Research article

The effect of modified Ohm’s and Fourier’s laws in generalized magneto-thermo viscoelastic spherical region

  • This paper is dealing the modified Ohm’s law, including the temperature gradient and charge thickness impacts, and the generalized Fourier’s law, including the current density impact, the conditions of generalized thermo-viscoelasticity for a thermally, isotropic and electrically leading unbounded body with a spherical cavity is given. The detailing is applied to the generalized thermo elasticity dependent on Green–Naghdi (G-N II) and (G-N III) theory, where there is an underlying magnetic field corresponding to the plane limit, because of the utilization of the magnetic field, it results an incited magnetic and electric fields in the medium. The state space investigation is applied to acquire the temperature, displacement, stresses, induced electric field, instigated magnetic field and current density. Application is utilized to our concern to get the arrangement in the total structure. The considered variables are introduced graphically and discussions are made.

    Citation: Alaa. K. Khamis, Allal Bakali, A. A. El-Bary, Haitham. M. Atef. The effect of modified Ohm’s and Fourier’s laws in generalized magneto-thermo viscoelastic spherical region[J]. AIMS Materials Science, 2020, 7(4): 381-398. doi: 10.3934/matersci.2020.4.381

    Related Papers:

    [1] Shishi Wang, Yuting Ding, Hongfan Lu, Silin Gong . Stability and bifurcation analysis of SIQR for the COVID-19 epidemic model with time delay. Mathematical Biosciences and Engineering, 2021, 18(5): 5505-5524. doi: 10.3934/mbe.2021278
    [2] Hongfan Lu, Yuting Ding, Silin Gong, Shishi Wang . Mathematical modeling and dynamic analysis of SIQR model with delay for pandemic COVID-19. Mathematical Biosciences and Engineering, 2021, 18(4): 3197-3214. doi: 10.3934/mbe.2021159
    [3] Sarita Bugalia, Jai Prakash Tripathi, Hao Wang . Mathematical modeling of intervention and low medical resource availability with delays: Applications to COVID-19 outbreaks in Spain and Italy. Mathematical Biosciences and Engineering, 2021, 18(5): 5865-5920. doi: 10.3934/mbe.2021295
    [4] Honghua Bin, Daifeng Duan, Junjie Wei . Bifurcation analysis of a reaction-diffusion-advection predator-prey system with delay. Mathematical Biosciences and Engineering, 2023, 20(7): 12194-12210. doi: 10.3934/mbe.2023543
    [5] Yuting Ding, Gaoyang Liu, Yong An . Stability and bifurcation analysis of a tumor-immune system with two delays and diffusion. Mathematical Biosciences and Engineering, 2022, 19(2): 1154-1173. doi: 10.3934/mbe.2022053
    [6] Fang Liu, Yanfei Du . Spatiotemporal dynamics of a diffusive predator-prey model with delay and Allee effect in predator. Mathematical Biosciences and Engineering, 2023, 20(11): 19372-19400. doi: 10.3934/mbe.2023857
    [7] Xin-You Meng, Yu-Qian Wu . Bifurcation analysis in a singular Beddington-DeAngelis predator-prey model with two delays and nonlinear predator harvesting. Mathematical Biosciences and Engineering, 2019, 16(4): 2668-2696. doi: 10.3934/mbe.2019133
    [8] Jinhu Xu, Yicang Zhou . Bifurcation analysis of HIV-1 infection model with cell-to-cell transmission and immune response delay. Mathematical Biosciences and Engineering, 2016, 13(2): 343-367. doi: 10.3934/mbe.2015006
    [9] Xinyu Liu, Zimeng Lv, Yuting Ding . Mathematical modeling and stability analysis of the time-delayed SAIM model for COVID-19 vaccination and media coverage. Mathematical Biosciences and Engineering, 2022, 19(6): 6296-6316. doi: 10.3934/mbe.2022294
    [10] Zuolin Shen, Junjie Wei . Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect. Mathematical Biosciences and Engineering, 2018, 15(3): 693-715. doi: 10.3934/mbe.2018031
  • This paper is dealing the modified Ohm’s law, including the temperature gradient and charge thickness impacts, and the generalized Fourier’s law, including the current density impact, the conditions of generalized thermo-viscoelasticity for a thermally, isotropic and electrically leading unbounded body with a spherical cavity is given. The detailing is applied to the generalized thermo elasticity dependent on Green–Naghdi (G-N II) and (G-N III) theory, where there is an underlying magnetic field corresponding to the plane limit, because of the utilization of the magnetic field, it results an incited magnetic and electric fields in the medium. The state space investigation is applied to acquire the temperature, displacement, stresses, induced electric field, instigated magnetic field and current density. Application is utilized to our concern to get the arrangement in the total structure. The considered variables are introduced graphically and discussions are made.


    Epidemic refers to the disease that can spread in a certain scale within a relatively short period of time. It caused a major hazard to human health. For example, the H1N1 influenza virus, which erupted in Mexico in 2009 and soon spread around the globe, is a dangerous virus and can develop to pneumonia. Some infectives may even appear respiratory failure, multiple organ damage, and finally be kicked to death. The H1N1 has killed about 16 thousand people throughout the world then and over 1.3 million people had been infected. Studying the transmission mechanism of epidemics can help reduce the spread of the diseases. Many researchers use the mathematical models to describe the transmission of epidemics [1], and various methods are derived to investigate the dynamics of the diseases [2,3].

    Researchers have proposed mathematical models of infectious diseases and studied their dynamical behaviours extensively [4,5]. The dynamical behaviours of ordinary differential equations have been widely studied, and abundant dynamical behaviours have been found, such as periodic oscillations, bifurcations, stable limit cycles and time-delay effects [6,7,8,9,10]. To investigate the dynamics of cholera which spread in the European Mediterranean regions, Capasso and Paveri-fontana proposed an epidemic model in 1979 as follows [6]:

    {˙u(t)=α1u(t)+av(t),˙v(t)=α2v(t)+g(u(t)), (1.1)

    where u and v represent the concentration of the infectious agent and the infected population, respectively. 1α1 denotes the average lifetime of the infectious agent, 1α2 denotes the average infectious period of infected people, thus α1, α2>0. a denotes the multiplicative factor of the virus caused by the increasing number of infected people. The function g(u) represents the infectivity of the virus to people, and can be affected by the concentration of the virus. They speculated that the infection process follow the nonlinear saturation pattern. They draw the conclusion of epidemic evolution by analyzing the equations in phase space, which provided a reference for public health policies. The validity of the model is verified by comparing it with available data from Bari town in Italy.

