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Amplitude death, oscillation death, and stable coexistence in a pair of VDP oscillators with direct–indirect coupling

  • In this paper, we investigated the dynamics of a pair of VDP (Van der Pol) oscillators with direct-indirect coupling, which is described by five first-order differential equations. The system presented three types of equilibria including HSS (homogeneous steady state), IHSS (inhomogeneous steady state) and NPSS (no-pattern steady state). Employing the corresponding characteristic equations of the linearized system, we obtained the necessary conditions for the pitchfork and Hopf bifurcations of the equilibria. Further, we illustrated one-dimensional bifurcation and phase diagrams to verify theoretical results. The results show that the system exhibited two types of oscillation quenching, i.e., amplitude death (AD) for HSS equilibria and oscillation death (OD) for IHSS equilibria. In some special regions of the parameters, the system proposed multiple types of stable coexistence including HSS and IHSS equilibria, periodic orbits or quasi-periodic oscillations.

    Citation: Xiaojun Huang, Zigen Song, Jian Xu. Amplitude death, oscillation death, and stable coexistence in a pair of VDP oscillators with direct–indirect coupling[J]. Electronic Research Archive, 2023, 31(11): 6964-6981. doi: 10.3934/era.2023353

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  • In this paper, we investigated the dynamics of a pair of VDP (Van der Pol) oscillators with direct-indirect coupling, which is described by five first-order differential equations. The system presented three types of equilibria including HSS (homogeneous steady state), IHSS (inhomogeneous steady state) and NPSS (no-pattern steady state). Employing the corresponding characteristic equations of the linearized system, we obtained the necessary conditions for the pitchfork and Hopf bifurcations of the equilibria. Further, we illustrated one-dimensional bifurcation and phase diagrams to verify theoretical results. The results show that the system exhibited two types of oscillation quenching, i.e., amplitude death (AD) for HSS equilibria and oscillation death (OD) for IHSS equilibria. In some special regions of the parameters, the system proposed multiple types of stable coexistence including HSS and IHSS equilibria, periodic orbits or quasi-periodic oscillations.



    Symmetry is a wonderful tool in explicating the laws of nature. One of the great stories of Lie's success was the initiation of a program to solve or at least to simplify the differential equation by using the Lie group theory on the analogy of ˊEvariste Galois's work to solve the algebraic equations having degrees two, three and four. But he was unable to give closed forms of roots of equations whose degrees are five or greater, by using only the arithmetic operation which are (+,,÷,×,) [1]. Lie's original ideas was to establish a general theory of integration of ordinary differential equation on the similar basis as Galois and Abel did for algebraic equations.

    At first the method could not attract much attention until last few decades when a revival of the interests in Lie's theory was observed which resulted in a significate progress. The main hinderance was the complicated system of many differential equations which we obtain in symmetry analysis. Today we have strong computer algebra systems like Maple and Mathematica to handle this problem. Physical laws of nature are best governed by the exploitation of symmetries involved in the system. Real world phenomenons can be transformed into mathematical language by using non-linear PDE's. The non-linear PDE's have attracted much attention and have wide range of applications in science and engineering like plasma waves, fluid mechanics, optics, biological systems, chemical physics and financial systems. One such immensely popular class of equation is Non-linear Evolution Equation (NLEE's). Complex systems can be characterized by NLDE's. One particularly famous NLDE is the KdV equation derived by korteweg and de Vries [2]. It can be give in the following form,

    ut+uux+uxxx=0. (1.1)

    It can be observed that this equation is a non-linear PDE in one dimension and describes the time-dependent motion of shallow water waves. Other equation is the Regularised Long wave equation (RLW) [3]. It is more common than KdV equation to depict the behaviour of non-linear dispersive waves. This equation can be written as,

    ut+ux+εuuxμuxxt=0. (1.2)

    Morrison et al. [4] introduced one dimensional non-linear Evolution Equation, in the given form,

    ut+uuxμuxxt=0. (1.3)

    It is also known as Equal Width wave (EW) equation [5,6,7,8,9,10] derived by using both KdV and RLW equation. Authors in [5] discussed solitary waves related to EW equation. Different methods have been adopted to find the solution of EW equations both numerically and analytically, see for instance [5,6,7,8,9,10]. Because of soliton solution with permanent speed and form, the wave has equal width for all wave amplitude. That's why it is called Equal Width wave equation. Here μ is a positive parameter, x, t denote space and time coordinate respectively, and u(x,t) represents wave amplitude with boundary condition u0 as x±. In plasma physics u represents ve of electrostatic potential and in fluid problems, u represents wave vertical displacement of water surface [11]. Solitary wave solutions can also be found in EW equation. Soliton is unique type of solitary waves that retain it's shape after colliding with other objects.

