
We discuss the dynamics of a fractional order discrete neuron model with electromagnetic flux coupling. The discussed neuron model is a simple one-dimensional map which is modified by considering flux coupling. We consider a discrete fractional order memristor to mimic the effects of electromagnetic flux on the neuron model. The bifurcation dynamics of the fractional order neuron map show an inverse period-doubling route to chaos as a function of control parameters, namely the fractional order of the map and the flux coupling coefficient. The bifurcation dynamics of the systems are derived both in the time and frequency domains. We present a two-parameter phase diagram using the Lyapunov exponent to categorize the various dynamics present in the system. In addition to the Lyapunov exponent, we use the entropy of the model to distinguish the various dynamics of the systems. To investigate the network behavior of the fractional order neuron map, a lattice array of N×N nodes is constructed and external periodic stimuli are applied to the network. The formation of spiral waves in the network and the impact of various parameters, like the fractional order, flux coupling coefficient and the coupling strength on the wave propagation are also considered in our analysis.
Citation: Janarthanan Ramadoss, Asma Alharbi, Karthikeyan Rajagopal, Salah Boulaaras. A fractional-order discrete memristor neuron model: Nodal and network dynamics[J]. Electronic Research Archive, 2022, 30(11): 3977-3992. doi: 10.3934/era.2022202
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Input-output |
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We discuss the dynamics of a fractional order discrete neuron model with electromagnetic flux coupling. The discussed neuron model is a simple one-dimensional map which is modified by considering flux coupling. We consider a discrete fractional order memristor to mimic the effects of electromagnetic flux on the neuron model. The bifurcation dynamics of the fractional order neuron map show an inverse period-doubling route to chaos as a function of control parameters, namely the fractional order of the map and the flux coupling coefficient. The bifurcation dynamics of the systems are derived both in the time and frequency domains. We present a two-parameter phase diagram using the Lyapunov exponent to categorize the various dynamics present in the system. In addition to the Lyapunov exponent, we use the entropy of the model to distinguish the various dynamics of the systems. To investigate the network behavior of the fractional order neuron map, a lattice array of N×N nodes is constructed and external periodic stimuli are applied to the network. The formation of spiral waves in the network and the impact of various parameters, like the fractional order, flux coupling coefficient and the coupling strength on the wave propagation are also considered in our analysis.
In recent years the problems of control and stabilisation for thermoelastic systems have been studied intensively. We refer for instance to [22], [42] in the context of controllability, and to [14], [18], [23], [24], [35] for stabilization, among others.
A closely related interesting issue is the asymptotic behaviour of a system consisting of two different materials joined together at the interface, one being purely elastic and the other thermoelastic (see Figure 1). In these systems, other than the intrinsic coupling effects of thermal and elastic components, typical in thermoelasticity, the thermoelastic and purely elastic rods are also coupled through the interface.
Exponential and polynomial decay properties for this kind of systems were proved by Rivera et al. in [12], [28], [31] and Messaoudi et al. in [30], using the energy multiplier method. Han and Xu in [15] got a sharp polynomial decay rate for a thermoelastic transmission problem with joint mass, based on a detailed spectral analysis and resolvent operator estimates. We also refer to [29] for the analysis of the large time behaviour of transmission problems of multi-dimensional thermoelasticity.
In this work, we consider similar transmission problems in multi-connected networks. More precisely, we are interested in the large time behaviour of star-shaped networks constituted by coupled thermoelastic and purely elastic rods (see Figure 2).
The aim of this work is to give a complete analysis on the large time behaviour of these systems proving exponential, polynomial and slow decay rates. The optimality of these results is also discussed.
In order to present the problems under consideration more precisely some notations are needed. We denote by
{uk,tt(x,t)−uk,xx(x,t)+αkθk,x(x,t)=0,x∈(0,ℓk),k=1,2,...,N1,t>0,θk,t(x,t)−θk,xx(x,t)+βkuk,tx(x,t)=0,x∈(0,ℓk),k=1,2,...,N1,t>0, | (1) |
and the other edges in the network are all purely elastic ones given by
uj,tt(x,t)−uj,xx(x,t)=0,x∈(0,ℓj),j=N1+1,⋯,N,t>0. | (2) |
Here and in the sequel
Assume that the exterior nodes of the network are all clamped, and the displacements and temperatures are all continuous at the common node. There is no heat exchange between thermoelastic components and purely elastic ones and the balance of forces at the common node is fulfilled. Thus, the boundary and transmission conditions read as follows:
{uj(ℓj,t)=0,j=1,2⋯,N,t>0,uj(0,t)=uk(0,t),∀j,k=1,2,3,⋯,N,t>0,θk(ℓk,t)=0,k=1,2,...,N1,t>0,θk(0,t)=θj(0,t),∀j,k=1,2,...,N1,t>0,N∑j=1uj,x(0,t)=N1∑j=1αjθj(0,t),N1∑j=1αjβjθj,x(0,t)=0,t>0 | (3) |
with initial conditions
u|t=0=u(0):=(u(0)j)Nj=1,ut|t=0=u(1):=(u(1)j)Nj=1,θ|t=0=θ(0):=(θ(0)j)N1j=1. | (4) |
Overall the thermoelastic-elastic network system under consideration reads as follows:
{uj,tt(x,t)−uj,xx(x,t)+αjθj,x(x,t)=0,x∈(0,ℓj),j=1,2,⋯,N1,t>0,θj,t(x,t)−θj,xx(x,t)+βjuj,tx(x,t)=0,x∈(0,ℓj),j=1,2,⋯,N1,t>0,uj,tt(x,t)−uj,xx(x,t)=0,x∈(0,ℓj),j=N1+1,⋯,⋯,N,t>0,uj(ℓj,t)=0,j=1,2,⋯,N,t>0,uj(0,t)=uk(0,t),∀j,k=1,2,3,⋯,N,t>0,θk(ℓk,t)=0,k=1,2,⋯,N1,t>0,θk(0,t)=θj(0,t),∀j,k=1,2,...N1,t>0,N∑j=1uj,x(0,t)=N1∑j=1αjθj(0,t),N1∑j=1αjβjθj,x(0,t)=0,t>0,u(t=0)=u(0),ut(t=0)=u(1),θ(t=0)=θ(0). | (5) |
Remark 1. The transmission problem in Figure 1 can be considered as a special case of the above network (
The natural energy of this system is as follows
E(t)=12N∑j=1∫ℓj0[u2j,t+u2j,x]dx+12N1∑j=1αjβj∫ℓj0θ2jdx, |
and a direct calculation yields the dissipation law
E′(t)=−N1∑k=1αkβk∫ℓk0θ2k,xdx≤0. | (6) |
Hence, the energy of system (5) is decreasing. Moreover, from (6), it is easy to see that the dissipation mechanism only acts in the thermoelastic rods. This motivates the problem of whether or not the dissipation is strong enough to make the total energy of the network decay to zero, and with which rate.
