x)‖2
α− ε, (17)
by (5), we have there exist a positive constant M0 which is independent of t, τ , suchthat ∫ t
τe−ε(t−s)M0(s)ds ≤M0
∫ t
τe−ε(t−s)ds, t ≥ τ, t, τ ∈ R. (18)
Set
r0 =
√δ0 +
M0
ε, (19)
where ε satisfies (15), δ0 is an any given given positive constant.The following Theorem concerns with the existence of Kernel Sections for {Sw(t, τ)}
and the estimation for radius of Kernel Sections which will be applied to estimate theHausforff dimension.
Theorem 2.7. Process {Sw(t, τ)} possesses Kernel K and Kernel SectionsK (τ), ∀τ ∈ R in E1, which satisfies
K (τ) ⊂ B0(0, r0), ∀τ ∈ R,
here B0(0, r0) denotes the open ball centered at the origin with radius is r0 whichsatisfies (19). Merge with (6), the Process Sw(t, τ) also possesses Kernel T−1
ε K andKernel Sections T−1
ε K (τ), ∀τ ∈ R.
Let
M1(t) = M1(f1, f) =β
3ε
∣∣∣∣f ′1(t) + 2εf1(t)
2β
∣∣∣∣2 +‖f(t, x)‖2
ε(Λ1
4
1α− ε). (20)
Merging with (5), we have there exist a positive constant M0 which is independent oft, τ , such that∫ t
τe−ε(t−s)M1(s)ds ≤M1
∫ t
τe−ε(t−s)ds, t ≥ τ, t, τ ∈ R. (21)
Set
r1 =
√δ0 +
M1
ε, (22)
8
here ε satisfies (16), δ0 is an any given given positive constant.It follows from the next Theorem that there exist Kernel and Kernel Sections for the
Process {Ss(t, τ)}. Moreover, the radius estimation of Kernel Section is also derived.
Theorem 2.8. Process {Ss(t, τ)} possesses Kernel K and Kernel SectionsK (τ), ∀τ ∈ R in E1, which satisfies
K (τ) ⊂ B0(0, r1), ∀τ ∈ R,
where B0(0, r1) denotes the open ball centered at the origin with radius is r1 whichsatisfies (22). Along with (6), the Process Ss(t, τ) also possesses Kernel T−1
ε K andKernel Sections T−1
ε K (τ), ∀τ ∈ R.
When no ambiguity is possible, the Kernel and Kernel Sections for Process {Sw(t, τ)}and {Ss(t, τ)} are denoted by same symbols, it must be kept in mind that they arenot the same. On the other hand, the forms of the Theorem 2.7 and Theorem 2.8are very similar, nevertheless, the proofs of them are different. Roughly speaking,the approach to obtain the existence of Kernel Section for {Sw(t, τ)} and {Ss(t, τ)}comprises two ingredients: the existence of uniform attracting set and verifying theuniformly asymptotical compact of system. There hardly exists significant differencebetween the methods to attain the existence of uniformly attracting set for the bothsystems. Since the presence of strongly damping, by using the splitting method [38]which is an classical approach in the investigation on NDS, the uniformly asymptoticalcompact of {Ss(t, τ)} can be verified, however, this method is invalid to {Sw(t, τ)}. Bythe techniques rely on the stabilization estimations [15], the uniformly asymptoticalcompactness of {Sw(t, τ)} can be achieved.
Let dH(K(τ)) be the Hausdorff dimension of K (τ),τ ∈ R. In light of the approachin [14], the coming results pertinent to the Hausdorff dimensions estimation of KernelSections for {Sw(t, τ)} and {Sw(t, τ)} can be accomplished.
Suppose r0 given in Theorem 2.7 satisfies
r20 <
ε(α− ε)4β
, (23)
then the next theorem holds.
Theorem 2.9. Let
d = min
{m ∈ Z+
∣∣∣∣∣ 1
m
m∑i=1
Λ− 1
2
i <Λ
1
2
1 k0
4β2r40
(ε
2− 2βr0
k0
)}, (24)
then
dH(K(τ)) < d, ∀τ ∈ R.
Moreover, if (23) does not hold, then finite Hausdorff dimensions of Kernel Sectionsfor {Sw(t, τ)} can not be achieved.
As for the Kernel Sections of {Ss(t, τ)}, the following holds.
9
Theorem 2.10. Let
d = min
{m ∈ Z+
∣∣∣∣∣ 1
m
(2βr2
1
α
m∑i=1
Λ− 1
4
i +2β2r4
1
Λ1
2
1
(Λ
1
4
1α− ε) m∑i=1
Λ− 1
2
i
<ε
2
}, (25)
then the following holds
dH(K (τ)) < d, ∀τ ∈ R.
If
2βr21
αΛ− 1
4
1 +2β2r4
1
Λ1(Λ1
4
1α− ε)≤ ε
2,
then dH(K (τ)) = 0, ∀τ ∈ R, it means that the system possesses a global stablecomplete trajectory.
The rest of this subsection is dedicated to numerically study the global dynamicsof the beam. It can be noticed that, when the Hausdorff dimension estimation of theKernel Sections is finite, the qualitative results provided by Theorem 2.7 and Theorem2.9, Theorem 2.8 and Theorem 2.10 are similar. Thus, the investigation is confined tothe beam with strongly damping. In term of the inertial manifold with delay [19] andnonlinear Galerkin method (NGM) [31], the model equations associated with system(3) can be achieved (See (A1) in Appendix A).
Let L = 4.5, k1 = 2, k2 = 2, invoking COMSOL [34], the eigenvalues {λ}4i=1 andeigenvectors {wi}4i=1 of A can be got, furthermore, the integration with respect tospace variable in (A2) in Appendix A can be attained by COMSOL with Matlab [34],then we catch the model equations, the solution of which can be get numerically.
where γ1, γ2, ω, ω0 are undetermined parameters, the motion of the position x = 2.2of beam is chosen to represent the dynamics of the beam.
Case I: fixed ω = 1, ω0 = 3, γ2 = 0.5.Case I(1): let γ1 = 1, Figure 2 denotes that the Basic Kernel Sections for the
system possesses only one point, which indicates that there solely exists one globalstability complete trajectory of system (3). Moreover, the Kernel Sections and BasicKernel Sections are coincident in this situation and the complete trajectory representthe periodic motion of the system which can be derived by the Figure 3.
Case I(2): let γ = 1.1, Basic Kernel Sections(see Figure 4) and Basic Kernel(seeFigure 5) assert the system possesses two complete trajectories which are local stability,in addition, they are all periodic motion. Figure 6 and Figure 7 are the sketches for theKernel Section and Kernel of system (3) from which a little information on dynamicsof the system can be catch excepted the radius of the Kernel. Therefore, the numericalresults of Kernel do not be listed in the in the sequel.
Case I(3): let γ = 4, there exist two complete trajectories for the system, which canbe verified by the Basic Kernel Sections (see Figure 8). The Kernel Sections describedby Figure 9 validates these complete trajectories are local stability. However, they aredouble period motion rather than period motion.