    The Modified Equal Width wave (MEW) equation is derived from EW equation and it has cubic non-linearity with dispersive wave form

    ut+u2uxμuxxt=0. (1.4)

    This equation has also been discussed on large scale recently [12,13,14,15,16,17,18]. Authors solved this equation numerically in [12,18]. Some B-spline methods have been adopted to solve this equation in [13,14,15,16,17]. A generalization of EW equation also known as Generalized Equal Width wave (GEW) equation [19,20,21,22,23] can be derived from EW equation in following form

    ut+unuxμuxxt=0, (1.5)

    with n+1 non linearity and dispersion wave having solitary solution, where n ϵ Z+. This equation can be related with Generalized KdV equation [24] and Generalized RLW equation [25,26]. Solitary waves of GEW has been discussed in [19]. Raslan used collocation method to solve GEW in [20]. Some other methods also have been employed on this equation, see [21,22,23].

    With the passage of time, application of DEs and it uses are increasing continuously. Many numerical as well as analytical method keep on emerging to solve PDE's. Some famous numerical method are Finite difference method [27], Multigrid method [28], Methods of lines [29], Domain decomposition method [30], Gradient discretization method [31], Mesh free method [32], Spectral method [33] and Quadratic B-spline method [34] etc. Some popular analytical methods are Daraboux transformation method [35,36], Inverse scattering method [37,38], Kudryashov method [39,40], Simplest equation method [41], Homogenous balance method [42], Tanh-coth method [43], Hirota bilinear method [44], Jacobi elliptic function expansion method [45,46], Sine-cosine method [47] and Lie symmetric method [48,49,50,51,52,53]. These methods give exact solutions of the PDE's. Many numerical and analytical method have been used to find the solution of EW equation, MEW equation and GEW equation. Zaki used the least square finite element method to find the numerical solution of EW equation [54]. Elein Yusufoglu et al. used He's variational iteration method to find the numerical solution of EW equation [55]. G. A Gardner et al. used Galerkin method to obtain numerical solution of EW equation [56]. B. G. Karakoc et al. used cubic B-spline lumped Galerkin method to find the numerical solution of MEW equation [57]. Khalique et al. used Jacobi elliptic expansion method to find the travelling wave solution of EW equation [58]. Evan et al. computed the solitary wave solution of GEW equation by using quadratic B-spline method [59]. S. Hamdi et al. discussed the exact solution of GEW equation [60]. Halil Zeybek obtained the numerical solution of GEW equation by using cubic B-spline Galerkin method [61]. S. B. G. Karakoc et al. used the septic B-spline collocation method to find the solution of GEW equation in [62] and derived numerical solution of GEW equation by using sextic B-spline finite element method in [63]. B. G. Karakoc et al. used the finite element method to find the numerical solution of GEW equation [64].

    In this article, we find Lie symmetries and travelling wave solution of GEW equation by using Sine-cosine method. Lie symmetry approach has not been used for GEW equation. We use this method to reduce the complexity of GEW equation. It is worth mentioning that Lie symmetry method is the most important approach for constructing analytical solutions of nonlinear PDEs. We prefer to use Lie symmetry analysis because it studies the invariance of differential equations (DEs) under a one-parameter group of transformations which transforms a solution to another new solution and is used to reduce the order such as the number of variables of DEs; moreover, the conservation laws can be constructed by using the symmetries of the DEs. Kumar et al. effectively used this method for (3+1)-dimensional generalized KP equation in [48], (3+1)-dimensional KdV-type equation in [49], (2+1)-dimensional BK equation with variable coefficient in [50] and CHKP equation in [51]. Liu et al. used this approach to find the invariant solutions of SP equation in [52]. Chauhan et al. used this method to find traveliing wave solutions of EW equation in [53].

    We describe a method to find the symmetry of non-linear PDEs in this section. For this, we consider the system of nth order non-linear PDEs having p independent variable X=(x1,x2,.....,xp) Rp and q dependent variable U=(u1,u2,.....,uq) Rq has form

    Δb(X,U(n))=0,b=1,2,..........l, (2.1)

    where U(n) = unxn.

    To construct the Lie symmetry, firstly we introduce a Lie group of transformation acting on both dependent and independent variable such as

    ˜xc=xc+εξc(X,U)+O(ε2),c=1,2,.....,p,˜ud=ud+εηd(X,U)+O(ε2),d=1,2,.....,q, (2.2)

    where ε taken as a very small parameter, usually taken as ϵ<<<1 and ξc and ηd are infinitesimal generators with respect to the independent and dependent variable respectively.

    The vector field of above transformation is

    V=pc=1ξc(X,U)xc+qd=1ηd(X,U)ud. (2.3)

    Thus the nth prolongation of vector field v is also the vector field, which is

    pr(n)V=V+qr=1SηSr(X,U(n))urS, (2.4)

    defined on the space M(n) X×U(n), where S=(s1,s2,....sk), with 1 sk p, 1 k n, here

    ηSr(x,u(n))=DS(ηrpc=1ξcurc)+pc=1ξcurS,c, (2.5)

    where urc = urxc, and urs,c = ursxc, where the total derivative Dx and Dy is defined as

    DxH=Hx+uxHu+uxxHux+uxyHuy+.......,DyH=Hy+uyHu+uxyHux+uyyHuy+......., (2.6)

    with condition

    Pr(n)v[Δb(X,U(n))]=0,b=1,2,..........l,wheneverΔb(X,U(n))=0. (2.7)