In this paper, the large time behaviour of system (5) is mainly discussed based on frequency domain analysis ([5], [6], [13], [19], [25], [26] and [34]). In [37], Shel showed the exponential stability of networks of thermoelastic and elastic materials for some special cases by similar methods. However, it was assumed that there was no heat exchange between the thermoelastic rods connected at the common nodes. In this paper, we allow for the heat exchange between thermoelastic rods at common nodes, which is a natural assumption.
By estimating the resolvent operator along the imaginary axis and employing multiplier techniques, we get a necessary and sufficient condition for system (5) to decay uniformly exponentially, namely that there is no more than one purely elastic rod entering in the network. If this condition fails, the system lacks exponential decay and we further show that the decay rate of the networks can not be faster than
To discuss the sharpness of slow decay rates it is useful to get explicit information on the spectrum of the system, and compare its real and imaginary parts (see [6] and [41]). However, spectra of PDE networks are often difficult to calculate. Thus, we prove the optimality by estimating the norm of the resolvent operator along the imaginary axis (see [1]). But resolvent estimates are hard to be achieved due to the thermoelastic coupling. Thus, we employ diagonalisation argument to deal with the resolvent problem. This allows building explicit approximations of solutions ensuring that the polynomial decay rates we get are nearly optimal.
The rest of the paper is organised as follows. In section 2, the main result of this paper is given. Section 3 is devoted to show the well-posedness and strong asymptotic stability of the system (5). In section 4, we prove the exponential and nearly optimal slow decay rates, under different conditions. Section 5 is devoted to discuss some more general slow decay rates for system (5). Especially, for the special case
The results in this paper contribute to the understanding of the decay properties of wave and thermoelastic wave networks, a topic in which important issues are still to be understood.
There has been an extensive literature on other closely related issues such as the large time behaviour and controllability properties of elastic networks with node and boundary feedback controls. We refer, among others (the present list of references is by no means complete), to Lagnese et al.[21] for the modelling and control of elastic networks; Ammari et al. [2], [3] and [4] and Nicaise et al. [32] for stabilisation problems on networks of wave and Euler-Bernoulli beams with star-shaped and tree-shaped configurations; Dáger and Zuazua [8], [9] and [10] for boundary controllability of wave networks; Xu et al. [16], [17] and [40] for the stabilisation and spectral properties of the wave networks.
This section is devoted to state the main result of this paper.
As we will see later, system (5) can be rewritten as an abstract Cauchy problem in an appropriate Hilbert space
dU(t)dt=AU(t), t>0; U(0)=U0, | (7) |
where
The problem of whether the energy of solutions tends to zero as time goes to infinity or not has a simple answer:
Theorem 2.1. Operator
1).
2).
We now obtain explicit decay rates for network (5).
It is well known that if all the components in the network are thermoelastic, that is
In fact, in Propositions 1 and 2 in section 4, we obtain the following necessary and sufficient condition for the exponential decay.
Theorem 2.2. The energy of system (5) decays to zero exponentially if and only if
Accordingly, when
Definition 2.3. ([36], [11]) Real numbers
The notation (S) for this condition was introduced in [11] to refer to the fundamental contribution by Schmidt [36], that defined this class of irrational numbers for the simultaneous approximation by rational ones. It should be noted that the condition (S) denotes a narrow class of irrationals, since the set of algebraic numbers is countable and has Lebesgue measure zero.
By a detailed frequency domain analysis, we have the following explicit polynomial decay rate for system (5).
Theorem 2.4. When
lim inft→∞tE(t)>0. | (8) |
Thus, we can not expect a decay rate which is beyond first order polynomial. Furthermore, if
E(t)≤Cϵt−11+ϵ‖(u(0),u(1),θ(0))‖2D(A),∀t≥0, | (9) |
for all
Remark 2. Using the method of proof of Theorem 2.4, we can obtain more general slow decay rates (polynomial, logarithmic or arbitrarily slow decay), which will be presented in section 5. Especially, for a very special case that there are two purely elastic rods involved in the network (
Remark 3. The optimality result in (8) is well-known for wave-like equations with velocity damping in the case of one single string with damping on an internal point, which is equivalent to a simple star-like network constituted only by two strings ([20], [11]).
Here we show the same lower bound on the decay rate for the more general system involving thermoelastic rods. However, for general cases that more than two purely elastic rods entering in the networks, it is still an open problem to find the condition to guarantee achieving a sharp polynomial decay rate.
This section is devoted to show the well-posedness of network (5) by the semigroup theory and prove Theorem 2.1, with a necessary and sufficient condition for strong stability of this system.
Let us first introduce an appropriate Hilbert space setting for the well-posedness of the system.