    In fact, system (1.1) can be modified to simulate other epidemics with oral-faecal transmission [6], such as typhoid fever, infectious hepatitis, poliomyelitis, etc. Moreover, system (1.1) can also be used to investigate the spreading of man-environment epidemics [6,11,12]. The man-environment epidemics refer that, the infected people may increase the concentration of the virus in the environment, thus people maybe infected by eating the contaminated food [11].

    Scientific researches show that the virus is highly sensitive to its environment. The concentration of the virus is not evenly distributed in space thus the effect of diffusion cannot be ignored. In 1997, Capasso and Wilson [11] studied the following diffusive epidemic model:

    {ut(x,t)=d1Δu(x,t)α1u(x,t)+av(x,t),x(0,l),t>0,vt(x,t)=α2v(x,t)+g(u),x(0,l),t>0,u(0,t)=u(l,t)=0,t>0,u(x,0)=u0(x),v(x,0)=v0(x),x(0,l), (1.2)

    where u(x,t),v(x,t) represent the spatial concentration of the bacteria and the infected people, respectively. α1, α2 are the same as they are in (1.1). For system (1.2), they performed a detailed analysis of the steady-state bifurcation at the endpoints of a one-dimensional interval based on the monotone techniques. They speculated that the system (1.2) has saddle point structure in the natural function space, similar to the ODE case in which the diffusivity of bacteria is also set to zero.

    In the transmission of the epidemics, an important factor that can not be ignored is the delay [13,14,15]. For example, there is time delay between the release of the virus from the infected person and its successful entry into the susceptible people. For those infected people, it also takes time from being infected to being able to release the virus. All of these can lead to time-delay feedback in the epidemics [16,17,18,19]. Time delay usually destroys the stability of the solutions, devotes to the complex dynamics of the system. Recently, systems with two delays have been used in characterizing the phenomena in various fields, such as power systems and predator-prey systems [20,21,22,23,24], etc.. Among them, Wu and Hsu [24] proposed a monostable epidemic model with double delays as follows:

    {ut(x,t)=d1Δu(x,t)α1u(x,t)+h(v(x,tτ1)),vt(x,t)=d2Δv(x,t)α2v(x,t)+g(u(x,tτ2)). (1.3)

    They studied the existence of entire solutions in the system (1.3) with and without the quasi-monotone condition. Some special cases of system (1.3) have been extensively investigated. For example, when τ1=d2=0, τ2, d1>0, Zhao and Wang [25] studied the properties of the existence and non-existence of the traveling wave fronts; when τ1=τ2=d2=0, d1>0, Xu and Zhao [26] proved the existence, uniqueness and global exponential stability of the bistable traveling wave fronts; Hsu et al. [27] studied the existence and exponential stability of traveling wave solutions in an extended form of system (1.3).

    Motivated by the previous works, we study an epidemic model with two delays subjecting to the Neumann boundary conditions in the following form:

    {u(x,t)t=d1Δu(x,t)α1u(x,t)+S(v(x,tτ1)),x(0,lπ),t>0,v(x,t)t=d2Δv(x,t)α2v(x,t)+T(u(x,tτ2)),x(0,lπ),t>0,u(x,t)x=0,v(x,t)x=0,x=0,lπ;t>0,u(x,t)=u0(x,t)0,v(x,t)=v0(x,t)0,x(0,lπ),t[¯τ,0], (1.4)

    where u(x,t) and v(x,t) denote the concentration of the virus and the number of the infected people, respectively. 1α1 denotes the average lifetime of the infectious agent, 1α2 denotes the average infectious period of infected people, S(v) represents the contribution of infected people to the growth of the virus, T(u) represents the infection rate of the virus to people. d1,d2>0 are the diffusion coefficients, τ1 represents the average time taken by the infected people from being infected to being able to release the virus, τ2 represents the average time taken by the virus from being released to entrying the susceptible people and ¯τ=max{τ1,τ2}.

    In the spread of an epidemic, the concentration of the virus would increase with the number of the infected people, and the susceptible people would more likely to be infected. Therefore, S(v) and T(u) are increasing functions of v and u respectively in general. However, the increasing number of the virus poses a threat to public health, people will take effective measures to avoid them from being infected by the virus, including isolating the infected people, regular disinfection, reminding people to pay attention to hygiene through media reports and so on [28,29]. Scientific studies have shown that these measures are effective in reducing infection rates [30]. Once the virus exceeds a certain concentration u0, even if the concentration continues to increase, people begin to take certain measures to reduce the infectivity of the virus. So in this paper, we assume that once if u>u0, T(u) is a decreasing function with respect to u.

    In this paper, we study the existence of the positive constant steady state. By analyzing the zeros of characteristic equations of (1.4), we investigate the local stability of constant steady state. We found that when the sum of two delays reaches a critical value, the system undergoes Hopf bifurcation. By taking one of two delays as a bifurcation parameter, we calculate the normal form on the center manifold near the Hopf singularity.

    This paper is organized as follows. In section 2, the stability of the positive constant steady state of the system (1.4) and the existence of Hopf bifurcation are established by analyzing the distribution of the eigenvalues. In section 3, the normal form on the center manifold near the Hopf bifurcation point is derived. In section 4, we carry out some numerical simulations to illustrate the theoretic results.

    In this section, we will study the local stability of the positive constant steady state and the existence of Hopf bifurcation by analyzing the distribution of the eigenvalues.

    For S(v) and T(u) in (1.4), we make the following assumption:

    (H1){S,TC3(R,R),S(0)=T(0)=0, there exists a unique positive constantvR,such thatT(S(v)/α1)=α2v.

    Normally speaking, the contribution function S(v) is sufficiently smooth with respect to the infected people v, the infection rate T(u) is sufficiently smooth with respect to u. For the research purposes, we assume S,TC3(R,R). Particularly, if no one is infected (v=0), then no infected people contribute to the growth of the virus, i.e., S(0)=0. Similarly, if the concentration of the virus is zero(u=0), then no one can be infected by the virus, i.e., T(0)=0. In the early stage of a transmission, the virus concentration is low but the number of the susceptible people is large, thus the epidemic would spread at a certain speed; as the epidemic continues to spread, although the virus concentration is high, a reduction in the number of the susceptible people will limit the spread of the epidemic. Thus the virus and the population may reach a coexistence state. In (H1), we denote the number of infected people in this coexistence state as v.