    Taking one parameter local Lie group of transformation having variable x, t and u are as follows:

    x=x+ες(t,x,u)+O(ε2),t=t+ε(t,x,u)+O(ε2),u=u+εφ(t,x,u)+O(ε2), (3.1)

    where εR is the group parameter. The vector field v of equation is defined as

    V=ς(t,x,u)x+(t,x,u)t+φ(t,x,u)u. (3.2)

    The third order prolongation is defined as:

    Pr(3)V=V+φxux+φtut+φxxuxx+φxtuxt+φttutt+φxxxuxxx+φxxtuxxt+φxttuxtt+φtttuttt. (3.3)

    Now by applying the third order prolongation to the Eq (1.5), we get invariance condition

    φt+nun1φux+unφxμφxxt=0, (3.4)

    by putting the value of coefficients [φt], [φx] and [φxxt] in above equation

    φt=Dt(φςuxut)+ςuxt+utt,
    φx=Dx(φςuxut)+ςuxt+utt, (3.5)
    φxxt=DxDxDt(φςuxut)+ςuxxxt+uxxtt,

    where Dt and Dx are total derivative. By applying the third order prolongation and use the value of φt, φx and φxxt, we get

    nun1φuxunςuu2xunuuxut+unφuuxunςxuxunxut+unφxςuuxutςtuxuu2ttut+φuut+φt+3μςuuuxuxxut+4μuuuxutuxtμφxxt+2μuxuxutt+μςuuuu3xut+μuuuu2xu2t+μuutu2xutμφuuuu2xut+3μςuuu2xuxt+μuuu2xutt+3μςutuxuxx+2μutuxuxt2μφuuuxuxt+μuuuxxu2t+μutuxxutμφuuuxxut+3μςuuxxuxt+μuuxxutt+μςuuxxxut+3μςuuxuxxt+2μuuxxtut+2μuuxuxtt+4μuxutuxt+2μςuuxu2xut+2μuuxuxu2t+2μuxtuxut+μςxxuuxut2μφuuxuxut+4μςuxuxuxt+2μςuxuxxut+μςuutu3xμφuutu2xμφutuxx+2μuu2xt+μςtuxxx+μtuxxtμφuuxxt+2μςxuxxt+2μxuxtt+μςxxtux+μxxtut2μφuxtuxμφuxxut+2μςxtuxx+2μxtuxt2μφuxuxt+μςxxuxt+μxxutt+2μςuxtu2x+μuxxu2t=0. (3.6)

    By putting the value of ut and put the coefficients of various monomial equal to zero, we get the system of equations in term of partial derivatives, which is

    ux:nun1φunςx+unt2μφuxt+μunφuxx=0,uxxt:2μςxμ2φuxx=0,uxt:2μφux+μςxx=0,uxuxt:φuu=0,uxx:φut=0,constant:φt+unφxμφxxt=0. (3.7)

    We can solve this system manually or by using software Mathematica or Maple, so by solving the system we get infinitesimal

    ς(t,x,u)=C3,(t,x,u)=C1t+C2,φ(t,x,u)=1nC1u,  (3.8)

    where C1, C2 and C3 are arbitrary constants. As a result, three vector fields spanned the Lie algebra of infinitesimal generators of Eq (1.5).

    H1=tt1nuu,H2=t,H3=x. (3.9)

    Thus the one-parameter Lie group Gi, (i=1,2,3) generated by the three vector fields H1, H2 and H3 which are

    G1=(t,x,u)(teϵ,x,ue1nϵ),G2=(t,x,u)(t+ϵ,x,u),G3=(t,x,u)(t,x+ϵ,u), (3.10)

    where ϵR is the group parameter.

    Since each group Gi is a symmetric group, so if u=h(t,x) is a solution of (1.5), so are the functions

    u(1)=e1nϵh(teϵ,x),u(2)=h(tϵ,x),u(3)=h(t,xϵ). (3.11)

    An optimal system of one parameter Lie group is the collection of all inequivalent one parameter Lie group that any other subgroup is conjugate to one of group in the collection. To compute the optimal system, first we compute the commutator table.

    The commutator table of Hi, (i=1,2,3) is

    Table 1.  Commutator table.
    H1 H2 H3
    H1 0 H2 0
    H2 H2 0 0
    H3 0 0 0

     | Show Table
    DownLoad: CSV

    Now we proceed to compute the adjoint table.

    The adjoint table for Hi, (i=1,2,3) is

    Table 2.  Adjoint table.
    [adj] H1 H2 H3
    H1 H1 eϵH2 H3
    H2 H1ϵH2 H2 H3
    H3 H1 H2 H3

     | Show Table
    DownLoad: CSV

    Consider a generator of the form

    H=a1H1+a2H2+a3H3, (3.12)

    where a1, a2 and a3 are arbitrary constant. We solve it by using different cases.