For
L2(Rn)={u|uj∈L2(0,ℓj),∀j=1,2,⋯,n}, |
Vn:={ϕ∈n∏j=1H1(0,ℓj)|ϕj(0)=ϕk(0),ϕj(ℓj)=0,∀k,j=1,2,⋯,n}. |
Set the state space
H=VN×L2(RN)×L2(RN1), |
equipped with inner product:
(W,˜W)H=N∑j=1∫ℓj0uj,x¯˜uj,xdx+N∑j=1∫ℓj0wj¯˜wjdx+N1∑k=1αkβk∫ℓk0θk¯˜θkdx, |
for
Then, define the system operator
A(uwθ)=(wuxx−αIN×N1θxθxx−βw1,x)=(0I0∂xx0−αIN×N1∂x0−βITN×N1∂x∂xx)[uwθ] |
where
D(A)={(u,w,θ)∈[VN∩∏Nj=1H2(0,ℓj)]×VN×∏N1j=1H2(0,ℓj)|N∑j=1uj,x(0)=N1∑k=1αjθj(0)θj(ℓj)=0,j=1,2,⋯,N1θj(0)=θk(0),j,k=1,2,⋯,N1N1∑j=1αjβjθj,x(0)=0}. |
Thus, system (5) can be rewritten as the evolution equation (7) in
It is easy to check that
Now, we focus on proving the strong stability property of the system in Theorem 2.1. The proof by contradiction is mainly used here.
Proof of Theorem 2.1.
Sufficiency. If the strong stability of system (5) does not hold, then by the Lyubich-Phóng strong stability theorem (see [27]), there exists at least one
˜W=((uj)Nj=1,˜λ(uj)Nj=1,(θk)N1k=1)T∈D(A) |
is an eigenvector of
0=ℜ˜λ‖˜W‖2H=ℜ(A˜W,˜W)H=−N1∑k=1αkβk∫ℓk0θ2k,xdx, |
which yields
{˜λ2uj(x)−uj,xx(x)=0,x∈(0,ℓj),j=1,2,⋯,N,βj˜λuj,x(x)=0,x∈(0,ℓj),j=1,2,⋯,N1,uj(ℓj)=0,j=1,2,⋯,N,uj(0)=uk(0),∀j,k=1,2,3,⋯,N,N∑j=1uj,x(0)=0. | (10) |
If
If
{uk=0,k=1,2,⋯,N1,uj=cjsinh˜λx,j=N1+1,N1+2,⋯,N, |
which satisfy
cjsinh˜λℓj=0,j=N1+1,N1+2,⋯,N, and N∑j=N1+1cj=0. |
Since
cj1,cj2≠0. |
Hence,
Necessity. If there exist
ˆui0(x)=sin(pπx/ℓi0),ˆuj0(x)=−sin(q0πx/ℓj0), |
ˆuj(x)=0,j=1,2,⋯,N,j≠i0,j0. |
This contradicts the strong stability property of system (5). Therefore,
Remark 4. For the proof of "Sufficiency", we also can use the unique continuation property for wave networks in Dager and Zuazua [11] (See Corollary 5.28, p.135). Indeed, (10), in the absence of thermal components, corresponds to the eigenproblem associated with the pure wave system and, according to the results in [11], its unique solution is the trivial one, which contradicts that
This section is devoted to achieve explicit decay rates of the total energy of solutions of system (5). The exponential decay and slow decay rates are deduced under different assumptions of the various components of the network.
In this subsection, we analyse the decay rate of network (5) when
Lemma 4.1. Let
iR⊂ρ(A) | (11) |
and
‖(iσI−A)−1‖H≤C,∀σ∈R. | (12) |
We are then in conditions to prove the following proposition, which is one of the main statements in Theorem 2.1:
Proposition 1. When
Proof. It is sufficient to show that the conditions in Lemma 4.1 are fulfilled. It should be noted that although the idea of this proof is similar to the one in [37], some different multipliers are employed to get certain estimates so as to deal with the transmission conditions in the present paper.
By the argument of the proof of Theorem 2.1, it is easy to see that, in the present case, there is no eigenvalue of
If condition (12) is not fulfilled, then there exists a sequence of real numbers
TnF=(iσnI−A)−1F=˜Ψn→∞,in H,n→∞. |
Thus,
(iσnI−A)˜Ψn‖˜Ψn‖H=F‖˜Ψn‖H→0,in H,n→∞. |
So there exists a sequence
limn→∞‖(iσnI−A)Φn‖H=0, |
namely,
iσnunj−vnj→0, inH1(0,ℓj),j=1,2,⋯,N, | (13) |
iσnvnj−unj,xx+αjθnj,x→0, inL2(0,ℓj),j=1,2,⋯,N−1, | (14) |
iσnθnj−θnj,xx+βjvnj,x→0, inL2(0,ℓj),j=1,2,⋯,N−1, | (15) |
iσnvnN−unN,xx→0, inL2(0,ℓN). | (16) |
Note that
θnj,x→0,inL2(0,ℓj),j=1,2,⋯,N−1 | (17) |
and hence by Poincaré inequality, we get
θnj(0)→0,j=1,2,⋯,N1. | (18) |
Removing
iσnvnj−unj,xx→0,inL2(0,ℓj),j=1,2,⋯,N−1. | (19) |
and taking the inner product of (19) with
(−σ2nunj,xunj,x)−(unj,xx,xunj,x)→0,j=1,2,⋯,N−1. |
Note that
2ℜ(−σ2nunj,xunj,x)=−σ2nunj(ℓj)¯ℓjunj(ℓj)−(−σ2nunj,unj)=−(−σ2nunj,unj), |
2ℜ(unj,xx,xunj,x)=unj,x(ℓj)¯ℓjunj,x(ℓj)−(unj,x,unj,x). |
Hence,
(−σ2nunj,(x−ℓj)unj,x)−(unj,xx,(x−ℓj)unj,x)→0,j=1,2,⋯,N−1. |
Hence,
2ℜ(−σ2nunj,(x−ℓj)unj,x)−2ℜ(unj,xx,(x−ℓj)unj,x)=−σ2nunj(0)¯ℓjunj(0)−(−σ2nunj,unj)−unj,x(0)¯ℓjunj,x(0)+(unj,x,unj,x). | (20) |