10
−0.2 −0.18 −0.16 −0.14 −0.12 −0.1
0.04
0.05
0.06
0.07
0.08
0.09
0.1
u
u
Figure 2. Basic Kernel Section in Case I(1)
−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
u
u
Figure 3. Basic Kernel in Case I(1)
Case I(4): let γ = 7.5, the result of Basic Kernel Sections (see Figure 10) whichis composed of two parts indicates that the systems do not possess isolated completetrajectories. The right component of Basic Kernel (see Figure 11) seems like a selfsimilarity structure, which along with the consequence of Basic Kernel reveals thatthe dynamics of system (3) become complex.
As indicated above, it is obtained that with the change of γ1 form 1 to 7.5, thedynamics of strongly damping Euler-Bernoulli beams can undergo the global stability,global bifurcation and complex dynamical behavior. Moreover, the local bifurcationalso appears in this process.
Case II: fixed γ2 = 0.5, γ1 = 1, ω = 1.Case II(1): let ω0 = 3, it is the same as Case I(1).Case II(2): let ω0 = 1.5, the fact that there exists uniqueness complete trajectory
of system (3)can be derived by the numerical result of Basic Kernel Sections (see Figure12). Furthermore, invoking the Basic Kernel (see Figure 13), we have this completetrajectory signifies the periodic motion of the system.
Case II(3): let ω0 = 1, the assertion that system possesses uniqueness period-doubling complete trajectory can be demonstrated by the Figure 14 and Figure 15.
Case II(4): let ω0 = 0.5, the Basic Kernel Sections (see Figure 16) denote thereexists none of isolated complete trajectory for the system. The complex dynamicalbehaviors of the system appear.
From the aforementioned numerical results, the following assertion can be derived.
11
−0.4 −0.3 −0.2 −0.1 0 0.10.02
0.04
0.06
0.08
0.1
u
u
Figure 4. Basic Kernel Section in Case I(2)
−0.5 0 0.5−0.3
−0.2
−0.1
0
0.1
0.2
u
u
Figure 5. Basic Kernel in Case I(2)
Let ω0 vary from 3 to 0.5, the dynamics of strongly damping Euler-Bernoulli beamundergoes global stability to complex behaviors. In additive, the local bifurcation alsooccurs in this process.
3. Preliminaries and main lemmas
3.1. Preliminaries
To begin with, the criteria for the existence of Kernel and Kernel sections of aProcess is introduced.
A set P ⊂ X is said to be a uniformly attracting set of Process {U(t, τ)}, if for anybounded set B ⊂ X,
limT→∞
supτ∈R
dist(U(T + τ, τ)B,P ) = 0,
where dist denotes the Hausdorff semidistance
dist(A,B) = supx∈A
infy∈B
d(x, y), A,B ⊂ X.
12
−0.1 0 0.1 0.2 0.3 0.40.045
0.05
0.055
0.06
0.065
0.07
0.075
u
u
Figure 6. Kernel Section in Case I(2)
Figure 7. Kernel in Case I(2)
The Process is uniformly asymptotically compact, if it possesses a compact uni-formly attracting set.
The next theorem contains the conditions under which the Process has Kernel andKernel sections
Theorem 3.1. [14] Let {U(t, τ)} be a uniformly asymptotically compact Process act-ing on E, with a compact uniformly attracting set P ⊂ E. Each mapping U(t, τ) :E → E is continuous. Then the Kernel of Process U(t, τ) is non-empty, the Kernelsections K(τ) is compact and given by
K(τ) =⋂s>0
⋃T>s
U(τ, τ − T )P . (26)
Theorem 3.1 can be applied to study the existence of Kernel Sections of system (9)directly, nevertheless, it is invalid to the study of system (7).
In order to investigate the Kernel Sections of system (7), we need stabilizationestimation of the system to verify the uniformly asymptotically compactness. Thecoming propositions of Kuratowski’s α-measure of non-compactness is significant toattained the stabilization estimation of the system (7).
13
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.5
0
0.5
u
u
Figure 8. Basic Kernel Section in Case I(3)
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
−2.2 −2 −1.8 −1.6
−0.5
0
0.5
1.6 1.8 2 2.2
−0.5
0
0.5
u
u
Figure 9. Basic Kernel Section in Case I(3)
Proposition 3.2. [15] Suppose B ⊂ X is any bounded set, let
α(B) = inf {d : B has a finite cover of diameter < d} ,
then α(B) is known as the Kuratowski’s α-measure of non-compactness of B, in short,α− measure of B, which have the following properties.
(i) α(B) = 0 if and only if B is precompact.(ii) α(A ∪B) ≤ max {α(A), α(B)}.(iii) α(A+B) ≤ α(A) + α(B).(iv) α(coB) = α(B), where α(coB) is the closed convex hull of B.(v) If B1 ⊃ B2 ⊃ B3 ⊃ · · · are nonempty closed sets in X such that α(Bn)→ 0 as
n→∞, then ∩n≥1Bn is nonempty and compact.
The approach which is intended to obtain the Hausdorff dimension of Kernel Sec-tions of system (7) and system (9) was proposed in Theorem 4.1, Section 4 of Ref[14].
3.2. Main lemmas
This subsection passes to accomplish some lemmas which are important to prove mainresults.
The next two Lemmas are frequently used in the main proofs.
14
−3 −2 −1 0 1 2 3−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
−2.8 −2.7 −2.6
1.3
1.4
1.5
1.6
2.7 2.75 2.8
−1.6
−1.55
−1.5
−1.45
u
u
Figure 10. Basic Kernel Section in Case I(4)
Figure 11. Basic Kernel in Case I(4)
Lemma 3.3. For any U = [U1, U2]T ∈ E1, the following holds
Proof. Similar to the proof of Lemma 3.5 and Lemma 3.6, the results can be got,therefore, omitted here.
Let M > 0 be a given constant, for any T > 0, τ ∈ R and ∀U,V ∈ E1, ‖U‖E1≤M ,
‖V‖E1≤M , t ∈ [τ, τ + T ], then
‖N(t, U1)−N(t, V1)‖
=∥∥∥(β ‖∇U1‖2 − f1(t)
)((−∆U1 − (−∆)V1)
+(β ‖∇U1‖2 − β ‖∇V1‖2
)(−∆)V1
∥∥∥≤
∣∣∣β ‖∇U1‖2 − f1(t)∣∣∣ ‖(−∆)U1 − (−∆)V1‖
+β (‖∇U1‖+ β ‖∇V1‖) ‖(−∆)V1‖×‖(−∆)U1 − (−∆)V1‖
≤ C(T, τ,M)‖U−V‖E1(37)
By (37), there exists a constant C1(T, τ,M) such that
‖F1(t,U)− F1(t,V)‖E1≤ C1(T, τ,M)‖U−V‖E1
, (38)
which gives that F1(t, U) : E1 → E1 satisfies Lipschitz condition.The following estimates for the nonlinear part of Euler-Bernoulli beams play mo-
mentous role in attaining the existence of uniformly attracting sets.([β ‖∇u‖2 − f1(t)
](−∆)u, ut
)=
1
4β
d
dt
[β ‖∇u‖2 − f1(t)
]2
+f′
1(t)
2β
[β ‖∇u‖2 − f1(t)
](39)
and ([β ‖∇u‖2 − f1(t)
](−∆)u, εu
)=
ε
β
[β ‖∇u‖2 − f1(t)
]2
+εf1(t)
β
[β ‖∇u‖2 − f1(t)
]. (40)
Combining with(39) and (40) yields([β ‖∇u‖2 − f1(t)
](−∆)u, ut + εu
)=
1
4β
d
dt
[β ‖∇u‖2 − f1(t)
]2
+ε
β
[β ‖∇u‖2 − f1(t)
]2
+
(f′
1(t) + 2εf1(t)
2β
)[β ‖∇u‖2 − f1(t)
].