    Case 1:

    Let a10 and a1=1, then

    H=H1+a2H2+a3H3. (3.13)

    By acting Adjea2H2 on H, the coefficient a2 disappear.

    ˊH=H1+a3H3. (3.14)

    Subcase 1: If a3<0, then

    ˊH=H1H3. (3.15)

    Subcase 2: If a3>0, then

    ˊH=H1+H3. (3.16)

    Case 2:

    Let a1=0 and a2=1, then

    H=H2+a3H3. (3.17)

    Subcase 1: If a3<0, then

    H=H2H3. (3.18)

    Subcase 2: If a3>0, then

    H=H2+H3. (3.19)

    Case 3:

    Let a1=0, a2=0 and a3=1, then

    H=H3. (3.20)

    Case 4:

    Let a3=0, then

    H=a1H1+a2H2. (3.21)

    By acting Adjea2a1H2 on H, we get

    ˊH=a1H1. (3.22)

    Case 5:

    Let a1=a3=0 and a20, then

    H=a2H2. (3.23)

    So by using the optimal system method, we get the optimal system of Eq (1.5), which is H1

    H2

    H3

    H1±H3

    H2±H3.

    In this section, we use the Lie symmetry method to find the exact solution of GEW equation. First by using the Lie symmetry, we convert PDEs into ODEs, then by using any appropriate method, we get the exact solution of ODEs. To obtain reduction form, we use subalgebra H1, H2, H3, H1+H3 and H2+H3.

    Case 1:

    The characteristic equation for H1=1nuu+tt is

    dx0=dtt=du1nu. (3.24)

    From this, we have

    x=r,s=ut1n, (3.25)

    where r and s are constant of integration. So

    u=F(r)t1n,and (3.26)
    ut=1nF(r)t1n+1,ux=F(r)t1n,uxxt=1nF(r)t1n+1.

    So by putting the value of ut, ux and uxxt in Eq (1.5), we get

    F(r)+nFn(r)F(r)+μF(r)=0, (3.27)

    which is a 2nd order non-linear ODE, which we can solved numerically.

    Case 2:

    The characteristics equation for H2=t is

    dx0=dt1=du0. (3.28)

    From this, we have

    x=r,u=s, (3.29)

    where r and s are constant of integration. So

    u=F(r),and (3.30)
    ut=0,ux=F(r),uxxt=0. (3.31)

    So by putting the value of ut, ux and uxxt in Eq (1.5), we get

    F(r)=0,and (3.32)
    F(r)=c1, (3.33)

    where c1 is the constant of integration.

    Case 3:

    The characteristics equation for H3=x is

    dx1=dt0=du0. (3.34)

    So, we have

    t=r,u=s, (3.35)

    where r and s are constant of integration. Thus we have

    u=F(r),and (3.36)
    ut=F(r),ux=0,uxxt=0. (3.37)

    By putting the value of ux, ut and uxxt in Eq (1.5), we have

    F(r)=0, (3.38)

    which implies that

    F(r)=c2, (3.39)

    where c2 is the constant of integration.

    Case 4:

    The characteristics equation for H1+H3=x+tt1nuu is

    dx1=dtt=du1nu. (3.40)

    So we have

    r=ext,s=ut1n, (3.41)

    where r and s are constant of integration. Thus we have

    u=F(r)t1n,and (3.42)
    ut=[F(r)+nrF(r)nt1n+1],ux=rF(r)t1n,uxxt=[nr3F(r)+(3n+1)r2F(r)+(n+1)rF(r)nt1n+1]. (3.43)

    By putting the value of ux, ut and uxxt in Eq (1.5), we get

    F(r)nrF(r)+nrFn(r)F(r)+μ(nr3F(r)+(3n+1)r2F(r)+(n+1)rF(r))=0, (3.44)

    which is a 3rd order non-linear ODE, which we can solve numerically.

    Case 5:

    The characteristic equation for H2+H3=t+x is

    dx1=dt1=du0. (3.45)

    So, we have

    r=xt,u=s, (3.46)

    where r and s are constant of integration. Thus we have

    u=F(r),and (3.47)
    ut=F(r),ux=F(r),uxxt=F(r). (3.48)

    By putting the value ut, ux and uxxt in Eq (1.5), we get a 3rd order non-linear ODE.

    F(r)+Fn(r)F(r)+μF(r)=0. (3.49)

    Travelling wave: A wave in which medium move in the direction of propagation of wave is called travelling wave. We can find the travelling wave solution of those equation which propagation the wave property. The wave occur in the form of u(x,t) = f(xct), where c is the wave speed which move in negative direction as c < 0 and in positive direction as c > 0.

    Procedure:

    We consider the non-linear PDE's in the form

    P(u,ux,ut,uxx,uxt,utt,..........)=0, (4.1)

    where u(x,t) is the travelling wave solution of non-linear PDE's. The wave variable ξ=xct is used to obtain the travelling wave solution so that u(x,t)=u(ξ), where ξ=xct.

    This allow us to make the following modification:

    t=cddξ,2t2=c2d2dξ2,x=ddξ,2x2=d2dξ2. (4.2)

    The above equation was used to transfer PDEs to ODEs.