    Define u=S(v)/α1, under the assumption (H1) system (4) has two constant steady states E0(0,0) and E(u,v). Obviously, E is a positive constant steady state. To investigate the dynamics of the epidemics, we focus on the properties of E.

    Sv=Sv|v,Tu=Tu|u,

    the linearized equation of system (1.4) at E is

    t(u(x,t)v(x,t))=(DΔ+A)(u(x,t)v(x,t))+B(u(x,tτ1)v(x,tτ1))+C(u(x,tτ2)v(x,tτ2)), (2.1)

    where

    D=(d100d2),A=(α100α2),B=(0Sv00),C=(00Tu0),

    and u(x,t), v(x,t) still satisfy the homogeneous Neumann boundary conditions.

    The characteristic equations of the system (2.1) are given by

    det(λI2MnABeλτ1Ceλτ2)=0,nN0, (2.2)

    where I2 is a 2 × 2 identity matrix, Mn=n2l2D. Since

    det(λI2MnABeλτ1Ceλτ2)=|λ+d1n2l2+α1Sveλτ1Tueλτ2λ+d2n2l2+α2|,

    denote m=Sv×Tu, Eq (2.2) becomes

    (λ+d1n2l2+α1)(λ+d2n2l2+α2)meλ(τ1+τ2)=0,nN0. (2.3)

    Denote τ=τ1+τ2, then Eq (2.3) becomes

    (λ+d1n2l2+α1)(λ+d2n2l2+α2)meλτ=0,nN0. (2.4)

    For convenience, denote k1=d1n2l2+α1,k2=d2n2l2+α2, then we have k1,k2>0. Now the characteristic Eq (2.4) with τ=0 is in the following form:

    λ2+(k1+k2)λ+k1k2m=0. (2.5)

    To ensure all the roots of Eq (2.5) have negative real parts, we further make the following assumption

    (H2)m<α1α2.

    Then we can conclude the following lemma.

    Lemma 2.1. When τ=0, all the roots of Eq (2.4) have negative real parts under the assumption (H2).

    Since the proof of Lemma 2.1 is intuitive, here we omit it.

    Now we will study the stability of E and investigate the existence of Hopf bifurcation of system (1.4). Assume that λ=iω(ω>0) is a root of (2.4). Substituting iω into (2.4) and separating the real and imaginary parts leads to

    {ω2+k1k2=mcosωτ,(k1+k2)ω=msinωτ. (2.6)

    This yields

    ω4+(k21+k22)ω2+k21k22m2=0.

    Denote z=ω2, then the above equation becomes

    z2+(k21+k22)z+k21k22m2=0, (2.7)

    the roots of (2.7) can be given by

    z1,2=12[(k21+k22)±(k21+k22)24(k21k22m2)].

    Clearly, z2<0. To maintain the existence of a positive ω that satisfies (2.6), z1>0 should hold. Obviously, z1>0 provided that m2>k21k22.

    If m2α21α22, then for all nN0, we have m2(d1n2l2+α1)2(d2n2l2+α2)2, and thus z10, all roots of (2.4) have negative real parts. Moreover, if m<α1α2 and m2α21α22 are satisfied, i.e., α1α2m<α1α2, then E is locally asymptotically stable. We can draw the following proposition.

    Proposition 1. If α1α2m<α1α2, all roots of (2.4) have negative real parts, then E is locally asymptotically stable.

    Now we further make the following assumption:

    (H3){NN0,s.t.m2>(d1n2l2+α1)2(d2n2l2+α2)2for0n<Nandm2(d1N2l2+α1)2(d2N2l2+α2)2.

    Then we study the existence of purely imaginary roots of (2.4) in the following lemma.

    Lemma 2.2. If (H3) is satisfied, then Eq (2.4) with n{0,1,,N1} has a pair of purely imaginary roots ±iωn when τ=τj,n, where

    ωn=12[(d1n2l2+α1)2(d2n2l2+α2)2+[(d1n2l2+α1)2(d2n2l2+α2)2]2+4m2]12,τj,n=1ωn[arccosωn2+(d1n2l2+α1)(d2n2l2+α2)m+2jπ],jN0;n=0,1,,N. (2.8)

    Proof. From (H3) we know that Eq (2.7) with n{0,1,,N1} has a positive root denoted by zn. Hence, ωn=zn>0 makes sense. Define

    τj,n=1ωn[arccosωn2+(d1n2l2+α1)(d2n2l2+α2)m+2jπ],j=0,1,2,.

    From the second equation of (2.6) one obtains that τ0,nωn(0,π]. Then (ωn,τj.n) solves (2.6), which implies that ±iωn is a pair of purely imaginary roots of Eq (2.4) with τ=τj,n.

    Lemma 2.3. If assumptions (H2) and (H3) are satisfied, then ωn is monotone decreasing with respect to n.

    Proof. By Lemma 2.2, we know that ωn exists for n=0,1,N1. Assumptions (H2) and (H3) imply that m<0. We know that

    zn=ωn2=12[(d1n2l2+α1)2(d2n2l2+α2)2+[(d1n2l2+α1)2+(d2n2l2+α2)2]2+4[m2(d1n2l2+α1)2(d2n2l2+α2)2]],
    dzndn=12{2(d1n2l2+α1)(2d1nl2)2(d2n2l2+α2)(2d2nl2)+122[(d1n2l2+α1)2+(d2n2l2+α2)2][2(d1n2l2+α1)(2d1nl2)+2(d2n2l2+α2)(2d2nl2)][(d1n2l2+α1)2+(d2n2l2+α2)2]2+4[m2(d1n2l2+α1)2(d2n2l2+α2)2]128[(d1n2l2+α1)(2d1nl2)(d2n2l2+α2)2+(d2n2l2+α2)(2d2nl2)(d1n2l2+α1)2][(d1n2l2+α1)2+(d2n2l2+α2)2]2+4[m2(d1n2l2+α1)2(d2n2l2+α2)2]}<12{2(d1n2l2+α1)(2d1nl2)2(d2n2l2+α2)(2d2nl2)+[(d1n2l2+α1)2+(d2n2l2+α2)2][2(d1n2l2+α1)(2d1nl2)+2(d2n2l2+α2)(2d2nl2)][(d1n2l2+α1)2+(d2n2l2+α2)2]24[(d1n2l2+α1)(2d1nl2)(d2n2l2+α2)2+(d2n2l2+α2)(2d2nl2)(d1n2l2+α1)2][(d1n2l2+α1)2+(d2n2l2+α2)2]2+4[m2(d1n2l2+α1)2(d2n2l2+α2)2]}=2[(d1n2l2+α1)(2d1nl2)(d2n2l2+α2)2+(d2n2l2+α2)(2d2nl2)(d1n2l2+α1)2][(d1n2l2+α1)2+(d2n2l2+α2)2]2+4[m2(d1n2l2+α1)2(d2n2l2+α2)2]<0.