Thus,
Dividing (15) by
(−θnj,xxiσn,unj,x)+(βjunj,x,unj,x)→0,j=1,2,⋯,N−1. | (21) |
Integrating the above by parts, we have
−θnj,x(ℓj)iσn¯unj,x(ℓj)+θnj,x(0)iσn¯unj,x(0)+(θnj,xiσn,unj,xx)+(βjunj,x,unj,x)→0,j=1,2,⋯,N−1. | (22) |
Note that dividing (14) by
(θnj,xiσn,unj,xx)→0,j=1,2,⋯,N−1. |
By the Gagliardo-Nirenberg inequality,
‖θnj,x‖L∞√iσn≤d1‖θnj,xxiσn‖12‖θnj,x‖12+d2‖θnj,x‖√iσn→0, | (23) |
we have
unj,x→0, inL2(0,ℓj), j=1,2,⋯,N−1. | (24) |
Thus,
(vnj,vnj)−(unj,xxiσn,vnj)→0,j=1,2,⋯,N−1, |
and hence
Integrating by parts, we have
(vnj,vnj)−unj,x(0)unj(0)−(unj,x,unj,x)→0,j=1,2,⋯,N−1. | (25) |
Note that
vnj→0,inL2(0,ℓj),j=1,2,⋯,N−1. | (26) |
Then from (20), we have
unj,x(0),σnunj(0)→0,j=1,2,⋯,N−1. | (27) |
On the other hand, on the segment
Taking the inner product of (16) with
(iσnunN,(x−ℓN)unN,x)−(unN,xx,(x−ℓN)unN,x)→0. | (28) |
Note that
2ℜ(iσnunN,(x−ℓN)unN,x)=−iσnunN(0)¯ℓNwnN(0)+(vnN,iσnunN)→(vnN,vnN), |
2ℜ(unN,xx,(x−ℓN)unN,x)=−unN,x(0)¯ℓNunN,x(0)−(unN,x,unN,x)→−(unN,x,unN,x). |
Thus,
ℜ(iσnvnN,(x−ℓN)unN,x)−ℜ(wnj,xx,(x−ℓN)unN,x)=(vnN,vnN)+(unN,x,unN,x)→0. | (29) |
Hence,
unN,x,vnN→0,inL2(0,ℓN). | (30) |
Thus, by (17), (24) and (26), we get
In this subsection, we shall show that if more than one purely elastic undamped rod is involved in the network, the exponential decay rate does not hold.
Proposition 2. If
Proof. Note that from Lemma 4.1, it is sufficient to show that the norm of the resolvent operator of system (5) along the imaginary axis is necessarily unbounded when
To do it we consider the resolvent problem
(λI−A)U=F,λ=−iσ,σ∈R, | (31) |
where
{λuj−vj=fj,j=1,2,⋯,N,λvj−(uj,xx−αjθj,x)=gj,j=1,2⋯,N1,λθj−(θj,xx−βjvj,x)=ηj,j=1,2⋯,N1,λvj−uj,xx=gj,j=N1+1,N1+2,⋯,N. | (32) |
Choose
{λuj−vj=0,j=1,2,⋯,N,λvj−(uj,xx−αjθj,x)=0,j=1,2⋯,N1,λθj−(θj,xx−βjvj,x)=0,j=1,2⋯,N1,λvj−uj,xx=gj,j=N1+1,N1+2,⋯,N. | (33) |
Using explicit representation formulas we get
uj(x)=u(0)sinσℓjsinσ(ℓj−x)+sinσ(ℓj−x)σsinσℓj∫ℓj0gj(ℓj−s)sinσ(ℓj−s)ds −1σ∫ℓj−x0gj(ℓj−s)sinσ(ℓj−x−s)ds,j=N1+1,N1+2,⋯,N. | (34) |
In order to calculate
{λuj−vj=0,j=1,2⋯,N1,λvj−(uj,xx−αjθj,x)=0,j=1,2⋯,N1,λθj−(θj,xx−βjvj,x)=0,j=1,2⋯,N1, |
that is,
{λ2uj−(uj,xx−αjθj,x)=0,j=1,2⋯,N1,λθj−(θj,xx−βjλuj,x)=0,j=1,2⋯,N1. | (35) |
We rewrite (35) in the vector form
dYj(x)dx=AjYj(x),j=1,2,⋯,N1, |
where
Set
Pj=[11˜bj−˜bj√dj,1λ−√dj,1λ˜bj√dj,2λ˜bj√dj,2λ˜aj−˜aj11˜aj√dj,1λ˜aj√dj,1λ√dj,2λ−√dj,2λ] |
and
˜aj=1αj√dj,1(−λ2+dj,1),˜bj=1β√dj,2(−1+dj,2λ). |
We then have
In this way we obtain the following system
Zj(x)=[e√dj,10000e−√dj,10000e√dj,20000e−√dj,2]Z(0), |
and
Yj(x)=PjZj(x)=Pj[e√dj,10000e−√dj,10000e√dj,20000e−√dj,2]P−1jYj(0). | (36) |
By the boundary and transmission conditions in (5), together with (34), we get the following estimate (the technical details are given as in Appendix):
σu(0)=N∑j=N1+1N∏k≠jk=N1+1sinσℓkcosσℓj∫ℓj0gj(ℓj−s)sinσ(ℓj−s)dsN∏k=N1+1sinσℓkN1∑j=1(icosh(iσ+αjβj2)ℓjsinh(iσ+αjβj2)ℓj+O(1√σ))−N∑j=N1+1N∏k≠jk=N1+1sinσℓkcosσℓj−N∑j=N1+1N∏k=N1+1sinσℓk∫ℓj0gj(ℓj−s)cosσ(ℓj−s)dsN∏k=N1+1sinσℓkN1∑j=1(icosh(iσ+αjβj2)ℓjsinh(iσ+αjβj2)ℓj+O(1√σ))−N∑j=N1+1N∏k≠jk=N1+1sinσℓkcosσℓj. | (37) |
Choosing
sinσnkℓj→0,j=N1+2,N1+3,⋯,N. |
In fact, note that there is always a sequence of rational numbers
sinσnkℓj=sin(σnkℓN1+1ℓjℓN1+1)=sin(π(N∏m=N1+2qkm)ℓjℓN1+1) →sin(π(N∏m=N1+2qkm)pkjqkj)=0,k→∞,j=N1+2,N1+3,⋯,N. |
Set
∫ℓN1+10gnkN1+1(ℓj−s)sinσnk(ℓj−s)ds→−ℓN1+12,nk→∞, |
∫ℓN1+10gnkN1+1(ℓj−s)cosσnk(ℓj−s)ds→0,nk→∞. |
Thus, from (37), we get
σnkunk(0)→ℓN1+12,nk→∞. |
Hence, by (34), we have
iσnkunkj(x)→∞,inL2(0,ℓj),j=N1+2,N1+3,⋯,N, |
due to the fact that
vnkj→∞,inL2(0,ℓj),j=N1+2,N1+3,⋯,N. |
Summarising the developments above we find a sequence
‖(iσnI−A)−1Fn‖H→∞,σn→+∞, |
which implies the lack of exponential decay rate of system (5). The proof is complete.