20
On the other hand,(f′
1(t) + 2εf1(t)
2β
)[β ‖∇u‖2 − f1(t)
]≤
∣∣∣∣f ′1(t) + 2εf1(t)
2β
∣∣∣∣ [β ‖∇u‖2 − f1(t)]
≤ 1
2ε1
∣∣∣∣f ′1(t) + 2εf1(t)
2β
∣∣∣∣2+ε1
2
[β ‖∇u‖2 − f1(t)
]2,
here ε1 > 0 is arbitrary constant. Therefore([β ‖∇u‖2 − f1(t)
](−∆)u, ut + εu
)≥ 1
4β
d
dt
[β ‖∇u‖2 − f1(t)
]2
+
(ε
β− ε1
2
)[β ‖∇u‖2 − f1(t)
]2
− 1
2ε1
∣∣∣∣f ′1(t) + 2εf1(t)
2β
∣∣∣∣2 . (41)
The next lemma shows that the system (7) possesses uniform attracting bounded setin E1.
Lemma 3.8. There exists an uniform attracting set P = B0(0, r0) ⊂ E1 for{Sw(t, τ)}, where B0(0, r0) denotes the open ball centered at the origin with radiusis r0, where r0 is defined by (19).
Proof. Taking the inner product of (7) by U = [U1, U2]T in E1. This gives
d
dt‖U‖2E1
= 2(A1U,U)E1+ 2(F1(U),U)E1
. (42)
Let ε1 = 3ε2β , substituting U = [u, v]T , v = ut + εu into (42), along with Lemma 3.3
and (41), we can obtain
d
dt
(‖u‖22,l + ‖v‖2 +
1
2β
[β ‖∇u‖2 − f1(t)
]2)
≤ −ε(‖u‖22,l + ‖v‖2 +
1
2β
[β ‖∇u‖2 − f1(t)
]2)
+β
3ε
∣∣∣∣f ′1(t) + 2εf1(t)
2β
∣∣∣∣2 +1
α− ε‖f(t, x)‖2. (43)
Let
E1(t, τ) = E 1(u, v) = ‖u‖22,l + ‖v‖2 +1
2β
[β ‖∇u‖2 − f1(t)
]2, (44)
by (17), (43) and (44), we have
d
dtE 1(u, v) ≤ −εE 1(u, v) +M(f1, f2). (45)
21
Applying (45), we have
E1(t, τ) ≤ e−ε(t−τ)E1(τ, τ) +
∫ t
τe−ε(t−s)M(s)ds. (46)
Merging with (18) and (46), we have that for any bounded set B ⊂ E1 and initialvalue [uτ , vτ ]T ∈ B, t ≥ τ , ∀τ ∈ R, the following holds
E1(t, τ) ≤ e−ε(t−τ)E1(τ, τ) +M0
ε
(1− e−ε(t−τ)
)Therefore, for ∀T ≥ 0, τ ∈ R, we have
E1(τ + T, τ) ≤ e−εTE1(τ, τ) +M0
ε
(1− e−εT
)(47)
Let P = B0(0, r0), where r0 is given by (19), together with (47), we have that for anygiven bounded set B ⊂ E1, the following holds
limT→+∞
supτ∈R
distE1(S(t, τ)B,P )→ 0.
By applying Theroem3.1, the S(t, τ) possesses uniform attracting bounded set P ⊂B0(0, r) ⊂ E1.
In term of (20), (21), (22) and the similar proof of Lemma 3.8, the following resultsfocus on the uniform attracting set for Ss(t, τ) can be achieved.
Lemma 3.9. The Process Ss(t, τ) possesses the uniform attracting set P =
B1(0, r1) ⊂ E1, where B0(0, r1) denotes the open ball centered at the origin withradius is r1, where r1 satisfies (22).
As for the solution of system system (7), from the proof of Lemma 3.8, we have
u ∈ L∞([τ, t], H2l (D));ut ∈ L∞([τ, t], L2(D));
[β ‖∇u‖2 − f1(s)
]∈ L∞([τ, t]).
Integrate both sides of (43) on [τ, t] with respect to time. This gives
u ∈ L2([τ, t], H2l (D)); ut ∈ L2([τ, t], L2(D));
[β ‖∇u‖2 − f1(s)
]∈ L2([τ, t]).
Analogously, the following results which are pertinent to the solutions of system (9)can be derived.
u ∈ L∞([τ, T ], H2l (D));ut ∈ L∞([τ, T ],D(A
1
8 ));
[β ‖∇u‖2 − f1(s)
2
]∈ L∞([τ, t]),
u ∈ L2([τ, T ], H2l (D)); ut ∈ L2([τ, T ],D(A
1
8 ));
[β ‖∇u‖2 − f1(s)
2
]∈ L2([τ, t]).
The way to obtain the uniformly asymptotical compactness of system (7) relies onthe stabilization estimations of the system, the proof of which depends on the resultsgiven in the next two Lemmas.
22
Lemma 3.10. Suppose u ∈ L∞((τ, t), H2l (D)), ut ∈ L∞((τ, t), L2(D)), f1(t) satisfies
(5). Then∫ t
τ
([β ‖∇u‖2 − f1(s)
]((−∆)u, ut)
)(s)ds =
(β
4‖∇u‖4 − f1‖∇u‖2
)(t)
−(β
4‖∇u‖4 − f1‖∇u‖2
)(τ)
−∫ t
τ
f′
1(s)
2‖∇u‖2ds (48)
Proof. By (5), with the similar proof of Lemma 3.7 given by Chen, Guirao, Cao etal in [13], this Lemma can be proved.
Lemma 3.11. Suppose {un}+∞n=1 is weakly star convergent to u inL∞((τ, t), H2
l (D), {unt }+∞n=1 is weakly star convergent to ut in L∞((τ, t), L2(D).f1(t) satisfies (5), let n,m ∈ N. Then
limn→∞
limm→∞
∫ t
τ
([β ‖∇un‖2 − f1
](−∆)un −
[β ‖∇um‖2 − f1
](−∆)um ,
unt − umt ) (s)ds = 0, (49)
limn→∞
limm→∞
∫ t
τ
([β ‖∇un‖2 − f1
](−∆)un −
[β ‖∇um‖2 − f1
](−∆)um ,
un − um) (s)ds = 0. (50)
Proof. Following the proof of Lemma 3.8 in Chen, Guirao, Cao et al [13], togetherwith Lemma 3.10, we can complete the proof.
According to Lemma 3.8, it is shown that the Process Sw(t, τ) possesses uniformattracting set P . Furthermore, from the proof of Lemma 3.8, we find there existsTP > 0 such that for ∀τ ∈ R, ∀T ≥ TP , the following holds
S(τ, τ − T )P ⊆ P. (51)
Merging with (51), the coming results on stabilization estimation of system (7) can bederived.