    Q(u,cu,u,u,c2u,........)=0, (4.3)

    where u=dudξ.

    Equation (4.3) is then integrated till all terms contain derivative. For the sake of simplicity, the integration constant has been set to zero.

    The solution of Sine-cosine method is in the forms

    u(t,x)=λcosβ(ξμ),|ξ|π2μ, (4.4)

    and

    u(t,x)=λsinβ(ξμ),|ξ|πμ, (4.5)

    where λ and β are parameters and μ is the wave number and c is the wave speed. From Eqs (4.4) and (4.5), we have

    (un)(ξμ)=nβλnμcosnβ1(ξμ)sin(ξμ),(un)(ξμ)=n2μ2β2λncosnβ(ξμ)+nμ2λnβ(nβ1)cosnβ2(ξμ), (4.6)

    and

    (un)(ξμ)=nβλnμsinnβ1(ξμ)cos(ξμ),(un)(ξμ)=n2μ2β2λnsinnβ(ξμ)+nμ2λnβ(nβ1)sinnβ2(ξμ). (4.7)

    By substitute Eqs (4.6) and (4.7) into (4.3), we get equation in the form of sinR(ξμ) or cosR(ξμ). Then to compute the parameter, we compare the exponent of each pair and then coefficient of equal power of cosk(ξμ) or sink(ξμ). Then we get the system of equation in μ, β and λ that will be determined.

    The Sine-cosine method reduce the size of computational work than any other method which we have mention before.

    The GEW equation is

    ut+unuxbuxxt=0, (4.8)

    here b is a parameter.

    By using u(x,t)=u(ξ), ξ=xct. Equation (4.8) is transferred to non-linear ODE

    cu+unu+cbu=0. (4.9)

    Integrating (4.9) one time and use the constant of integration to be zero for the sake of simplicity, we find that

    cu+un+1n+1+cbu=0. (4.10)

    By using u(x,t)=λcosβ(ξμ) into (4.10), we get

    cλcosβ(ξμ)+1n+1λn+1cos(n+1)β(ξμ)+cbμ2λβ(β1)cos(β2)(ξμ)cbβ2μ2λcosβ(ξμ)=0. (4.11)

    By comparing the exponent of each pair and coefficient of equal power of cosk(ξμ), we get system of algebraic equation in β, μ and ξ, which is

    β10,β(n+1)=β2,cλcbβ2μ2λ=0,1n+1λn+1=cbμ2λβ(β1). (4.12)

    By figuring out the system, we get

    β=2n,μ=1b(n2),λ=[c2(n+1)(n+2)]1n, (4.13)

    by putting the value of β, μ and λ in (4.4), we get the following periodic solution for b<0 as shown in Figure 1.

    u1(x,t)=[c2(n+1)(n+2)sec2(1b(n2)(xct))]1n,|1b(n2)(xct)|<π2. (4.14)
    Figure 1.  Periodic solution, u1(x,t) of GEW with wave speed c=1, b=1, n=2 and x=4..4, t=4..4.

    When we take b>0, we get the following soliton solution as shown in Figure 2.

    u2(x,t)=[c2(n+2)(n+1)sech2(1b(n2)(xct))]1n,0<1b(n2)(xct)<π. (4.15)
    Figure 2.  Soliton solution u2(x,t) of GEW with wave speed c=1, b=1, n=2 and x=6..6, t=6..6.

    If we put ansatz u(x,t)=λsinβ(μξ), then for b<0, we get periodic solution as shown in Figure 3.

    u3(x,t)=[c2(n+2)(n+1)csc2(1b(n2)(xct))]1n. (4.16)
    Figure 3.  Periodic solution u3(x,t) of GEW with wave speed c=1, b=1, n=2 and x=6..6, t=6..6.

    And for b>0, we have soliton solution as shown in Figure 4.

    u4(x,t)=[c2(n+2)(n+1)csch2(1b(n2)(xct))]1n. (4.17)
    Figure 4.  Soliton solution u4(x,t) of GEW with wave speed c=1, b=1, n=2 and x=6..6, t=6..6.

    In this section we present the graphs of solution of the GEW equation. In Figure 1, we obtain periodic solution of GEW which is given in (4.14). It has wave speed value c=1.

    In the Figure 2, we give a soliton solution of GEW equation for given values of parameters c=1, b=1, n=2 and x=6..6, t=6..6.

    In the Figure 3, we give a periodic solution of GEW for given values of parameters c=1, b=1, n=2 and x=6..6, t=6..6.

    In the Figure 4, we give a soliton solution of GEW equation for given values of parameters c=1, b=1, n=2 and x=6..6, t=6..6.

    In the present article, we discussed the solution of Generalized Equal Width wave equation. We analysed the solution of GEW using two methods. At the first step transformed this PDE into ODE using Lie symmetry analysis. We used Lie symmetry analysis to reduce the complexity of the equations actually. It is worth-mentioning that GEW equation has not been discussed from the point of view of Lie symmetries. In the second step, we also used method of Sine-cosine to evaluate the exact solutions of this equation. We also analyzed the graphs of the solutions and found how they behave depending upon the parameters involved.