This subsection is devoted to get the lower bounds on the polynomial decay rate as stated in Theorem 2.4.
We need the following result from [6] (see also [26]).
Lemma 4.2. A
‖etAU0‖≤Ct−1ℓ‖U0‖D(A),∀U0∈D(A),t→∞ |
for some constant
Proposition 3. The decay rate of the energy of system (5) can be, at most, polynomial of order
Proof. From Lemma 4.2, it is sufficient to show that there exists at least one sequence
‖(iσnI−A)−1Fn‖>˜Cσ2n, | (38) |
where
For simplicity, we consider the case
Step 1). Choose
Since
sinσℓN1+1=sinσ(ℓN1+1+ℓN1+2−ℓN1+2) =sinσ(ℓN1+1+ℓN1+2)cosσ(ℓN1+2)−cosσ(ℓN1+1+ℓN1+2)sinσ(ℓN1+2), |
we have
sinσnℓN1+1=±sinσnℓN1+2,cosσnℓN1+1=±cosσnℓN1+2. |
Step 2). Choose
Substituting the above into (37), we get
σnu(0)=±sinσnℓN1+2cosσnℓN1+1∫ℓN1+10sinσnssinσn(ℓN1+1−s)ds(sinσnℓN1+2)2N1∑j=1(icosh(iσn+αjβj2)ℓjsinh(iσn+αjβj2)ℓj+O(1√σn))±(sinσnℓN1+2)2∫ℓN1+10sinσnscosσn(ℓN1+1−s)ds(sinσnℓN1+2)2N1∑j=1(icosh(iσn+αjβj2)ℓjsinh(iσn+αjβj2)ℓj+O(1√σn)). | (39) |
Note that
∫ℓN1+10sinσnssinσn(ℓN1+1−s)ds→−ℓN1+1cosσnℓN1+12,n→∞, |
∫ℓN1+10sinσnscosσnk(ℓj−s)ds→ℓN1+1sinσnℓN1+12,n→∞, |
and
N1∑j=1(icosh(iσn+αjβj2)ℓjsinh(iσn+αjβj2)ℓj+O(1√σn))=O(1)∉R. |
Hence,
|σnu(0)|=|ℓN1+1cos2σnℓN1+22sinσnℓN1+2±ℓN1+1sinσnℓN1+22|O(1). | (40) |
Step 3). Note that we have chosen in Step 1) that
sinσnℓN1+2=sin(ℓN1+2ℓN1+1+ℓN1+2(ℓN1+1+ℓN1+2)σn)=sin(ℓN1+2ℓN1+1+ℓN1+2nπ). |
By Dirichlet theorem, we can always find an infinite subsequence
|||ℓN1+2ℓN1+1+ℓN1+2nk|||<1nk, |
and hence
Thus, there exists a positive constant
|σnku(0)|>Cnk. | (41) |
Note that from (33) and (34), we get
uN1+2(x)=u(0)sinσℓN1+2sinσ(ℓN1+2−x)+sinσ(ℓN1+2−x)σsinσℓN1+2∫ℓN1+20gN1+2(ℓN1+2−s)sinσ(ℓN1+2−s)ds−1σ∫ℓN1+2−x0gN1+2(ℓN1+2−s)sinσ(ℓN1+2−x−s)ds. |
Hence, by (41), there exists a positive constant
‖vnkN1+2‖>ˆCn2k. |
Summarizing above, we have found a sequence
σnk=1ℓN1+1+ℓN1+2nkπ,Fnk=F=((fj)Nj=1,(gj)Nj=1,(ηj)N1j=1), |
in which
‖(iσnkI−A)−1Fnk‖>˜Cσ2nk,σnk→∞, |
where
Remark 5. For general case
In subsection 4.2 we proved that if there are more than one purely elastic rods involved in the network (5), the system can not achieve the exponential decay. We have also shown that the decay rate can be, at most, polynomial of order
The mutual-irrationality of the radii of
Now, we prove effective and explicit polynomial decay results when
Proof of Theorem 2.4. Similarly to the proof of Theorem 2.2, we argue by contradiction, on the basis of Lemma 4.2.