Lemma 3.12. For any given T0 ∈ [0,∞) and any given bounded sequence {U(0)n }+∞n=1 ∈
P , let n,m ∈ N, then for ∀ε > 0, there exists TP (ε) ≥ TP , T = max{TP , T0} such thatthe following holds
according to the Lemma3.8, the {Un}+∞n=1 possesses subsequence weakly star conver-gent in L∞([τ − T, τ ], E1), still denoted by {Un}+∞n=1.
Taking the inner product of (8) by Un−Um in E1, which together with the Lemma3.3 gives
d
dt‖Un −Um‖2E1
(t, τ) ≤ −ε‖Un −Um‖2E1+ 2K(Um,Un), (53)
where
K(Um,Un)(s) =([β ‖∇un‖2 − f1(s)
](−∆)un
−[β ‖∇um‖2 − f1(s)
](−∆)um, unt − umt
)+
([β ‖∇un‖2 − f1(s)
](−∆)un
−[β ‖∇um‖2 − f1(s)
](−∆)um, ε(un − um)
). (54)
Thus
‖Un −Um‖2E1(τ, τ − T ) ≤ e−εT ‖U(0)
n −U(0)m ‖2E1
+ 2
∫ τ
τ−Te−ε(τ−s)K(Um,Un)(s)ds.
Since U(0)n −U
(0)m ∈ P , then sup
τ∈Re−εT ‖U(0)
n −U(0)m ‖2E1
→ 0, (T →∞), let
TP (ε) = max
{−1
εlog
(ε
2‖P‖2E1
), TP
}.
then, for ∀ε > 0, there exists
T = max {TP (ε), T0}
such that
supτ∈R‖Un −Um‖2E1
(τ, τ − T ) ≤ ε+ 2 supτ∈R
∫ τ
τ−Te−ε(τ−s)K(Um,Un)(s)ds.
Let
Kε,P,T,τ (U(0)m ,U(0)
n ) = 2
∫ τ
τ−Te−ε(τ−s)K(Um,Un)(s)ds,
and f(s) = e−ε(τ−s), then f(s) ∈ L∞([τ − T, τ ]), merging with (54) and Lemma3.11,we have
lim infn→+∞
lim infm→+∞
supτ∈R
Kε,P,T,τ (U(0)m ,U(0)
n ) = 0.
24
Inspired by Chueshov and Lasiecka [15], in term of Lemma 3.12, the next The-orem which is crucial to verify the asymptotical compactness of system (7) can beimplemented.
Theorem 3.13. Suppose T0 ∈ [0,∞) is any given constant, B ⊂ E1 is any bounded
set, {U(0)n }+∞n=1 ∈ B is any given sequence, and for ∀n,m ∈ N, ∀ε > 0, there exists
TB(ε) ≥ 0, T = max {TB(ε), T0} satisfies
supτ∈R‖Sw(τ, τ − T )U(0)
m − Sw(τ, τ − T )U(0)n ‖E1
≤ ε+ supτ∈R
Kε,B,T,τ (U(0)m ,U(0)
n ), (55)
where
lim infn→+∞
lim infm→+∞
supτ∈R
Kε,B,T,τ (U(0)m ,U(0)
n ) = 0.
Then, K (τ) = limT→+∞
S(τ, τ − T )B is relative compactness.
Proof. In order to verify K (τ) is relative compactness, it is sufficient to prove that
limT→+∞
αK(S(τ, τ − T )B) = 0, (56)
where αK(·) is α− measure, see proposition 3.2.We follow the argument in Proposition 7.1.12 of Ref [15] to verify (56) holds. Proved
by contradiction, if (56) is not true, then there exists ε0 > 0, for ∀T 1 ≥ 0, there existsT 0 > T 1such that αK(S(τ, τ − T 0)B) ≥ 3ε0.
Let ε0 = ε0, T 1 = TB(ε), T0 = T 0, T = max {TB(ε), T0}, obviously, T = T 0, thenαK(S(τ, τ − T )B) ≥ 3ε0, which reveals that there exists a sequence {U(0)}∞n=1 in Bsuch that
supτ∈R‖Sw(τ, τ − T )U(0)
m − Sw(τ, τ − T )U(0)n ‖E1
≥ 2ε0,m 6= n, n,m = 1, 2, · · ·
This contradicts (55), then the suppose is not true, alternatively, (56) holds. Thusconclude the proof.
The approach to derive the uniformly asymptotical compactness of system (7) isnormal, be known as splitting method, the validity of which depends the regularityor asymptotical regularity of solution for the considered system. It was dedicated toinvestigate the long-time behavior of many mathematical physics problem, such asheat conduction equations, Sine-Gordon equations, for more detail, one can refer to[38, 9]. The proofs relate to obtain the asymptotical compactness of system (7) areonly sketched here.
Let U = U + U, where U = [u, v]T , U = [u, v]T is the solution of the followingsystems respectively {
Ut = A2U,t = τ : U = Uτ .
(57)
25
and {Ut = A2U + F2(t,U)
t = τ : U = 0.(58)
Set S(t, τ) be the solution mapping induced by the system described by (57), mapping
related to (58) represented by S(t, τ). Obviously,
Ss(t, τ) = S(t, τ) + S(t, τ). (59)
There exists the following result for S(t, τ).
Lemma 3.14. For any bounded set B ⊂ E1, we have
limT→∞
supτ∈R‖S(t, τ)B‖E1
→ 0. (60)
Proof. Taking the inner product of (57) by U = [u, v]T in E1 and taking into accountLemma (3.4), we obtain
d
dt‖U‖2E1
= 2(A2U,U)E1≤ −ε‖U‖2E1
. (61)
Obviously, (60) can be derived by (61).
With respect to S(t, τ), let the next lemma holds.
Lemma 3.15. There exists a constant K > 0 such that
limT→∞
supτ∈R‖S(τ + T, τ)B‖E1+ 1
8
≤ K, ∀τ ∈ R. (62)
Proof. Taking the inner product of (57) by U = [u, v]T in E2. This gives
1
2
d
dt‖U‖2E1+ 1
8
= (A2U, U)E1+ 18
− [β‖∇u‖2 − f1(t)]A1
8 (−∆)u,A1
8 v) + (A1
8 f(t, x), A1
8 v),(63)
where u = u+ u, v = v + v. By Lemma 3.4 and Young equality, we have
[β‖∇u‖2 − f1(t)](A1
8 (−∆)u,A1
8 v) ≤ α
4‖A
1
4 v‖2 +
([β‖∇u‖2 − f1(t)]‖A
1
2u‖)2
α, (64)
where r1 is given by (22). Similar to proof of Lemma (3.4), we find
(A2U, U)E1+ 18
≤ −ε2‖U‖2E1+ 1
8
− α
4‖A
1
4U2‖2 − (Λ
1
4
1α
2− ε
2)‖A
1
8 v‖2. (65)
Using Cauchy inequality and Holder inequality, we get
(A1
8 f(t, x), A1
8 v) ≤ 2Λ3
4
1
Λ1
4
1α− ε‖A
1
8 f(t, x)‖2 +Λ
1
4
1α− ε2
‖A1
8 v‖2. (66)
26
Substituting (64)–(66) into (63) implies
d
dt‖U‖2E1+ 1
8
≤ −ε‖U‖2E1+ 18
+4Λ
3
4
1
Λ1
4
1α− εm2 +
([β‖∇u‖2 − f1(t)]‖A
1
2u‖)2
α. (67)
Let K0 = 4Λ341
Λ141 α−ε
m2 + 2r41α , here r1 is given by (22), thus K0 ≥ 0. On the other hand,
since the initial value of (58) is Zero, thus, let T enough large, we have
‖U‖2E1+s(τ + T, τ) ≤ e−ε(T+τ)
∫ T+τ
τeεsK0ds ≤
K0
ε≡ K.