    Hence, zn is monotone decreasing with respect to n. For i=0,1,,N1 we have zi>zi+1. Thus ωn is monotone decreasing with respect to n.

    Lemma 2.4. If assumption (H2) and (H3) are satisfied, then τ0,nωn is monotone increasing with respect to n. Moreover, τ0,0=min{τ0,n}0nN1.

    Proof. From Lemma 2.3, for n=0,1,,N2, we have ωn>ωn+1 and

    τ0,nωn=arccosωn2+(d1n2l2+α1)(d2n2l2+α2)m.

    By ωn2<ω2n+1 and m<0, we have ωn2+(d1n2l2+α1)(d2n2l2+α2)m is monotone decreasing with respect to n. Since the anti-trigonometric function y=arccosx is monotone decreasing with respect to x, from the properties of composite functions, we can obtain that τ0,nωn is monotone increasing with respect to n. The last conclusion follows from ωn>ωn+1.

    Under the assumption (H2) and (H3), let λ(τ)=α(τ)+iβ(τ) be the root of Eq (2.4) with n{0,1,,N1} satisfying α(τj,n)=0 and β(τj,n)=ωn.

    Lemma 2.5. Suppose that (H2) and (H3) are satisfied. Then

    α(τ)|τ=τj,n>0.

    Proof. Since Eq (2.4) can be rewritten in the following form:

    λ2+(k1+k2)λ+k1k2+meλτ=0. (2.9)

    Substituting λ(τ) into (2.9) and taking the derivative of both sides of Eq (2.9) with respect to τ gives

    (dλdτ)1=τλ+2λ+(k1+k2)λ[λ2(k1+k2)λk1k2]. (2.10)

    Then we have

    (dλdτ)1|τ=τj,n=τωi+i[2ω3(k1+k2)ω(k1+k2)(ω2k1k2)](k1+k2)2ω4+ω2(ω2k1k2)2+2ω2(ω2k1k2)+(k1+k2)2ω2(k1+k2)2ω4+ω2(ω2k1k2)2.

    Hence,

    sign[α(τj,n)]=sign[(dReλdτ)1|τ=τj,n]=sign(2ω2(ω2k1k2)+(k1+k2)2ω2(k1+k2)2ω4+ω2(ω2k1k2)2)=sign(2ω4+(k21+k22)ω2(k1+k2)2ω4+ω2(ω2k1k2)2)>0.

    This completes the proof. By Lemmas 2.1, 2.4, 2.5 and applying Corollary 2.4 in [31], we have the following conclusions on the distribution of the roots of the characteristic Eq (2.4).

    Lemma 2.6. Suppose that (H2) and (H3) are satisfied. Then there exists a sequence values of {τj,n}(j=0,1,2,;n=0,1,,N1), such that all the roots of Eq (2.4) have negative real parts when τ1+τ2[0,τ0,0); When τ1+τ2=τ0,0, all the other roots of Eq (2.4) have negative real parts, except for a pair of pure imaginary roots ±iω0; When τ1+τ2>τ0,0, Eq (2.4) has at least a couple of roots with positive real parts.

    From Lemma 2.6, we have the following the conclusions on the stability of the positive equilibrium of the system (1.4) and the existence of Hopf bifurcations directly.

    Theorem 2.7. Suppose that (H1),(H2) and (H3) are satisfied. Then the positive constant steady state E of system (1.4) is asymptotically stable when τ1+τ2[0,τ0,0), unstable when τ1+τ2>τ0,0, and system (1.4) undergoes a Hopf bifurcation at E when τ1+τ2=τj,n, j=0,1,2,;n=0,1,,N1.

    In this section, we will calculate the normal form of Hopf bifurcation on the center manifold. To investigate the dynamical behaviours of system (1.4) near the Hopf bifurcation point, we will study the normal form of Hopf bifurcation by making use of the normal form method of partial functional differential equations [32,33].

    Without loss of generality, we assume that τ1>τ2, let ¯u(x,t)=u(x,τ1t)u, ¯v(x,t)=v(x,τ1t)v and take out of the bars, the system (1.4) becomes

    t(u(x,t)v(x,t))=τ1(DΔ+A)(u(x,t)v(x,t))+τ1B(u(x,t1)v(x,t1))+τ1C(u(x,tτ2τ1)v(x,tτ2τ1))+τ1(f1f2),x(0,lπ),t>0 (3.1)

    with the boundary and initial conditions:

    {u(x,t)x=0,v(x,t)x=0,x=0,lπ,t>0,u(x,t)=u0(x,t)0,v(x,t)=v0(x,t)0,x(0,lπ),t[1,0], (3.2)

    where

    f1=S(v(x,t1))(Sv)v(x,t1),f2=T(u(x,tτ2τ1))(Tu)u(x,tτ2τ1).

    We define the real-valued Hilbert space

    Y={(u,v)TH2(0,lπ)×H2(0,lπ):ux=vx=0atx=0,lπ}

    and the complexification of Y

    YC=YiY={U1+iU2:U1,U2Y}

    with the general complex-value L2 inner product

    (U,V)=lπ0(¯u1v1+¯u2v2)dx

    for U=(u1,u2)T,V=(v1,v2)TYC.

    Let C:=C([1,0],YC) represents the phase space with the sup norm. We write utC for ut(θ)=u(t+θ), 1θ0.