Let
limn→∞σ2(1+ϵ)n‖(iσnI−A)Φn‖H=0, |
where
σ2(1+ϵ)n(iσnunj−vnj)=fnj→0,inH1(0,ℓj),j=1,2,⋯,N, | (42) |
σ2(1+ϵ)n(iσnvnj−unj,xx+αjθnj,x)=gnj→0,inL2(0,ℓj),j=1,2,⋯,N1, | (43) |
σ2(1+ϵ)n(iσnθnj−θnj,xx+βjvnj,x)=ynj→0,inL2(0,ℓj),j=1,2,⋯,N1, | (44) |
and
σ2(1+ϵ)n(iσnvnj−unj,xx)=gj→0,inL2(0,ℓj),j=N1+1,N1+2,⋯,N. | (45) |
Note that
Hence,
σ(1+ϵ)nθnj,x→0,inL2(0,ℓj),j=1,2,⋯,N1. | (46) |
Thus, by Poincaré inequality,
σ(1+ϵ)nθnj→0,inL2(0,ℓj),j=1,2,⋯,N1. |
Similar to the proof of Theorem 2.2, we can get that
σ(1+ϵ)nunj,x(x),σ(1+ϵ)nvnj(x),σ(1+ϵ)nθnj(x)→0,inL2(0,ℓj),j=1,2,⋯,N1, | (47) |
σ(1+ϵ)nunj,x(0),σ(1+ϵ)nσnunj(0),σ(1+ϵ)nθnj(0)→0,j=1,2,⋯,N1. | (48) |
Thus, by the transmission conditions at the common node, we get
σ(1+ϵ)nσnunj(0)→0,j=N1+1,N1+2,⋯,N, | (49) |
σ(1+ϵ)nN∑j=N1+1unj,x(0)→0. | (50) |
Then we will prove that
Case 1). Assume that
By (42), (45), we can easily get that
Thus a direct calculation yields
unj(x)=un(0)sinσnℓjsinσn(ℓj−x) +sinσn(ℓj−x)σnsinσnℓj∫ℓj0gnj(ℓj−s)+iσnfnj(ℓj−s)σ2(1+ϵ)nsin(σn(ℓj−s))ds −1σn∫ℓj−x0gnj(ℓj−s)+iσnfnj(ℓj−s)σ2(1+ϵ)nsinσn(ℓj−x−s)ds, j=N1+1,N1+2,⋯,N. |
Hence,
unj,x(0)=−σnun(0)cosσnℓjsinσnℓj −cosσnℓjsinσnℓj∫ℓj0gnj(ℓj−s)+iσnfnj(ℓj−s)σ2(1+ϵ)nsin(σn(ℓj−s))ds +∫ℓj0gnj(ℓj−s)+iσnfnj(ℓj−s)σ2(1+ϵ)ncosσn(ℓj−s)ds, j=N1+1,N1+2,⋯,N. | (51) |
It is easy to show that
∫ℓj0[gnj(ℓj−s)+iσnfnj(ℓj−s)]sinσn(ℓj−s)ds→0,∫ℓj0[gnj(ℓj−s)+iσnfnj(ℓj−s)]cosσn(ℓj−s)ds→0. | (52) |
Set
γn=maxj=N1+1,N1+2,⋯,NN∏i=N1+1,i≠j|sin(σnℓi)|,∀n≥1. | (53) |
When
γn≥cϵσ1+ϵn,∀ϵ>0. | (54) |
For any given sequence
γn=N∏i=N1+1,i≠jn0|sin(σnℓi)|. |
Hence, for any
|sinσnℓj|≥γn, |
which implies that
1|sinσnℓj|≤1γn≤1cϵσ1+ϵn. |
Therefore, by (49) and (52) together with (51), we get that
unj,x(0)→0,N1+1≤j≤N,j≠jn0,n→∞. | (55) |
Then by the transmission condition (50), we obtain that
unj,x(0)→0,N1+1≤j≤N,n→∞. | (56) |
Taking the inner product of (45) with
(iσnvnj,(x−ℓj)unj,x)−(unj,xx,(x−ℓj)unj,x)→0,j=N1+1,N1+2,⋯,N. | (57) |
We have
2ℜ(iσnvnj,(x−ℓj)unj,x)=iσnvnj(0)¯ℓjunj(0)+(vnj,iσnunj)→(vnj,vnj), |
2ℜ(unj,xx,(x−ℓj)unj,x)=unj,x(0)¯ℓjunj,x(0)−(unj,x,unj,x)→−(unj,x,unj,x). |
Hence,
ℜ(iσnvnj,(x−ℓj)unj,x)−ℜ(unj,xx,(x−ℓj)unj,x)=(vnj,vnj)+(unj,x,unj,x)→0. | (58) |
Therefore,
unj,x,vnj→0,inL2(0,ℓj),j=N1+1,N1+2,⋯,N. | (59) |
Thus, by (47) and (59), we have obtained
Φn=((unj)Nj=1,(vnj)Nj=1,(θnj)N1j=1)→0,inH,n→∞, |
which contradicts
Now let us continue to consider the second step:
Case 2). Assume that there exists
Note that there exists at most one
γn=N∏i=N1+1,i≠jn0|sin(σnℓi)|, |
where
unj,x(0)→0,N1+1≤j≤N,j≠jn0,n→∞, |
which together with the transmission condition (50), implies
unj,x(0)→0,N1+1≤j≤N,n→∞. |
Then by (57)-(59) in
unj,x,vnj→0,inL2(0,ℓj),j=N1+1,N1+2,⋯,N. |
Hence, the same contradiction holds as in
Therefore, by Lemma 4.2, we get the polynomial decay rate of system (5), that is
E(t)≤Cϵt−11+ϵ‖(u(0),u(1),θ(0))‖2D(A),∀t≥0. |
Note that in Proposition 3, we have proved that the polynomial decay order of the energy of system (5) is at most
In the last section, we have shown that, when
In fact, the discussion of the proof of Theorem 2.4 (see subsection 4.3), shows that the slow decay rate of the system is determined by
To do this, let us introduce the following definition on the irrational sets (see p. 209 in [11]), which is deduced from [7] (see Theorem Ⅰ, p. 120).
Definition 5.1. (Theorem [7], [11]) 1. Set
2. Set
We have the following result:
Corollary 1. For any
E(t)≤C˜st−1˜s‖(u(0),u(1),θ(0))‖2D(A),∀t≥0, | (60) |
where
Proof. Since the proof is similar to the one of Theorem 2.4, we only give a sketch of it.
If the decay rate is not fulfilled, there exists a sequence
limn→∞σ2˜sn‖(iσnI−A)Φn‖H=0, |
where
By Diophantine approximation (see [11]), different estimates can be gotten for
(1) if
(2) if
Thus, proceeding as in the discussion of the proof of Theorem 2.4, finally we can get
Φn→0,inH,n→∞. |
This contradicts
Remark 6. By Corollary 1, it is easy to see that when
More generally, by the similar proof as the one for Theorem 2.4, together with the so called
Corollary 2. Set
If
γn≥1√M(σn),n>0, |
where
γn=maxj=N1+1,N1+2,⋯,NN∏i=N1+1,i≠j|sin(σnℓi)|,∀σn→∞, |
then
E(t)≤C1(M−1log(ct))2‖(u(0),u(1),θ(0))‖2D(A),∀t≥0, | (61) |
where
Especially, if
Proof. We can still use the proof of Theorem 2.4 to derive
lim sup|σ|→∞1(M(σ))‖(iσ−A)−1‖<∞, |
which together with
Remark 7. From Corollary 2, we see that in order to obtain an explicit decay rate, it is very important to estimate the lower bound of
This section is devoted to present some numerical simulations on the dynamical behaviour of system (5) to support the results obtained above.