Then, (62) holds.
The Lemma 3.14 shows that the solution of S(t, τ) that starts from any bounded
set E1 evolves to 0 as the time tends to ∞. The fact that solution of S(t, τ) withoutinitial energy is bounded in E1+ 1
8is asserted by Lemma 3.15. Take into account the
compact embedding theorem and (59), we can conclude that the Ss(t, τ) is uniformlyasymptotical compact.
The remaining of this subsection is dedicated to give some consequences employedin the procedure to achieve the Hausdorff dimensions of Kernel Sections.
Before estimating the Hausdorff dimension of the Kernel Sections of system (7) andsystem (9), the uniformly qusidifferentiable of the systems must be checked. The Thefollowing Lemma is significant in verifying the uniformly quasidifferentiable of ProcessSw(t, τ) and Ss(t, τ). Since F12(t,U), F12(t,U) is independent to v, when no ambiguity
according to Theorem 3.8, we get∣∣β‖∇u‖2 − f1(t)∣∣ ≤ r0,
which together with (69) gives
‖F12(t, u)‖L (D(A
12 ),H)
≤ c2(r0)‖A1
2 %‖
Then, (71) is satisfied.
If replacing the r0 in Lemma 3.16 by the r1 given by (22), the above results alsoholds.
The uniformly qusidifferentiable of Sw(t, τ) can be checked in the ensuring manner.Suppose U(1) = [u1, v1],U(2) = [u2, v2] are the solution of system (7) with initial
values U(1)0 ∈ K (τ),U
(2)0 ∈ K (τ) respectively, where U
(2)0 = U
(1)0 + h, h = [h1, h2] ∈
E1, then U(1) − U(2) satisfies (8). Let U be the solution of system (10) in which
U = U(1) with initial value t = 0 : U0 = h. Define U = DS(t, τ,U(1))h, then
DS(t, τ,U(1)) is the solution mapping induce by system (10). Moreover, set Ψ =
[Ψ1,Ψ2]T = U(1) −U(2) − U.Taking the inner product of (8) by U(1) −U(2) in E1, this gives
1
2
d‖U(1) −U(2)‖2E1
dt=
(A1(U(1) −U(2)),U(1) −U(2)
)E1
+ (F12(u1)− F12(u2), v1 − v2) . (72)
28
Since
F12(t, u1)− F12(t, u2) =
∫ 1
0F12(t, su1 + (1− s)u2)(u1 − u2)ds,
merging with (71) in Lemma 3.16, we have
‖F12(t, u1)− F12(t, u2)‖ ≤ c2(r0)‖A1
2 (u1 − u2)‖,
furthermore, by Cauchy inequality and Young inequality, we obtain
(F12(t, u1)− F12(t, u2), v1 − v2) ≤ c2(r0)(‖A
1
2 (u1 − u2)‖2 + ‖v1 − v2‖)
≤ c2(r0)‖U (1) − U (2)‖2E1, (73)
substituting (73) into (72), together with Lemma 3.3 gives
d‖U(1) −U(2)‖2E1
dt≤ −ε‖U(1) −U(2)‖2E1
+ 2c2(r0)‖U(1) −U(2)‖2E1
≤ 2c2(r0)‖U(1) −U(2)‖2E1. (74)
Hence
‖U(1) −U(2)‖2E1≤ e2c2(r0)(t−τ)‖h‖2E1
. (75)
Since U(1) ∈ K (τ), τ ∈ R, invoking Theorem 2.7, we have ‖U(1)‖E1≤ r. On the other
hand, (A1U, U
)E1
≤ −ε2‖U‖ − k0
2‖U2‖2,
and(F1(t,U(1))U, U
)E1
= −([β ‖∇u‖2 − f1(t)
](−∆)U1 + 2β(∇u,∇U1)(−∆)u, U2
)≤
√2βr0‖∆U1‖‖U2‖+ 2βΛ
− 1
4
1 r20‖A
1
4 U1‖‖U2‖
≤ k0
2‖U2‖2 +
2βr20
k0‖A
1
2 U1‖2 +4β2r4
0
Λ1
2
1 k0
‖A1
4 U1‖2,
which together with (10) gives
1
2
d‖U1‖2E1
dt≤ 2βr2
0
k0‖A
1
2 U1‖2 +4β2r4
0
Λ1
2
1 k0
‖A1
4 U1‖2
≤ p‖U1‖2E1, (76)
29
where
p = max
{4βr2
0
k0,8β2r4
0
Λ1
2
1 k0
}, (77)
merging with Theorem 2.7, we find p is bounded. In addition, employing (76), we have
‖U‖E1≤ ep(t−τ)‖h‖E1
. (78)
Let C1(t−τ) = ep(t−τ), since U = DS(t, τ,U(1))h, along with (77) and (78), we attain
‖DS(t, τ,U(1))‖L (E1,E1) ≤ C1(t− τ) <∞. (79)
On the other hand, Ψ satisfies
dΨ
dt= A1Ψ + F1(t,U(1))Ψ + l (80)
with the initial of t = τ : Ψ = 0, where l = [0, l1],
l1 =
∫ 1
0
(F12(su2 + (1− s)u1) −F12(u1)
)(u1 − u2)ds.
It follows from Lemma 3.16 that
‖l1‖ ≤ c1(r0)‖A1
2u1 −A1
2u2‖2. (81)
Taking the inner product of (80) by Ψ in E1, along with Lemma 3.3, we have
d‖Ψ‖2E1
dt≤ −ε‖Ψ‖2E1
+ 2(F1(t,U(1))Ψ,Ψ) + 2(l,Ψ)
= −ε‖Ψ‖2E1+ 2(F12(u1)Ψ2,Ψ2) + 2(l1,Ψ2). (82)
Invoking Lemma 3.16 yields
(F (t,U(1))Ψ,Ψ) ≤ c2(r0)‖Ψ‖2. (83)
By (81), it can be find
(l1,Ψ2) ≤ c1(r0)(‖A
1
2u1 −A1
2u2‖4 + ‖Ψ2‖2). (84)
Substituting (83) and (84) into (82), we attain
d‖Ψ‖2E1
dt≤ 2(c1(r0) + c2(r0))‖Ψ‖2E1
+ 2c1(r0)‖A1
2u1 −A1
2u2‖4.
30
Since the initial value of Ψ is 0, we get
‖Ψ‖2E1(t, τ) ≤ 2c1(r0)
∫ t
τe2(c1(r0)+c2(r0))(t−s)‖A
1
2u1 −A1
2u2‖4(s)ds.
Thus,
‖Ψ‖2E1(t, τ)
≤ 2c1(r0)e2(c1(r0)+c2(r0))(t−τ)
∫ t
τ‖A
1
2u1 −A1
2u2‖4(s)ds,
merging with (75), we find that there exist constant c3 > 0, c4 > 0 such
‖Ψ‖E1(t, τ) ≤ c4e
c3(t−τ)‖h‖2E1, (85)
where
c3 = 3c1(r0) + c2(r0), c4 =
√c1(r0)
2c2(r0).