    In the previous section, we have obtained that when τ=τ0,0, all the roots of Eq (2.4) have negative real parts except ±iω0. We denote the Hopf bifurcation point by (τ1,τ2) with τ1+τ2=τ0,0, and introduce parameter μ by setting τ1=τ1+μ with fixing τ2=τ2 and |μ| sufficiently small. Then we have τ=τ1+τ2=τ1+τ2+μ=τ0,0+μ, and μ=0 is the Hopf bifurcation value of system (3.1).

    Denote U(t)=(u(x,t),v(x,t))T and substitute τ1=τ1+μ,τ2=τ2 into (3.1). We have

    dU(t)dt=DμΔU(t)+Lμ(Ut)+f(μ,Ut), (3.3)

    where

    Dμ=(τ1+μ)D,Lμ(Ut)=(τ1+μ)(AUt(0)+BUt(1)+CUt(τ2τ1+μ))

    and

    f(μ,Ut)=(τ1+μ)(f1f2)=(τ1+μ)(S(v(x,t1))(Sv)v(x,t1)T(u(x,tτ2τ1+μ))(Tu)u(x,tτ2τ1+μ)).

    Consider the linearized system of (3.3)

    dU(t)dt=DμΔU(t)+Lμ(Ut). (3.4)

    In Y, the eigenvalues of DΔ are d1n2l2 and d2n2l2, nN0, with the corresponding normalized eigenvectors: β(1)n(x)=γn(x)e1, β(2)n(x)=γn(x)e2, where

    γn(x)=cosnlxcosnlxL2={1lπ,n=0,2lπcosnlx,n1,

    and e1=(1,0)T,e2=(0,1)T are the unit coordinate vectors of R2.

    Define the subspace of C as Bn, by

    Bn:=span{(v(),β(1)n)β(1)n,(v(),β(2)n)β(2)n|vC},

    we write (v(),βn)=((v(),β(1)n),(v(),β(2)n))T for simplification. Then on Bn, the linear Eq (3.4) is equivalent to retarded functional differential equation on R2:

    ˙x(t)=n2l2Dμx(t)+Lμxt,nN0. (3.5)

    By Riese representation theorem, there exists a 2×2 matrix function ηn(μ,θ), θ[1,0], whose entries are of bounded variation such that

    n2l2Dμϕ(0)+Lμ(ϕ)=01dηn(μ,θ)ϕ(θ),ϕC.

    In fact, we can choose

    ηn(μ,θ)={(τ1+μ)B,θ=1,0,θ(1,τ2τ1+μ],(τ1+μ)C,θ(τ2τ1+μ,0),(τ1+μ)(A+Cn2l2D),θ=0.

    For simplicity, we write ηn(θ) for ηn(0,θ), define A as the infinitesimal generator of the semigroup generated by Eq (3.5) with μ=0 and n=0. Define the adjoint operator A of A on C:=C([0,1],(R2)) as

    Aψ(s)=˙ψ(s)+X0(s)[01ψ(θ)dη0(θ)+˙ψ(0)],

    with the bilinear form

    ψ,φ=¯ψ(0)φ(0)01θξ=0¯ψ(ξθ)dη0(θ)φ(ξ)dξ, (3.6)

    Let

    q(θ)=q(0)eiω0τ1θ(θ[1,0]),q(s)=q(0)eiω0τ1s(s[0,1])

    be the eigenvectors of A and A corresponding to iω0τ1 and iω0τ1 respectively. Thus

    q(0)=(1,q1)T,q(0)=¯M(q2,1),

    where

    q1=iω0+α1(Sv)eiω0τ1,q2=iω0+α2(Sv)eiω0τ1.

    We choose M as

    M=[q1+¯q2+τ1q1¯q2(Sv)eiω0τ1+τ2(Tu)eiω0τ2]1

    which assures that q(s),q(θ)=1.

    Let

    BC={ϕ:[1,0]YC,ϕiscontinouson[1,0),limθ0ϕ(θ)YC},

    and define Aμ as the infinitesimal generator of the C0 semi-group of solution maps of the linear equation of (3.3):

    Aμ:C10BCBC,Aμϕ=˙ϕ+X0[DμΔϕ(0)+Lμ(ϕ)˙ϕ(0)]

    with

    dom(Aμ)={ϕC:˙ϕC,ϕ(0)dom(Δ)},

    where

    X0(θ)={0,1θ<0,I,θ=0.

    On BC, Eq (3.3) can be rewritten as an abstract ordinary differential equation

    dUtdt=AμUt+X0f(μ,Ut). (3.7)

    Using the same notations in Wu [34], we compute the coordinates to describe the center manifold C0 at μ=0. Let Ut be the solutions of Eq (3.3) with μ=0. Define

    z(t)=q,(Ut,β0),W(t,θ)=Ut(θ)2Re{z(t)q(θ)}.

    On the center manifold C0, we have

    W(t,θ)=(z(t),¯z(t),θ),

    where

    z=(z1,z2)T,W(z,¯z,θ)=W20(θ)z22+W11(θ)z¯z+W02(θ)¯z22+W30(θ)z36+,

    z and ˉz are local coordinates for center manifold C0 in the direction of q and ¯q respectively. Notice that W is real if Ut is real, we consider real solutions.

    For the solution UtC0 of the system (3.3) with μ=0,

    ˙z(t)=iω0τ1z+q(θ),f(0,W+2Re{z(t)q(θ)})=iω0τ1z+¯q(0)f(0,W(z,¯z,0)+2Re{z(t)q(0)})=iω0τ1z+¯q(0)F0(z,¯z).

    It can be denoted as

    ˙z(t)=iω0τ1z(t)+g(z,¯z), (3.8)

    where

    g(z,¯z)=¯q(0)f(0,W(z,¯z,0)+2Re{z(t)q(0)})=g20z22+g11z¯z+g02¯z22+g21z2¯z2+.