    The fourth author thanks Prince Sultan University for funding this paper through the TAS research group.

    The authors declare no conflict of interest.



    [1] A. Pikovsky, M. Rosenblum, J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge Nonlinear Science Series, Cambridge University Press, 2010. https://doi.org/10.1017/CBO9780511755743
    [2] K. Kaneko, Theory and applications of coupled map lattices, in Nonlinear Science: Theory and Applications, Wiley–Blackwell, 1993. Available from: https://cir.nii.ac.jp/crid/1573105973923422464.
    [3] E. Ott, K. Wiesenfeld, Chaos in Dynamical Systems, Phys. Today, 47 (1994). https://doi.org/10.1063/1.2808369 doi: 10.1063/1.2808369
    [4] G. Saxena, A. Prasad, R. Ramaswamy, Amplitude death: the emergence of stationarity in coupled nonlinear systems, Phys. Rep., 521 (2012), 205–228. https://doi.org/10.1016/j.physrep.2012.09.003 doi: 10.1016/j.physrep.2012.09.003
    [5] P. Kumar, A. Prasad, R. Ghosh, Stable phase-locking of an external-cavity diode laser subjected to external optical injection, J. Phys. B: At. Mol. Opt. Phys., 41 (2008), 135402. https://doi.org/10.1088/0953-4075/41/13/135402 doi: 10.1088/0953-4075/41/13/135402
    [6] B. Gallego, P. Cessi, Decadal variability of two oceans and an atmosphere, J. Clim., 14 (2001), 2815–2832. https://doi.org/10.1175/1520-0442(2001)014<2815:DVOTOA>2.0.CO;2 doi: 10.1175/1520-0442(2001)014<2815:DVOTOA>2.0.CO;2
    [7] H. Zhang, D. Xu, C. Lu, E. Qi, J. Hu, Y. Wu, Amplitude death of a multi-module floating airport, Nonlinear Dyn., 79 (2015), 2385–2394. https://doi.org/10.1007/s11071-014-1819-x doi: 10.1007/s11071-014-1819-x
    [8] T. Banerjee, D. Biswas, Amplitude death and synchronized states in nonlinear time-delay systems coupled through mean-field diffusion, Chaos, 23 (2013), 043101. https://doi.org/10.1063/1.4823599 doi: 10.1063/1.4823599
    [9] D. Ghosh, T. Banerjee, Transitions among the diverse oscillation quenching states induced by the interplay of direct and indirect coupling, Phys. Rev. E: Stat. Nonlinear Soft Matter Phys., 90 (2014), 062908. https://doi.org/10.1103/PhysRevE.90.062908 doi: 10.1103/PhysRevE.90.062908
    [10] G. B. Ermentrout, N. Kopell, Oscillator death in systems of coupled neural oscillators, SIAM J. Appl. Math., 50 (1990), 125–146. https://doi.org/10.1137/0150009 doi: 10.1137/0150009
    [11] A. Koseska, E. Volkov, J. Kurths, Oscillation quenching mechanisms: amplitude vs. oscillation death, Phys. Rep., 531 (2013), 173–199. https://doi.org/10.1016/j.physrep.2013.06.001 doi: 10.1016/j.physrep.2013.06.001
    [12] R. Curtu, Singular Hopf bifurcations and mixed-mode oscillations in a two-cell inhibitory neural network, Physica D, 239 (2010), 504–514. https://doi.org/10.1016/j.physd.2009.12.010 doi: 10.1016/j.physd.2009.12.010
    [13] A. Koseska, E. Volkov, J. Kurths, Parameter mismatches and oscillation death in coupled oscillators, Chaos, 20 (2010), 023132. https://doi.org/10.1063/1.3456937 doi: 10.1063/1.3456937
    [14] N. Suzuki, C. Furusawa, K. Kaneko, Oscillatory protein expression dynamics endows stem cells with robust differentiation potential, PLoS One, 6 (2011), e27232. https://doi.org/10.1371/journal.pone.0027232 doi: 10.1371/journal.pone.0027232
    [15] D. Biswas, N. Hui, T. Banerjee, Amplitude death in intrinsic time-delayed chaotic oscillators with direct–indirect coupling: the existence of death islands, Nonlinear Dyn., 88 (2017), 2783–2795. https://doi.org/10.1007/s11071-017-3411-7 doi: 10.1007/s11071-017-3411-7
    [16] A. H. Nayfeh, D. T. Mook, Nonlinear Oscillations, John Wiley & Sons, 2008.
    [17] A. Anees, Z. Ahmed, A technique for designing substitution box based on van der pol oscillator, Wireless Pers. Commun., 82 (2015), 1497–1503. https://doi.org/10.1007/s11277-015-2295-4 doi: 10.1007/s11277-015-2295-4
    [18] G. Juárez, M. Ramírez-Trocherie, Á. Báez, A. Lobato, E. Iglesias-Rodríguez, P. Padilla, et al., Hopf bifurcation for a fractional van der Pol oscillator and applications to aerodynamics: implications in flutter, J. Eng. Math., 139 (2023), 1–15. https://doi.org/10.1007/s10665-023-10258-7 doi: 10.1007/s10665-023-10258-7
    [19] S. Dutta, N. R. Cooper, Critical response of a quantum van der Pol oscillator, Phys. Rev. Lett., 123 (2019), 250401. https://doi.org/10.1103/PhysRevLett.123.250401 doi: 10.1103/PhysRevLett.123.250401
    [20] S. Wirkus, R. Rand, The dynamics of two coupled van der Pol oscillators with delay coupling, Nonlinear Dyn., 30 (2002), 205–221. https://doi.org/10.1023/A:1020536525009 doi: 10.1023/A:1020536525009
    [21] E. Camacho, R. Rand, H. Howland, Dynamics of two van der Pol oscillators coupled via a bath, Int. J. Solids Struct., 41 (2004), 2133–2143. https://doi.org/10.1016/j.ijsolstr.2003.11.035 doi: 10.1016/j.ijsolstr.2003.11.035
    [22] K. Konishi, Experimental evidence for amplitude death induced by dynamic coupling: van der Pol oscillators, in 2004 IEEE International Symposium on Circuits and Systems (ISCAS), 4 (2004), 792–795. https://doi.org/10.1109/ISCAS.2004.1329123
    [23] T. Endo, S. Mori, Mode analysis of a ring of a large number of mutually coupled van der Pol oscillators, IEEE Trans. Circuits Syst., 25 (1978), 7–18. https://doi.org/10.1109/TCS.1978.1084380 doi: 10.1109/TCS.1978.1084380
    [24] V. Resmi, G. Ambika, R. E. Amritkar, General mechanism for amplitude death in coupled systems, Phys. Rev. E: Stat. Nonlinear Soft Matter Phys., 84 (2011), 046212. https://doi.org/10.1103/PhysRevE.84.046212 doi: 10.1103/PhysRevE.84.046212
    [25] D. Ghosh, T. Banerjee Mixed-mode oscillation suppression states in coupled oscillators, Phys. Rev. E: Stat. Nonlinear Soft Matter Phys., 92 (2015), 052913. https://doi.org/10.1103/PhysRevE.92.052913 doi: 10.1103/PhysRevE.92.052913
    [26] C. O. Weiss, R. Vilaseca, Dynamics of lasers, NASA STI/Recon Tech. Rep. A, 92 (1991), 39875. Available from: https://ui.adsabs.harvard.edu/abs/1991STIA...9239875W/abstract.
    [27] K. A. Robbins, A new approach to subcritical instability and turbulent transitions in a simple dynamo, in Mathematical Proceedings of the Cambridge Philosophical Society, Cambridge University Press, 82 (1997), 309–325. https://doi/org/10.1017/S0305004100053950
    [28] B. Ermentrout, XPPAUT 5.0-the Differential Equations Tool, University of Pittsburgh, Pittsburgh, 2001.
    [29] E. X. DeJesus, C. Kaufman, Routh-Hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations, Phys. Rev. A: At. Mol. Opt. Phys., 35 (1987), 5288. https://doi.org/10.1103/PhysRevA.35.5288 doi: 10.1103/PhysRevA.35.5288
    [30] G. Saxena, A. Prasad, R. Ramaswamy, Amplitude death: the emergence of stationarity in coupled nonlinear systems, Phys. Rep., 521 (2012), 205–228. https://doi.org/10.1016/j.physrep.2012.09.003 doi: 10.1016/j.physrep.2012.09.003
    [31] A. Sharma, M. D. Shrimali Amplitude death with mean-field diffusion, Phys. Rev. E: Stat. Nonlinear Soft Matter Phys., 85 (2012), 057204. https://doi.org/10.1103/PhysRevE.85.057204 doi: 10.1103/PhysRevE.85.057204
    [32] A. Sharma, K. Suresh, K. Thamilmaran, A. Prasad, M. D. Shrimali, Effect of parameter mismatch and time delay interaction on density-induced amplitude death in coupled nonlinear oscillators, Nonlinear Dyn., 76 (2014), 1797–1806. https://doi.org/10.1007/s11071-014-1247-y doi: 10.1007/s11071-014-1247-y
    [33] T. Banerjee, D. Ghosh, Experimental observation of a transition from amplitude to oscillation death in coupled oscillators, Phys. Rev. E: Stat. Nonlinear Soft Matter Phys., 89 (2014), 062902. https://doi.org/10.1103/PhysRevE.89.062902 doi: 10.1103/PhysRevE.89.062902
    [34] N. K. Kamal, P. R. Sharma, M. D. Shrimali, Suppression of oscillations in mean-field diffusion, Pramana, 84 (2015), 237–247. https://doi.org/10.1007/s12043-015-0929-4 doi: 10.1007/s12043-015-0929-4
    [35] A. Zakharova, I. Schneider, Y. N. Kyrychko, K. B. Blyuss, A. Koseska, B. Fiedler, et al., Time delay control of symmetry-breaking primary and secondary oscillation death, Europhys. Lett., 104 (2013), 50004. https://doi.org/10.1209/0295-5075/104/50004 doi: 10.1209/0295-5075/104/50004
    [36] D. V. R. Reddy, A. Sen, G. L. Johnston, Time delay induced death in coupled limit cycle oscillators, Phys. Rev. Lett., 80 (1998), 5019. https://doi.org/10.1103/PhysRevLett.80.5109 doi: 10.1103/PhysRevLett.80.5109
    [37] D. V. R. Reddy, A. Sen, G. L. Johnston, Experimental evidence of time-delay-induced death in coupled limit-cycle oscillators, Phys. Rev. Lett., 85 (2000), 3381. https://doi.org/10.1103/PhysRevLett.85.3381 doi: 10.1103/PhysRevLett.85.3381
    [38] F. M. Atay, Distributed delays facilitate amplitude death of coupled oscillators, Phys. Rev. Lett., 91 (2003), 094101. https://doi.org/10.1103/PhysRevLett.91.094101 doi: 10.1103/PhysRevLett.91.094101
    [39] W. Zou, D. V. Senthilkumar, A. Koseska, J. Kurths, Generalizing the transition from amplitude to oscillation death in coupled oscillators, Phys. Rev. E: Stat. Nonlinear Soft Matter Phys., 88 (2013), 050901. https://doi.org/10.1103/PhysRevE.88.050901 doi: 10.1103/PhysRevE.88.050901
    [40] R. Karnatak, R. Ramaswamy, A. Prasad, Amplitude death in the absence of time delays in identical coupled oscillators, Phys. Rev. E: Stat. Nonlinear Soft Matter Phys., 76 (2007), 035201. https://doi.org/10.1103/PhysRevE.76.035201 doi: 10.1103/PhysRevE.76.035201
    [41] A. Sharma, P. R. Sharma, M. D. Shrimali, Amplitude death in nonlinear oscillators with indirect coupling, Phys. Lett. A, 376 (2012), 1562–1566. https://doi.org/10.1016/j.physleta.2012.03.033 doi: 10.1016/j.physleta.2012.03.033
    [42] C. R. Hens, O. I. Olusola, P. Pal, S. K. Dana, Oscillation death in diffusively coupled oscillators by local repulsive link, Phys. Rev. E: Stat. Nonlinear Soft Matter Phys., 88 (2013), 034902. https://doi.org/10.1103/PhysRevE.88.034902 doi: 10.1103/PhysRevE.88.034902
    [43] B. K. Bera, C. Hens, D. Ghosh, Emergence of amplitude death scenario in a network of oscillators under repulsive delay interaction, Phys. Lett. A, 380 (2016), 2366–2373. https://doi.org/10.1016/j.physleta.2016.05.028 doi: 10.1016/j.physleta.2016.05.028
    [44] N. K. Kamal, P. R. Sharma, M. D. Shrimali, Oscillation suppression in indirectly coupled limit cycle oscillators, Phys. Rev. E: Stat. Nonlinear Soft Matter Phys., 92 (2015), 022928. https://doi.org/10.1103/PhysRevE.92.022928 doi: 10.1103/PhysRevE.92.022928
    [45] P. R. Sharma, N. K. Kamal, U. K. Verma, K. Suresh, K. Thamilmaran, M. D. Shrimali, Suppression and revival of oscillation in indirectly coupled limit cycle oscillators, Phys. Lett. A, 380 (2016), 3178–3184. https://doi.org/10.1016/j.physleta.2016.07.041 doi: 10.1016/j.physleta.2016.07.041
    [46] A. Sharma, U. K. Verma, M. D. Shrimali, Phase-flip and oscillation-quenching-state transitions through environmental diffusive coupling, Phys. Rev. E: Stat. Nonlinear Soft Matter Phys., 94 (2016), 062218. https://doi.org/10.1103/PhysRevE.94.062218 doi: 10.1103/PhysRevE.94.062218
    [47] J. Choi, P. Kim, Reservoir computing based on quenched chaos, Chaos, Solitons Fractals, 140 (2020), 110131. https://doi.org/10.1016/j.chaos.2020.110131 doi: 10.1016/j.chaos.2020.110131
    [48] E. Ullner, A. Zaikin, E. I. Volkov, J. García-Ojalvo, Multistability and clustering in a population of synthetic genetic oscillators via phase-repulsive cell-to-cell communication, Phys. Rev. Lett., 99 (2007), 148103. https://doi.org/10.1103/PhysRevLett.99.148103 doi: 10.1103/PhysRevLett.99.148103
    [49] A. Takamatsu, Spontaneous switching among multiple spatio-temporal patterns in three-oscillator systems constructed with oscillatory cells of true slime mold, Physica D, 223 (2006), 180–188. https://doi.org/10.1016/j.physd.2006.09.001 doi: 10.1016/j.physd.2006.09.001
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