Therefore, we conclude the following results.
Theorem 3.17. The Process {Sw(t, τ)} is uniformly qusidifferentiable in Kernel.
→ 0, by (36) in Ref [14], we have Process Sw(t, τ)is uniformly qusidifferentiable in Kernel.
Similarly, the following results can be proved, we omitted the proof here.
Theorem 3.18. The Process {Ss(t, τ)} is uniformly qusidifferentiable in Kernel.
4. Main proofs
According to the results given in subsection 3.2, this Section is devoted to prove themain results proposed in subsection 2.2.
4.1. Proof of Theorem 2.5
For ∀U ∈ E1, by Lemma 3.3, we have
<(A1U,U) ≤ 0, (87)
where < means take the Real part. Lemma 3.5 implied the operator A−11 : E1 → E1
is compact, then spectrum of A1 comprise no more than the eigenvalues and zero. Let
31
σ(A1) be the spectrum of A1, by Lemma 3.6, we have < (σ(A1)) ≤ 0, then
<(I −A1) = E1. (88)
According to Lemma 2.2.3 in Ref [43], along with (87) and (88), we obtain A1 cangenerate a linear semigroup of contractions, signified by eA1t, t ≥ 0. On the otherhand, (38) shows that F1 : E1 → E1 satisfies the Lipshitz condition. Theroefore, byusing the Theorem 2.5.1 in Ref [43], the results of this theorem can be derived.
4.2. Proof of Theorem 2.6
Since F1 = F2, by Lemma 3.4 and Lemma 3.7, similar to the proof of the Theorem2.5, the results in this Theorem can be accomplished.
4.3. Proof of Theorem 2.7
The Lemma 3.8 together with its proof gives that there exits TP > 0, for any τ ∈R, T ≥ TP , the following holds
S(τ, τ − T )P ⊆ P.
Let T0 ≥ TP and
Kn(τ) =⋃
s≥nT0
Sw(τ, τ − s)P , n ∈ N
According to Ref [14], we have
Kn+1(τ) ⊂ Kn(τ), n ∈ N, (89)
and Kn(τ), ∀n ∈ N is non-empty. Let
K∞(τ) = limn→∞
Kn(τ),K(τ) = limT→+∞
S(τ, τ − T )P,
then
K(τ) = K ∞(τ)
Applying Lemma 3.12 and Lemma 3.13, we find αK(K(τ)) = 0, thereforeαK (K∞(τ)) = 0, it means lim
n→∞αK(Kn(τ)) = 0, which together with (89) and Propo-
sition 3.2 gives the following results.Let
K (τ) =⋂N≥0
⋃n≥N
S(τ, τ − nT0)P ,
and K =⋃τ∈R
K (τ), then K (τ) is compact. By Theorem 3.1, we can find that the
Process Sw(t, τ) possesses Kernel K and Kernel Sections K (τ), τ ∈ R. By the trans-
32
formation between system (2) and system (7), we can conclude that system (2) alsopossesses Kernel T−1
ε K and Kernel Sections T−1ε K (τ), ∀τ ∈ R.
4.4. Proof of Theorem 2.8
Lemma 3.9 sets forth that the Process Ss(t, τ) possesses uniform attracting set in E1.Combing Lemma 3.14 and Lemma 3.15, we have Ss(t, τ) is uniformly asymptoticallycompact. Thus, by Theorem 3.1, the Process Ss(t, τ) possesses Kernel and KernelSections, Moreover, Kernel Sections K (τ), τ ∈ R are indicated by (26) and Kernel
is K =⋃τ∈R
K (τ). Furthermore, system (3) also possesses Kernel T−1ε K and Kernel
Sections T−1ε K (τ), ∀τ ∈ R.
4.5. Proof of Theorem 2.9
We follow the procedure given by Theorem 4.1 in [14] to derive the Hausdorff dimen-sions estimation.
By Lemma 3.17, We have Sw(t, τ) is uniformly qusidifferentiable in Kernel.According to (79), we find that the condition 38 of Theorem 4.1 in [14] can be
satisfied.Let U = Sw(t, τ)Uτ and U1, · · · , Um are the solutions of (10) with initial val-
ues U = U01, · · · , U0m respectively. Set Qm(s) = Qm(s, τ,Uτ ; U01, · · · , U0m)is the
orthogonal projector in E1 onto the space spanned by U1(t), · · · , Um(t). For any giv-en time s, Let Φi(s) = {ξi(s), ζi(τ)}, i = 1, · · · ,m denote an orthonormal basis ofQm(s)E1, then
Tr(A1 + F1(s,U)) ◦Qm(s) =
∞∑i=1
((A1 + F1(s,U)) ◦Qm(s)Φi(s),Φi(s)
)E1
=
m∑i=1
((A1 + F1(s,U))Φi(s),Φi(s)
)E1
. (90)
Since
(A1Φi(s),Φi(s))E1≤ −ε
2‖Φi‖2E1
− k0
2‖ζi‖2,
and(F1(s,U)Φi(s),Φi(s)
)E1
= −([β ‖∇u‖2 − f1(s)
]A
1
2 ξi +2β(∇u,∇ξi)(−∆)u, ζi)
≤√
2βr0‖A1
2 ξi‖‖ζi‖+ 2βΛ− 1
4
1 r20‖A
1
4 ξi‖‖ζi‖
≤ k0
2‖ζi‖2 +
2βr20
k0‖A
1
2 ξi‖2 +4β2r4
0
Λ1
2
1 k0
‖A1
4 ξi‖2.
Thus, we get
Tr(A1 + F1(s,U)) ◦Qm(s) ≤ −(ε
2− 2βr2
0
k0
)m+
4β2r40
Λ1
2
1 k0
m∑i=1
Λ− 1
2
i ,
33
Let
qm = lim infT→+∞
supτ∈R
supuτ∈K(τ)
1
T
∫ T+τ
τTr(A1 + F1(s,U)) ◦Qm(s)ds.
then
qm ≤ −(ε
2− 2βr2
0
k0
)m+
4β2r40
Λ1
2
1 k0
m∑i=1
Λ− 1
2
i . (91)
By the Lemma 6.3, Chapter VI in [38] and the compactness of operator A, we have
limn→∞
1
m
m∑i=1
Λ− 1
2
i = 0. (92)
If ε2−
2βr2
k0≤ 0, combing with (91) and (92) gives that, the d defined by (24) is infinite,
which means Hausdorff dimension estimation is infinite.When (23) holds, together with 0 < k0 ≤ α− ε, we have
ε
2− 2βr2
0
k0> 0,
merging (91) with (92), we have the minimum integer m which satisfies qm < 0 isfinite, it means d defined by (24) is finite, Thus, by Theorem 4.1 in [14], the Hausdorffdimension estimation of Kernel Sections less than d.