    Base on (3.7) and (3.8), similar to the method used in Zhao and Wei [35], we obtain

    ˙W=˙Ut˙zq˙¯z¯q={A0W2Re{¯q(0)F0q(θ)},θ[1,0),A0W2Re{¯q(0)F0q(θ)}+F0,θ=0,=A0W+H(z,¯z,θ),

    where

    H(z,¯z,θ)=H20(θ)z22+H11(θ)z¯z+H02(θ)¯z22+. (3.9)

    By expanding the above series and comparing the coefficients, we have

    {(A02iω0τ1)W20(θ)=H20(θ),A0W11(θ)=H11(θ),(A0+2iω0τ1)W02(θ)=H02(θ),. (3.10)

    Notice that

    q(0)=¯M(q2,1),v(t1)=q1eiω0τ1z+¯q1eiω0τ1¯z+W(1)(t,1),u(tτ2τ1)=eiω0τ2z+eiω0τ2¯z+W(2)(t,τ2τ1),

    where

    W(1)(t,1)=W(1)20(1)z22+W(1)11(1)z¯z+W(1)02(1)¯z22+,W(2)(t,τ2τ1)=W(2)20(τ2τ1)z22+W(2)11(τ2τ1)z¯z+W(2)02(τ2τ1)¯z22+,

    and

    F0=12τ1(S(v)v2(t1)T(u)u2(tτ2τ1))+16τ1(S(v)v3(t1)T(u)u3(tτ2τ1))+.

    We have

    g(z,¯z)=¯q(0)F0=M2τ1[¯q2(S(v)v2(t1)+S(v)3v3(t1))+T(u)u2(tτ2τ1)+T(u)3u3(tτ2τ1)]=M2τ1{¯q2[S(v)(q21e2iω0τ1z2+¯q21e2iω0τ1¯z2+2q1¯q1z¯z)+(S(v)[¯q1eiω0τ1W(1)20(1)+2q1eiω0τ1W(1)11(1)]+S(v)q21¯q1eiω0τ1)z2¯z]+T(u)(e2iω0τ2z2+e2iω0τ2¯z2+2z¯z)+(T(u)[eiω0τ2W(2)20(τ2τ1)+2eiω0τ2W(2)11(τ2τ1)]+T(u)eiω0τ2)z2¯z}.

    Comparing the coefficients with (3.9), we obtain that

    g20=τ1M¯q2S(v)q21e2iω0τ1+τ1MT(u)e2iω0τ2,g11=2τ1M¯q2S(v)q1¯q1+τ1MT(u),g02=τ1M¯q2S(v)¯q21e2iω0τ1+τ1MT(u)e2iω0τ2,g21=τ1M{S(v)[¯q1eiω0τ1W(1)20(1)+2q1eiω0τ1W(1)11(1)]+S(v)q21¯q1eiω0τ1+T(u)[eiω0τ1W(2)20(τ2τ1)+2eiω0τ2W(2)11(τ2τ1)]+T(u)eiω0τ2}. (3.11)

    We need to compute W20(θ) and W11(θ) for θ[1,0), since

    H(z,¯z,θ)=2Re{¯z(0)F0q(θ)}=gq(θ)¯g¯q(θ)=(g20z22+g11z¯z+g02¯z22+)q(θ)(¯g20¯z22+¯g11z¯z+¯g02z22+)¯q(θ).

    Comparing the coefficients with (3.9), we obtain

    H20(θ)=g20q(θ)¯g02¯q(θ),
    H11(θ)=g11q(θ)¯g11¯q(θ).

    From (3.10), it can be given that

    ˙W20(θ)=2iω0τ1W20(θ)+g20q(0)eiω0τ1θ+¯g02¯q(0)eiω0τ1θ.

    Solving for W20(θ), we have

    W20(θ)=g20iω0τ1q(0)eiω0τ1θ¯g023iω0τ1¯q(0)eiω0τ1θ+E1e2iω0τ1θ, (3.12)

    and similarly

    W11(θ)=g11iω0τ1q(0)eiω0τ1θ¯g11iω0τ1¯q(0)eiω0τ1θ+E2, (3.13)

    where E1 and E2 can be determined by setting θ=0 in H. Since

    H(z,¯z,0)=2Re{¯q(0)F0q(0)}+F0,

    we have

    H20(0)=g20q(0)¯g02¯q(0)+(S(v)q21e2iω0τ1T(u)e2iω0τ2), (3.14)
    H11(0)=g11q(0)¯g11¯q(0)+(S(v)q1¯q1T(u)). (3.15)

    From (3.10) and the definition of A0, we have

    (α100α2)W20(0)+(0Sv00)W20(1)+(00Tu0)W20(τ2τ1)=2iω0τ1W20(0)H20(0), (3.16)
    (α100α2)W11(0)+(0Sv00)W11(1)+(00Tu0)W11(τ2τ1)=H11(0).

    Noting that

    (1iω0Sveiω0τ1Tueiω0τ21iω0)q(0)=0,

    substituting (3.12) into (3.16), we obtain

    (12iω0τ1Sve2iω0τ1Tue2iω0τ212iω0τ1)E1=g20q(0)¯g02¯q(0)H20(0). (3.17)

    Substituting (3.14) into this, we have

    ((1+2iω0τ1)Sve2iω0τ1Tue2iω0τ2(1+2iω0τ1))E1=(S(v)q21e2iω0τ1T(u)e2iω0τ2).

    Solving Eq (3.17) for (E(1)1,E(2)2)T=E1, we get

    E(1)1=(1+2iω0τ1)S(v)q21e2iω0τ1+SvT(u)e2iω0(τ1+τ2)(1+2iω0τ1)2SvTue2iω0(τ1+τ2),
    E(2)1=e2iω0τ2[TuS(v)q21e2iω0τ1+T(u)(1+2iω0τ1)](1+2iω0τ1)2SvTue2iω0(τ1+τ2).

    Similarly, we can get

    (1SvTu1)E2=(S(v)q1¯q1T(u)),

    and hence,

    E(1)2=S(v)q1¯q1+T(u)Sv1SvTu,E(2)2=S(v)Tuq1¯q1+T(u)1SvTu.

    Then g21 can be confirmed. Therefore, we can calculate the following quantities

    c1(0)=i2ω0τ1(g20g112|g11|213|g02|2)+12g21,μ2=Re(c1(0))Reλ(τ1),β2=2Re(c1(0)),T2=1ω0τ1(Im(c1(0))+μ2Im(λ(τ1))). (3.18)

    By the general theory in Hassard [36], we obtain that μ2 determines the direction of the Hopf bifurcation: When μ2>0(<0), the direction of the Hopf bifurcation is forward (backward), in other words, for τ1>τ1(τ1<τ1), there exist the bifurcating periodic solutions; β2 determines that whether the bifurcating periodic solutions are stable: When β2<0(>0), the bifurcating periodic solutions are orbitally asymptotically stable (unstable); T2 determines the period of the bifurcating periodic solutions: When T2>0(<0), the period of the bifurcating periodic solutions increases (decreases).