The backward Euler method in time (time step:
For simplicity, we assume that the star-shaped network consists of three edges, the lengths of which are given as
ℓ1=1,ℓ2=2,ℓ3=1. |
The following cases are considered:
Case A.
In this case, all the three edges of the network are constituted by thermoelastic ones. First, choose the initial conditions as follows:
{u1(x,0)=5sin(πx),u2(x,0)=5sin(πxℓ2),u3(x,0)=−5sin(πx),u1,t(x,0)=9sin(πx),u2,t(x,0)=4sin(πxℓ2),u3,t(x,0)=−9sin(πx),θ1(x,0)=3sin(12πx+π2),θ2(x,0)=3sin(π2ℓ2x+π2),θ3(x,0)=3sin(12πx+π2). | (62) |
and the parameters in system (5):
α1=1,β1=1,α2=2,β2=1,α1=3,β3=1. | (63) |
The dynamical behaviour of
Case B.
In this case, the network consists of purely elastic rods, that is, no thermal-damping dissipates energy. The initial conditions are the same as in Case A. Solutions are plotted in Figure B-1, 2, 3, which confirms the conservative character of the system.
Case C.
In this case, the network is constituted by two thermoelastic rods and one purely elastic one. The initial conditions and parameters in system (5) are chosen as (62) and (63). We get the dynamical behaviour in Figure C-1, 2, 3, 4, 5.
In these figures, we can see that the behaviour of each
Case D.
In this case, the network consists of one thermoelastic rod and two purely elastic ones. The lengths of rods are still given as
Case E.
In this case, the network is still constituted by one thermoelastic rod and two purely elastic ones. This time the length
In Figure E-1, 2, 3, 4, we observe a very slow decay rate. It implies that the energy of this system decays to zero but lacks exponential growth rate.
Moreover, we presented Figure F-1, 2 to compare the decay rate of the energy for each case. Figure F-1 shows the dynamical behaviours of the logarithmic scale of the energy for Case A, B and C. Figure F-2 shows the dynamical behaviours of the energies for Case D and E. From these two figures, we can clearly see the behaviours of energies for each case respect time, which are consistent with our theoretical results obtained in this paper.
This appendix is devoted to show how to get (37).
Note that
Thus, by (36), for
uj(x)=u(0)−√dj,2+˜aj˜bj√dj,1[−√dj,2cosh√dj,1x+˜aj˜bj√dj,1cosh√dj,2x] +uj,x(0)√dj,1−˜aj˜bj√dj,2[sinh√dj,1x−˜aj˜bjsinh√dj,2x] +θ(0)√dj,1−˜aj˜bj√dj,2[−˜bj√dj,2sinh√dj,1x+˜bj√dj,1sinh√dj,2x] +θj,x(0)−√dj,2+˜aj˜bj√dj,1[˜bjcosh√dj,1x−˜bjcosh√dj,2x],θj(x)=u(0)−√dj,2+˜aj˜bj√dj,1[−√dj,2˜ajsinh√dj,1x+˜aj√dj,1sinh√dj,2x] +uj,x(0)√dj,1−˜aj˜bj√dj,2[˜ajcosh√dj,1x−˜ajcosh√dj,2x] +θ(0)√dj,1−˜aj˜bj√dj,2[−˜aj˜bj√dj,2cosh√dj,1x+√dj,1cosh√dj,2x] +θj,x(0)−√dj,2+˜aj˜bj√dj,1[˜aj˜bjsinh√dj,1x−sinh√dj,2x]. |
Thus, by the boundary condition
0=u(0)−√dj,2+˜aj˜bj√dj,1[−√dj,2cosh√dj,1ℓj+˜aj˜bj√dj,1cosh√dj,2ℓj] +uj,x(0)√dj,1−˜aj˜bj√dj,2[sinh√dj,1ℓj−˜aj˜bjsinh√dj,2ℓj] +θ(0)√dj,1−˜aj˜bj√dj,2[−˜bj√dj,2sinh√dj,1ℓj+˜bj√dj,1sinh√dj,2ℓj] +θj,x(0)−√dj,2+˜aj˜bj√dj,1[˜bjcosh√dj,1ℓj−˜bjcosh√dj,2ℓj],0=u(0)−√dj,2+˜aj˜bj√dj,1[−√dj,2˜ajsinh√dj,1ℓj+˜aj√dj,1sinh√dj,2ℓj] +uj,x(0)√dj,1−˜aj˜bj√dj,2[˜ajcosh√dj,1ℓj−˜ajcosh√dj,2ℓj] +θ(0)√dj,1−˜aj˜bj√dj,2[−˜aj˜bj√dj,2cosh√dj,1ℓj+√dj,1cosh√dj,2ℓj] +θj,x(0)−√dj,2+˜aj˜bj√dj,1[˜aj˜bjsinh√dj,1ℓj−sinh√dj,2ℓj]. |
Then a direct calculation yields, for
(uj,x(0)θj,x(0))=(Cj11Cj12Cj21Cj22)(u(0)θ(0)), | (A.1) |
where
Cj11=[(√dj,1−˜aj˜bj√dj,2)(−˜aj˜bjsinh√dj,1ℓjcosh√dj,2ℓj+cosh√dj,1ℓjsinh√dj,2ℓj)]/Dj,Cj12=[−˜aj˜b2j√dj,2−˜bj√dj,1+˜bj√dj,1(−˜aj˜bjsinh√dj,1ℓjsinh√dj,2ℓj+cosh√dj,1ℓjcosh√dj,2ℓj) +˜bj√dj,2(−sinh√dj,1ℓjsinh√dj,2ℓj+˜aj˜bjcosh√dj,1ℓjcosh√dj,2ℓj)]/Dj,Cj21=[−˜a2j˜bj√dj,1−˜aj√dj,2+˜aj√dj,1(˜aj˜bjcosh√dj,1ℓjcosh√dj,2ℓj−sinh√dj,1ℓjsinh√dj,2ℓj) +˜aj√dj,2(cosh√dj,1ℓjcosh√dj,2ℓj−˜aj˜bjsinh√dj,1ℓjsinh√dj,2ℓj)]/Dj,Cj22=[(−√dj,2+˜aj˜bj√dj,1)(˜aj˜bjcosh√dj,1ℓjsinh√dj,2ℓj−sinh√dj,1ℓjcosh√dj,2ℓj)]/Dj, |
and
Dj=−2˜aj˜bj−(1+˜a2j˜b2j)sinh√dj,1ℓjsinh√dj,2ℓj+2˜aj˜bjcosh√dj,1ℓjcosh√dj,2ℓj. |
Then by the transmission condition
N1∑j=1αjβj(Cj21u(0)+Cj22θ(0))=0, | (A.2) |
which implies that
θ(0)=−u(0)N1∑j=1αjβjCj21N1∑j=1αjβjCj22. |
Hence by (A.1),
uj,x(0)=Cj11u(0)+Cj12θ(0)=u(0)(Cj11−Cj12N1∑j=1αjβjCj21N1∑j=1αjβjCj22),j=1,2,⋯,N1. | (A.3) |
On the other hand, by (34),
uj,x(0)=−σu(0)cosσℓjsinσℓj−cosσℓjsinσℓj∫ℓj0gj(ℓj−s)sinσ(ℓj−s)ds +∫ℓj0gj(ℓj−s)cosσ(ℓj−s)ds,j=N1+1,N1+2,⋯,N. | (A.4) |
Then by the transmission condition
u(0)(N1∑j=1(Cj11−Cj12N1∑j=1αjβjCj21N1∑j=1αjβjCj22+αjN1∑j=1αjβjCj21N1∑j=1αjβjCj22)−N∑j=N1+1σcosσℓjsinσℓj)=N∑j=N1+1cosσℓjsinσℓj∫ℓj0gj(ℓj−s)sinσ(ℓj−s)ds −N∑j=N1+1∫ℓj0gj(ℓj−s)cosσ(ℓj−s)ds. | (A.5) |
Hence,
σu(0)=N∑j=N1+1(cosσℓjsinσℓj∫ℓj0gj(ℓj−s)sinσ(ℓj−s)ds−∫ℓj0gj(ℓj−s)cosσ(ℓj−s)ds)1σN1∑j=1(Cj11−Cj12N1∑j=1αjβjCj21N1∑j=1αjβjCj22+αjN1∑j=1αjβjCj21N1∑j=1αjβjCj22)−N∑j=N1+1cosσℓjsinσℓj. | (A.6) |
As
dj,1=(iσ)2+αjβj(iσ)+αjβj+O(1σ),dj,2=iσ−αjβj+O(1σ). |
Hence,
√dj,1=iσ+αjβj2+O(1σ),√dj,2=√iσ−αjβj2√iσ+O(1σ3/2), |
˜aj=βj+O(1σ),˜bj=−αj(iσ)−32+O(1σ5/2). |
and
−√dj,2+˜aj˜bj√dj,1=−√iσ−αjβj√iσ+O(1σ3/2), |
√dj,1−˜aj˜bj√dj,2=iσ+αjβj2+αjβj√iσ+O(1σ). |
Substituting the above into (A.1), we get that
Cj11=(√dj,1−˜aj˜bj√dj,2)cosh√dj,1ℓj(1+˜a2j˜b2j)sinh√dj,1ℓj+O(1σ3/2)=(iσ+αjβj2+αjβj√iσ)cosh√dj,1ℓjsinh√dj,1ℓj+O(1σ), | (A.7) |
Cj12=(˜bj√dj,1)cosh√dj,1ℓj(1+˜a2j˜b2j)sinh√dj,1ℓj+O(1σ3/2)=−αj√iσcosh√dj,1ℓjsinh√dj,1ℓj+O(1σ), | (A.8) |
Cj21=−(˜aj√dj,1)(1+˜a2j˜b2j)+O(1σ3/2)=−iβjσ+αjβ2j2+O(1σ), | (A.9) |
Cj22=−(−√dj,2+˜aj˜bj√dj,1)(1+˜a2j˜b2j)+O(1σ3/2)=√iσ+αjβj√iσ+O(1σ). | (A.10) |
Then by (A.6),
σu(0)=N∑j=N1+1(cosσℓjsinσℓj∫ℓj0gj(ℓj−s)sinσ(ℓj−s)ds−∫ℓj0gj(ℓj−s)cosσ(ℓj−s)ds)N1∑j=1(icosh(iσ+αjβj2)ℓjsinh(iσ+αjβj2)ℓj+O(1√σ))−N∑j=N1+1cosσℓjsinσℓj=N∑j=N1+1(cosσℓjsinσℓj∫ℓj0gj(ℓj−s)sinσ(ℓj−s)ds−∫ℓj0gj(ℓj−s)cosσ(ℓj−s)ds)N1∑j=1(icosh(iσ+αjβj2)ℓjsinh(iσ+αjβj2)ℓj+O(1√σ))−N∑j=N1+1cosσℓjsinσℓj=N∑j=N1+1N∏k≠jk=N1+1sinσℓkcosσℓj∫ℓj0gj(ℓj−s)sinσ(ℓj−s)dsN∏k=N1+1sinσℓkN1∑j=1(icosh(iσ+αjβj2)ℓjsinh(iσ+αjβj2)ℓj+O(1√σ))−N∑j=N1+1N∏k≠jk=N1+1sinσℓkcosσℓj −N∑j=N1+1N∏k=N1+1sinσℓk∫ℓj0gj(ℓj−s)cosσ(ℓj−s)dsN∏k=N1+1sinσℓkN1∑j=1(icosh(iσ+αjβj2)ℓjsinh(iσ+αjβj2)ℓj+O(1√σ))−N∑j=N1+1N∏k≠jk=N1+1sinσℓkcosσℓj. |
Thus, (37) has been obtained.
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