4.6. Proof of Theorem 2.10
Let the solution mapping grenerated by (11) denoted by DS(t, τ,Uτ ), similar to theprocess to obtain (79), we have
supUτ∈K (τ)
DS(t, τ,Uτ ) ≤ C1(t− τ) <∞, t ≥ τ, τ ∈ R.
then the condition (38) of Theorem 4.1 in [14] can be satisfied.Theorem 3.18 shows that Sw(t, τ) is uniformly qusidifferentiable in Kernel.Similar to the process to achieve (91), we have
qm ≤ −ε
2m+
2βr21
α
m∑i=1
Λ− 1
4
i +2β2r4
1
Λ1
2
1 (Λ1
4
1α− ε)
m∑i=1
Λ− 1
2
i ,
together with (92), we find that d defined by (25) is finite. Thus, by Theorem 4.1 in[14], we have dH(K (τ)) < d, ∀τ ∈ R. Especially, if
2βr21
αΛ− 1
4
1 +2β2r4
1
Λ1(Λ1
4
1α− ε)<ε
2,
34
then dH(K (τ)) = 0, which means that there exists one point in Kernel Section andKernel comprises one complete trajectory of the system (9).
5. Conclusions
In light of Theorem 2.7 and Theorem 2.8, the nonautonomous Euler-Bernoulli beamequations possess Kernel Sections, no matter the form of damping is strongly or not.Alternatively, we can conclude that the form of damping do not affect the existenceof Kernel Sections. Nevertheless, the ways to derive the related results depend on dif-ferent techniques.
Theorem 2.9 indicates that, if the form of damping is weak and the coefficient ofdamping is too small, then the big intensity of the external excitation as well as lon-gitudinal time-vary loading leads to the infinite Hausdorff dimensions estimation, it isno sense. In the case of strongly damping, Theorem 2.10 expounds that the Hausdorffdimensions estimation of the beam is always finite.
Kernel Sections for NDS generated by PDE are rather crude objects which attractall bounded sets. For the application in the study on global dynamics of the nonlin-ear systems in mechanics, Basic Kernel Sections are the useful objects to illustratethe global dynamics in both theoretical and numerical aspect, see Proposition 2.4, formore detail, one can refer to Birnir [9]. In this paper, we do not consider the structureof Basic Kernel Sections for nonautonomous Euler-Bernoulli beam equations analyti-cally which will be addressed in the further. From the numerical standpoint, in light ofProposition 2.4 and the algorithm proposed by Keller and Ochs[29], the enough longnumerical integration with respect to time for the solution of the considered systemleads to the structure of Basic Kernel Sections which represent the stable completetrajectory of the systems. Nevertheless, we must keep in mind that the existence ofBasic Kernel Section is derived from the assertion that the NDS possesses Kernel Sec-tions, which indicates that proof of the existence of Kernel Sections plays the primaryrole in the investigation of dynamics of the system by the structure of Kernel sectionsnumerically.
Let f1(t) = γ1 + γ2 sin(ω0t), by (20), (21), (22) and Theorem (2.10), we find thatwith the increasing of γ1, the Hausdorff dimension of the Kernel Sections also increas-es, which means that the global dynamics may become complex. From the engineeringviews, γ1 denotes the mean of axial force of the Euler-Bernoulli beams, the numericalresults about the global dynamics on strongly damping Euler-Bernoulli beams showsthat the system possesses a global stability complete trajectory which is a periodicmotion with the small mean of axial force. Increasing the mean of axial force to leadsto the global bifurcation behaviors of the system. Continue to increase the mean ofaxial force to bigger one (such as, the value in Case I(3)), the phenomenon of glob-al bifurcation still exists, nevertheless, the qualitative of local dynamics change. Thecomplex dynamical behaviors look like to appear with the mean of axial force variesto enough big value. Thus, the numerical results about the global dynamics are inagreement with the qualitative results. In the practical application, ω0 represent fre-quency of axial excitation, from the numerical results on global dynamics, the globalstability periodic motion occurs with a certain frequency. Decreasing the frequency toa lower value, the global stability periodic motion is interrupted to a global stabilityperiod-doubling motion, this phenomenon can be in turn by change the value of toeven small, nevertheless, the enough low frequency may result in the complex dynam-ical behavior of the system. But we can not get these information from the theoretical
35
result. The reason is that, with respect to frequency of axial excitation, (21) and (22) istoo conservative. In addition, less information about the complex dynamical behavior(such as, CaseI(4) and Case II(4) in subsection 2.2) is introduced in this paper.
It is well known that the local Lyapunov exponents [2] are efficient quantities toaccount for local dynamics of nonlinear systems, there exist many relevant composi-tions in engineering applications, one can refer to Wolf et.al [40] and the referencestherein. Global Lyapunov exponents introduced by Constantin and Foias [16] shouldbe the nature tool to investigate the global dynamics, nonetheless, to our best knowl-edge, there hardly exists investigation on global dynamics relies on global Lyapunovexponents. The obstacle is that, almost of the approaches to estimate the Hausdorffdimension which are strongly related to global Lyapunov exponents depend on theradius of the global attractor. Sometimes, the radius of absorbing set depends on thetotally energy function which may not only contain Kinetic and Potential energy but
also include other energy, such as the term 12β
[β ‖∇u‖2 − f1(t)
]2in this paper, which
may leads to that the radius of global attractor obtained is bigger than the actualvalue. At last, the Hausdorff dimension achieved by the method in [14] is an estimatedvalue which is conservative. However, if we only consider the previous order modalequations related to the beams, since these equations are finite ordinary differentialequations, the troubles can be handled. Thus, according to approach of deriving theHausdorff dimension estimation of the Kernel Section, the maximum and other orderglobal Lyapunov exponents which can interpret the phenomena of global bifurcationand Chaos may be attained. We will pursue this problem in the further.
Appendix A. The model equation
Suppose u =k∑i=1
li(t)wi, k = k1 + k2, k1, k2 ∈ N, according to the inertial manifold
with delay [19] and the nonlinear Galerkin method (NGM) [31] the following modelequations associated with EBS can be achieved
where l = (l1, · · · , lk1)T = (l1, · · · , lk1)T which represent the low-frequency modal and
the other is high-frequency modal l = (l1, · · · , lk2)T = (lk1 , · · · , lk1+k2)T , li is the value
of l at time i, h is step size of numerical integration, τ ∈ N is an undetermined constant.
Λ1 =
Λ
1
4
1 0 0
0. . . 0
0 0 Λ1
4
k1
, A1 =
Λ1 0 0
0. . . 0
0 0 Λk1
,F1(t) =
F 1(t)...
F k1(t)
, F1(t, l, l) =
F1(t, l)...
Fk1(t, l)
36
and
Λ2 =
Λ
1
4
k1+1 0 0
0. . . 0
0 0 Λ1
4
k
, F2(t) =
F k1+1(t)...
F k(t)
,A2 =
Λk1+1 0 0
0. . . 0
0 0 Λk
,F2(t, l, l) =
Fk1+1(t, l)...
Fk(t, l)
,
here
F(t) =
F 1(t)...
F k(t)
=
(f(t, x), w1)...
(f(t, x), wk)
,F(t, l) =
F1(t, l)...
Fk(t, l)
=
([β
∥∥∥∥∇ k∑i=1
liwi
∥∥∥∥2
− f1(t)
](−∆)
k∑i=1
liwi, w1
)...([
β
∥∥∥∥∇ k∑i=1
liwi
∥∥∥∥2
− f1(t)
](−∆)
k∑i=1
liwi, wk
)
.
(A2)
Acknowledgements
This work is partly supported by the Key Project of National Natural Science Foun-dation of China (No. 11732005).
References
[1] Ludwig Arnold, Random dynamical systems, Springer-Verlag, Berlin, 1998.[2] Ludwig Arnold, Hans Crauel and Jean-Pierre Eckmann, Lyapunov exponents: proceedings
of a conference held in Oberwolfach, May 28-June 2, 1990, Springer, 2006.[3] Jan Awrejcewicz, AV Krysko, IE Kutepov, NA Zagniboroda, V Dobriyan and VA Krysko,
Chaotic dynamics of flexible Euler-Bernoulli beams, Chaos: An Interdisciplinary Journalof Nonlinear Science 23 (2013), 043130.
[4] JM Ball, Initial-boundary value problems for an extensible beam, Journal of MathematicalAnalysis and Applications 42 (1973), 61–90.
[5] JM Ball, Stability theory for an extensible beam, Journal of Differential Equations 14(1973), 399–418.
[6] Amin Barari, Hamed Dadashpour Kaliji, Mojtaba Ghadimi and G Domairry, Non-linearvibration of Euler-Bernoulli beams, Latin American Journal of Solids and Structures 8(2011), 139–148.
37
[7] M Bayat, Amin Barari and M Shahidi, Dynamic response of axially loaded Euler-Bernoullibeams, Mechanics 17 (2011), 172–177.
[8] Bjorn Birnir, Global attractors and basic turbulence, Nonlinear coherent structures inphysics and biology, Springer, 1994, pp. 321–334.
[9] Bjorn Birnir, Basic Attractors and Control, Springer-Verlag, New York, 2015.[10] Zdzis law Brzezniak, Bohdan Maslowski and Jan Seidler, Stochastic nonlinear beam equa-
tions, Probability theory and related fields 132 (2005), 119–149.[11] T Caraballo, JA Langa and J Valero, The dimension of attractors of nonautonomous
partial differential equations, The ANZIAM Journal 45 (2003), 207–222.[12] Alexandre Carvalho, Jose A Langa and James Robinson, Attractors for infinite-
dimensional non-autonomous dynamical systems, Springer Science & Business Media,2012.
[13] Huatao Chen, Juan Luis Garcia Guirao, Dengqing Cao, Jingfei Jiang and Xiaoming Fan,Stochastic Euler-Bernoulli beam driven by additive white noise: global random attractorand global dynamics, Nonlinear Analysis-Theory Method & Application 185 (2019), 216–246.
[14] V Chepyzhov and M Vishik, A Hausdorff dimension estimate for kernel sections of non-autonomous evolution equations, Indiana University Mathematics Journal 42 (1993),1057–1076.
[15] Igor Chueshov and Irena Lasiecka, Von Karman Evolution Equations: Well-posedness andLong Time Dynamics, Springer Science & Business Media, 2010.
[16] P Constantin and C Foias, Global lyapunov exponents, kaplan-yorke formulas and thedimension of the attractors for 2D navier-stokes equations, Communications on pure andapplied mathematics 38 (1985), 1–27.
[17] Hans Crauel and Peter E Kloeden, Nonautonomous and random attractors, Jahresberichtder Deutschen Mathematiker-Vereinigung 117 (2015), 173–206.
[18] Constantine M Dafermos, Semiflows associated with compact and uniform processes,Mathematical systems theory 8 (1974), 142–149.
[19] Arnaud Debussche and Roger Temam, Some new generalizations of inertial manifolds,Discrete & Continuous Dynamical Systems-A 2 (1996), 543–558.
[20] Michael Dellnitz and Andreas Hohmann, A subdivision algorithm for the computation ofunstable manifolds and global attractors, Numerische Mathematik 75 (1997), 293–317.
[21] A Eden and AJ Milani, Exponential attractors for extensible beam equations, Nonlinearity6 (1993), 457.
[22] Xiaoming Fan and Shengfan Zhou, Kernel sections for non-autonomous strongly dampedwave equations of non-degenerate Kirchhoff-type, Applied mathematics and computation158 (2004), 253–266.
[23] Zaichun Feng and Stephen Wiggins, On the existence of chaos in a class of two-degree-of-freedom, damped, strongly parametrically forced mechanical systems with brokenO (2)symmetry, Zeitschrift fur angewandte Mathematik und Physik ZAMP 44 (1993), 201–248.
[24] Bao-Zhu Guo and Kun-Yi Yang, Dynamic stabilization of an Euler–Bernoulli beam equa-tion with time delay in boundary observation, Automatica 45 (2009), 1468–1475.
[25] Seon M Han, Haym Benaroya and Timothy Wei, Dynamics of transversely vibratingbeams using four engineering theories, Journal of Sound and vibration 225 (1999), 935–988.
[26] CS Hsu, A theory of cell-to-cell mapping dynamical systems, Journal of Applied Mechanics47 (1980), 931–939.
[27] Jum-Ran Kang, Global attractor for an extensible beam equation with localized nonlineardamping and linear memory, Mathematical Methods in the Applied Sciences 34 (2011),1430–1439.
[29] Hannes Keller and Gunter Ochs, Numerical approximation of random attractors, Stochas-tic dynamics, Springer, 1999, pp. 93–115.
[30] To Fu Ma and Vando Narciso, Global attractor for a model of extensible beam withnonlinear damping and source terms, Nonlinear Analysis: Theory, Methods & Applications73 (2010), 3402–3412.
[31] Martine Marion and Roger Temam, Nonlinear galerkin methods, SIAM Journal on Nu-merical Analysis 26 (1989), 1139–1157.
38
[32] Mark M Meerschaert and Charles Tadjeran, Finite difference approximations for two-sided space-fractional partial differential equations, Applied numerical mathematics 56(2006), 80–90.
[33] Omer Morgul, Dynamic boundary control of a Euler-Bernoulli beam, IEEE Transactionson automatic control 37 (1992), 639–642.
[34] AB Multiphysics, COMSOL Multiphysics 3.5 a Reference Manual, PDE mode equationbased modeling, Multiphysics Ltd, Stohkholm,Sweden, 2008.
[35] Solange Kouemou Patcheu, On a global solution and asymptotic behaviour for the gener-alized damped extensible beam equation, Journal of Differential Equations 135 (1997),299–314.
[36] Bjorn Schmalfuss, Attractors for the non-autonomous dynamical systems, Equadiff 99: (In2 Volumes), World Scientific, 2000, pp. 684–689.
[37] Hui-Shen Shen, A two-step perturbation method in nonlinear analysis of beams, plates andshells, John Wiley & Sons, Hoboken, 2013.
[38] Roger Temam, Infinite-dimensional dynamical systems in mechanics and physics,Springer-Verlag, New York, 1997.
[39] Stephen Wiggins, Global bifurcations and chaos: analytical methods, 73, Springer Science& Business Media, Heidenburg, 2013.
[40] Alan Wolf, Jack B Swift, Harry L Swinney and John A Vastano, Determining Lyapunovexponents from a time series, Physica D: Nonlinear Phenomena 16 (1985), 285–317.
[41] Zhijian Yang, On an extensible beam equation with nonlinear damping and source terms,Journal of Differential Equations 254 (2013), 3903–3927.
[42] Wei Zhang, Fengxia Wang and Minghui Yao, Global bifurcations and chaotic dynamicsin nonlinear nonplanar oscillations of a parametrically excited cantilever beam, NonlinearDynamics 40 (2005), 251–279.