    In this section, we choose

    S(vτ1)=K1v(x,tτ1),T(uτ2)=K3u(x,tτ2)(1u(x,tτ2)K2), (4.1)

    and take the following data:

    (A)l=3,α1=α2=0.8,d1=d2=1,K1=40,K2=20,K3=0.1.

    Then system (1.4) with (4.1) and (A) has a unique positive constant steady state E=(16.8,0.336). By direct calculation, we have Sv=40, Tu=0.068 and m=2.72. Thus the assumptions (H1) and (H2) are satisfied, and E is asymptotically stable when τ1=τ2=0.

    By calculation, we get that when N=3,m2>(d1n2l2+α1)2(d2n2l2+α2)2 for 0n<Nandm2(d1N2l2+α1)2(d2N2l2+α2)2, thus assumption (H3) is satisfied. From Eq (2.8), we get that

    ω01.4422,τj,00.7023+π0.7211j,ω11.3747,τj,10.8515+π0.6874j,ω20.5411,τj,21.5799+π0.2706j.

    And Eq (2.7) has no positive roots for n3. Clearly, τ0,00.7023.

    From Theorem 2.7, we know that the positive constant steady state E is asymptotically stable when τ1+τ2[0,τ0,0), and unstable when τ1+τ2>τ0,0. Meanwhile, (1.4) undergoes a Hopf bifurcation at E when τ1+τ2=τj,n, n=0,1,2;jN0. The bifurcation set on (τ1,τ2) plane is shown by Figure 1.

    Figure 1.  τ1+τ2=τ0,0 is the stability switching curve.

    When (τ1,τ2) are chosen as P1(0.4,0.2), we have τ1+τ2<0.7023, thus E is asymptotically stable, this is illustrated in Figure 2.

    Figure 2.  E is asymptotically stable when τ1+τ2<τ0,0, where τ1=0.4, τ2=0.2 and initial values are u0(x,t)=14+10cosx and v0(x,t)=0.3+0.3cosx.

    We choose the point P3(0.55115,0.15115) on the curve τ=τ0,0, that is τ1=0.55115 and τ2=0.15115. According to the algorithm given in the previous section, we can obtain that c1(0)=0.001550.00290i, λ(τ1)=0.920630.94716i, μ2=0.00168>0, β2=0.0031<0 and T2=0.00565>0. This implies that the direction of the Hopf bifurcation is forward, the bifurcating periodic solutions are stable and the period of the bifurcating periodic solutions increases, which are illustrated by Figure 3.

    Figure 3.  The bifurcating period solutions are stable, where τ1=0.6, τ2=0.15115, τ1+kgeτ2>kgeτ0,0, and the initial values are u0(x,t)=14+10cosx and v0(x,t)=0.3+0.3cosx.

    In fact, with the help of the norm form calculated in section 2, we get that when τ1 and τ2 satisfying τ1+τ2=τ0,0 and τ1>τ2, the corresponding Rec1(0) would be negative, we list some values of c1(0) in Table 1.

    Table 1.  Some values of c1(0) on the curve τ1+τ2=τ0,0.
    Point (τ1,τ2) The value ofc1(0)
    1 (0.7023, 0) -0.00210-0.00389i
    2 (0.6023, 0.1) -0.00175-0.00323i
    3 (0.5023, 0.2) -0.00134-0.00261i
    4 (0.4023, 0.3) -0.00089-0.00211i

     | Show Table
    DownLoad: CSV

    In this paper, we study the dynamics of epidemics by using a diffusive model with two delays. First we perform the stability analysis at the constant steady state in the system. Then we present the conditions for the existence of the purely imaginary roots of the characteristic equations. Moreover, we investigate the existence of Hopf bifurcation by taking two delays as parameters. Theoretical result shows that the sum of two delays affect the stability of the steady state and the existence of Hopf bifurcation. Considering the sum of two delays as one parameter, we verify that τ1+τ2=τ0,0 is the critical Hopf bifurcation curve and the stability switching curve of this system. To calculate the normal form on the center manifold near the Hopf singularity, we fix τ2 as a constant and take τ1 as a parameter. Finally, numerical simulations are carried out to verify the theoretical results. The simulations show that when τ1+τ2<τ0,0, E is locally asymptotically stable; when τ1+τ2>τ0,0, the bifurcating periodic solutions are stable. Both theoretical verifications and numerical simulations reveal that when τ1+τ2<τ0,0, the system is locally stable at E; when τ1+τ2>τ0,0 and close enough to τ0,0, E is unstable, the periodic solutions generated by the Hopf bifurcation is stable. All the analysis shows that the sum of two delays plays an important role in the dynamics of the system.

    The authors are grateful to the anonymous referees for their helpful comments and valuable suggestions which have improved the presentation of the paper. This research is supported by National Natural Science Foundation of China (No.11771109).

    The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.



    [1] Biot MA (1956) Thermo elasticity and irreversible thermo-dynamics. J Appl Phys 27: 240-253. doi: 10.1063/1.1722351
    [2] Lord HW, Shulman Y (1976) A generalized dynamical theory of thermo elasticity. J Mech Phys Solid 15: 299-309.
    [3] Müller MM, Kaiser E, Bauer P, et al. (1976) Lipid composition of the rat kidney. Nephron 17: 41-50. doi: 10.1159/000180709
    [4] Green AE, Laws N (1972) On the entropy production inequality. Arch Ration Mech An 45: 47-53. doi: 10.1007/BF00253395
    [5] Green AE, Lindsay KA (1972) Thermo elasticity. J Elasticity 2: 1-7. doi: 10.1007/BF00045689
    [6] Shuhubi E (1957) Thermo elastic solid, In: Eringen AC, Continuum Physics, New York: Academic Press.
    [7] Green AE, Naghdi PM (1991) A re-examination of the basic postulate of thermo-mechanics. P Roy Soc A-Math Phy 432: 171-194.
    [8] Green AE, Naghdi PM (1993) Thermoelasticity without energy dissipation. J Elasticity 31: 189-208. doi: 10.1007/BF00044969
    [9] Green AE, Naghdi PM (1992) An unbounded heat wave in an elastic solid. J Therm Stresses 15: 253-264. doi: 10.1080/01495739208946136
    [10] Illyushin AA, Pobedria BE (1970) Fundamentals of the mathematical theory of thermal viscoelasticity.
    [11] Biot MA (1954) Theory of stress-strain relations in anisotropic viscoelasticity and relaxation phenomena. J Appl Phys 25: 1385-1391. doi: 10.1063/1.1721573
    [12] Biot MA (1955) Variational principles in irreversible thermodynamics with application to viscoelasticity. Phys Rev 97: 1463-1469. doi: 10.1103/PhysRev.97.1463
    [13] Morland LW, Lee EH (1960) Stress analysis for linear viscoelastic materials with temperature variation. J Rheol 4: 233-263.
    [14] Tanner RI (1988) Engineering Rheology, Oxford: Oxford University Press.
    [15] Drozdov AD (1996) A constitutive model in thermoviscoelasticity. Mech Res Commun 23: 543-548. doi: 10.1016/0093-6413(96)00055-9
    [16] Bland DR (1960) The Theory of Linear Viscoelasticity, Oxford: Pergamon Press.
    [17] Lion A (1997) On the large deformation behavior of reinforced rubber at different temperatures. J Mech Phys Solids 45: 1805-1834. doi: 10.1016/S0022-5096(97)00028-8
    [18] Liao Z, Hossain M, Yao X, et al. (2020) Thermo-viscoelastic experimental characterization and numerical modelling of VHB polymer. Int J Nonlin Mech 118: 103263. doi: 10.1016/j.ijnonlinmec.2019.103263
    [19] Niyonzima I, Jiao Y, Fish J (2019) Modeling and simulation of nonlinear electro-thermo-mechanical continua with application to shape memory polymeric medical devices. Comput Method Appl M 350: 511-534. doi: 10.1016/j.cma.2019.03.003
    [20] Mehnert M, Hossain M, Steinmann P (2017) Towards a thermo-magneto-mechanical coupling framework for magneto-rheological elastomers. Int J Solids Struct 128: 117-132. doi: 10.1016/j.ijsolstr.2017.08.022
    [21] Mehnert M, Hossain M, Steinmann P (2016) On nonlinear thermo-electro-elasticity. P Roy Soc A-Math Phys 472: 20160170. doi: 10.1098/rspa.2016.0170
    [22] Youssef HM, El-Bary AA, Elsibai KA (2014) Vibration of gold nano beam in context of two-temperature generalized thermoelasticity subjected to laser pulse. Lat Am J Solids Stru 11: 2460-2482. doi: 10.1590/S1679-78252014001300008
    [23] Ezzat MA, El-Bary AA (2014) Two-temperature theory of magneto-thermo-viscoelasticity with fractional derivative and integral orders heat transfer. J Electromagnet Wave 28: 1985-2004. doi: 10.1080/09205071.2014.953639
    [24] Ezzat MA, El-Bary AA (2015) State space approach to two-dimensional magneto-thermoelasticity with fractional order heat transfer in a medium of perfect conductivity. Int J Appl Electrom 49: 607-625.
    [25] Ismail MAH, Khamis AK, El-Bary AA, et al. (2017) Effect of rotation of generalized thermoelastic layer subjected to harmonic heat: state-space approach. Microsyst Technol 23: 3381-3388. doi: 10.1007/s00542-016-3137-3
    [26] Khamis AK, Ismail AH, Youssef HM, et al. (2017) Thermal shock problem of two-temperature generalized thermoelasticity without energy dissipation with rotation. Microsyst Technol 23: 4831-4839. doi: 10.1007/s00542-017-3279-y
    [27] Youssef HM, Elsibai KA, El-Bary AA (2017) Effect of the speed, the rotation and the magnetic field on the Q-factor of an axially clamped gold micro-beam. Meccanica 52: 1685-1694. doi: 10.1007/s11012-016-0498-8
    [28] Ezzat MA, El-Karamany AS, El-Bary AA (2017) Thermoelectric viscoelastic materials with memory-dependent derivative. Smart Struct Sys 19: 539-551. doi: 10.12989/sss.2017.19.5.539
    [29] El-Karamany AS, Ezzat MA, El-Bary AA (2018) Thermodiffusion with two time delays and Kernel functions. Math Mech Solids 23: 195-208. doi: 10.1177/1081286516676870
    [30] Ezzat MA, El-Bary AA (2018) Thermoelectric spherical shell with fractional order heat transfer. Microsyst Technol 24: 891-899. doi: 10.1007/s00542-017-3400-2
    [31] El-Bary AA, Atef H (2016) On effect of viscous fractional parameter on infinite thermo Viscoelastic medium with a spherical cavity. Journal of computational and theoretical. Nanoscience 13: 1-5. doi: 10.1166/jctn.2016.4099
    [32] El-Bary AA, Atef M (2016) Modified approach for stress strain equation in the linear Kelvin-Voigt solid based on fractional order. J Comput Theor Nanos 13: 1027-1036. doi: 10.1166/jctn.2016.4332
    [33] Amin MM, El-Bary A, Atef H (2018) Effect of viscous fractional parameter on generalized magneto thermo-viscoelastic thin slim strip exposed to moving heat source. Mater Focus 7: 814-823. doi: 10.1166/mat.2018.1591
    [34] Amin MM, El-Bary AA, Atef HM (2018) Modification of Kelvin-Voigt model in fractional order for thermoviscoelastic isotropic material. Mater Focus 7: 824-832. doi: 10.1166/mat.2018.1592
    [35] Sharma K, Kumar P (2013) Propagation of plane waves and fundamental solution in thermoviscoelastic medium with voids. J Therm Stresses 36: 94-111. doi: 10.1080/01495739.2012.720545
  • This article has been cited by:

    1. Yongli Song, Yahong Peng, Tonghua Zhang, The spatially inhomogeneous Hopf bifurcation induced by memory delay in a memory-based diffusion system, 2021, 300, 00220396, 597, 10.1016/j.jde.2021.08.010
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(3628) PDF downloads(242) Cited by(3)

Figures and Tables

Figures(9)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog