Research article Special Issues

Two-dimensional discrete-time laser model with chaos and bifurcations

  • We explore the local dynamical characteristics, chaos and bifurcations of a two-dimensional discrete laser model in R2+. It is shown that for all a, b, c and p, model has boundary fixed point P0y(0,pc), and the unique positive fixed point P+xy(apbcab,ba) if p>bca. Further, local dynamical characteristics with topological classifications for the fixed points P0y(0,pc) and P+xy(apbcab,ba) have explored by stability theory. It is investigated that flip bifurcation exists for the boundary fixed point P0y(0,pc), and also there exists a flip bifurcation if parameters vary in a small neighborhood of the unique positive fixed point P+xy(apbcab,ba). Moreover, it is also explored that for the fixed point P+xy(apbcab,ba), laser model undergoes a Neimark-Sacker bifurcation, and in the meantime stable invariant curve appears. Numerical simulations are implemented to verify not only obtain results but also exhibit complex dynamics of period 2, 3, 4, 5, 8 and 9. Further, maximum lyapunov exponents along with fractal dimension are computed numerically to validate chaotic behavior of the laser model. Lastly, feedback control method is utilized to stabilize chaos exists in the model.

    Citation: Abdul Qadeer Khan, Mohammed Bakheet Almatrafi. Two-dimensional discrete-time laser model with chaos and bifurcations[J]. AIMS Mathematics, 2023, 8(3): 6804-6828. doi: 10.3934/math.2023346

    Related Papers:

    [1] Ibraheem M. Alsulami . On the stability, chaos and bifurcation analysis of a discrete-time chemostat model using the piecewise constant argument method. AIMS Mathematics, 2024, 9(12): 33861-33878. doi: 10.3934/math.20241615
    [2] Yao Shi, Zhenyu Wang . Bifurcation analysis and chaos control of a discrete fractional-order Leslie-Gower model with fear factor. AIMS Mathematics, 2024, 9(11): 30298-30319. doi: 10.3934/math.20241462
    [3] Limei Liu, Xitong Zhong . Analysis and anti-control of period-doubling bifurcation for the one-dimensional discrete system with three parameters and a square term. AIMS Mathematics, 2025, 10(2): 3227-3250. doi: 10.3934/math.2025150
    [4] Abdul Qadeer Khan, Ayesha Yaqoob, Ateq Alsaadi . Neimark-Sacker bifurcation, chaos, and local stability of a discrete Hepatitis C virus model. AIMS Mathematics, 2024, 9(11): 31985-32013. doi: 10.3934/math.20241537
    [5] A. Q. Khan, Ibraheem M. Alsulami, S. K. A. Hamdani . Controlling the chaos and bifurcations of a discrete prey-predator model. AIMS Mathematics, 2024, 9(1): 1783-1818. doi: 10.3934/math.2024087
    [6] Ali Al Khabyah, Rizwan Ahmed, Muhammad Saeed Akram, Shehraz Akhtar . Stability, bifurcation, and chaos control in a discrete predator-prey model with strong Allee effect. AIMS Mathematics, 2023, 8(4): 8060-8081. doi: 10.3934/math.2023408
    [7] Xiaoming Su, Jiahui Wang, Adiya Bao . Stability analysis and chaos control in a discrete predator-prey system with Allee effect, fear effect, and refuge. AIMS Mathematics, 2024, 9(5): 13462-13491. doi: 10.3934/math.2024656
    [8] Xiongxiong Du, Xiaoling Han, Ceyu Lei . Dynamics of a nonlinear discrete predator-prey system with fear effect. AIMS Mathematics, 2023, 8(10): 23953-23973. doi: 10.3934/math.20231221
    [9] Figen Kangalgil, Seval Isșık . Effect of immigration in a predator-prey system: Stability, bifurcation and chaos. AIMS Mathematics, 2022, 7(8): 14354-14375. doi: 10.3934/math.2022791
    [10] A. Q. Khan, Ibraheem M. Alsulami . Complicate dynamical analysis of a discrete predator-prey model with a prey refuge. AIMS Mathematics, 2023, 8(7): 15035-15057. doi: 10.3934/math.2023768
  • We explore the local dynamical characteristics, chaos and bifurcations of a two-dimensional discrete laser model in R2+. It is shown that for all a, b, c and p, model has boundary fixed point P0y(0,pc), and the unique positive fixed point P+xy(apbcab,ba) if p>bca. Further, local dynamical characteristics with topological classifications for the fixed points P0y(0,pc) and P+xy(apbcab,ba) have explored by stability theory. It is investigated that flip bifurcation exists for the boundary fixed point P0y(0,pc), and also there exists a flip bifurcation if parameters vary in a small neighborhood of the unique positive fixed point P+xy(apbcab,ba). Moreover, it is also explored that for the fixed point P+xy(apbcab,ba), laser model undergoes a Neimark-Sacker bifurcation, and in the meantime stable invariant curve appears. Numerical simulations are implemented to verify not only obtain results but also exhibit complex dynamics of period 2, 3, 4, 5, 8 and 9. Further, maximum lyapunov exponents along with fractal dimension are computed numerically to validate chaotic behavior of the laser model. Lastly, feedback control method is utilized to stabilize chaos exists in the model.



    Differential and difference equations govern a wide range of physical models. In recent years, discrete models represented by difference equations have been better examined than continuous models. During the last few decades, mathematical models of physics, ecology, engineering, physiology, psychology, chemistry and social sciences have given birth to key research areas. Furthermore, like the biological model, physical model plays an active role in all fields of engineering and science; particularly laser model having enormous applications such as in industries for manufacturing purposes, e.g., for drilling, welding, cutting, marking, hardening, ablating, engraving and micromachining etc. The lasers can also be utilized for the applications of optical data storage, e.g., in DVDs and compact disks (CDs) etc. With regard to communications, the lasers are employed for long-distance optical data transmission, e.g., for inter-satellite communications. On medical side, the lasers are used for vision correction (LASIK) and eye surgery, dermatology, dentistry (e.g., photodynamic therapy of cancer). In optical metrology, it is used for extremely precise position measurements and optical surface profiling. Similarly, in energy technology, it is used for electricity generation and laser-induced nuclear fusion etc. Moreover, in recent times few high-power lasers are developed for potential use as directed energy weapons in the battle field or for destroying missiles, mines and projectiles etc. [1,2,3,4,5,6,7].

    The term laser is used for the phenomena of light production by emission of radiation. Laser is an appliance that is used for the production of coherent and single wavelength of light called monochromatic light. The basic principle for the working of laser based upon the theory of light proposed by Einstein in 1916, and developed by G. Gould in 1957. The first working ruby laser was introduced in 1960 by Theodore Maiman. One can differentiate laser light from ordinary light easily as it is very directional and monochromatic, and works on the principles of spontaneous emission, spontaneous absorption, population inversion and stimulated emission of the light. In nature, an atom exists in different energy states, when it absorb light it jumps from low energy state E1 to higher energy state E2. This condition is known as spontaneous absorption. Spontaneous emission represents the state of an atom as in excited state E2 it does not remains for a long period of the time about 108 is a life time of an atom in a excited state and it jumps down from level E2 to the lower lever E1 by loosing energy in the form of photon of energy h(v) is incoherent and moves in any direction. During this situation if another photon interacts with another atom which is in excited state, then it emits a pair of photon in same direction and same phase, and again it goes to its ground state. For the laser formation, it is necessary to amplify light in the cavity. For achieving population inversion, the atoms of gain medium accelerate to the high energy state by means of pumping. The chain of photon in cavity is obtained by population inversion and this chain become so intense that at the end of cavity reflected mirror cannot reflect them, and hence laser is obtained [8]. So, now we will give the mathematical formulation of the desired two-dimensional discrete-time laser model. As it is pointed out in [8], the number of laser photons in the cavity changes mainly for the following two reasons:

    (i) Due to stimulated emission, laser photons are continuously being added.

    (ii) Mirror transmission, absorption or scattering at the mirrors result continuously lost of laser photons.

    Thus, the rate of change of number of laser photon including loss and gain represented by the equation as follows:

    dxdt=gainloss=axybx. (1.1)

    The two rates, the rate of increase by means of stimulated emission as well as the rate of decrease by means of imperfect mirror indicates the rate at which the number of laser photons changes. In Eq (1.1), x denotes number of laser photon, y is number of atom in excited state, a is gain coefficient and b is the transmission coefficient. Moreover, equation (1.1) represents that due to stimulated emission the gain of laser photon is not only proportional to the number of photons already in cavity but also to the number of atoms. Thus, the type of atom used and other factors collectively indicate the efficiency of the stimulated emission while the rate of loss of laser photon is simplify proportional to the number of laser photon present. Now, one has the following equation for y, that is, the number atom in excited state where both spontaneous and emission causes y to decrease, and the pump p causes y to increase at some rate

    dydt=axycy+p, (1.2)

    where c is the spontaneous emission. Note that with opposite sign the first term appears in both Eqs (1.1) and (1.2). This shows the increase of x corresponds precisely to the decrease of y and represents the role of stimulated emission in laser. Furthermore, equations (1.1) and (1.2) describe the laser action. These equations show the relation of number of laser photon in the cavity and the number of loosing atoms but do not indicate what happens to the atoms when their electrons jumps to some other level or the photons that leave the cavity. So, equations (1.1) and (1.2) give the following model equations of two-dimensional continuous-time laser model [8,9,10]:

    dxdt=axybx, dydt=axycy+p. (1.3)

    Now, by Euler's forward scheme, the continuous-time laser model (1.3) becomes

    xn+1=(1bh)xn+ahxnyn, yn+1=(1ch)ynahxnyn+ph. (1.4)

    The goal of this paper is to investigate local dynamical properties, chaos and bifurcations of the laser model (1.4). Precisely the rest of the paper is structured as follows: Linearized form and existence for the fixed points of the laser model (1.4) are explored in Section 2. In Section 3, local dynamics for the fixed points of the laser model (1.4) is explored. The existence of bifurcation for the fixed points is explored in Section 4. The detailed bifurcation analysis for the fixed points is explore in Section 5. In Section 6, theoretical results are verified numerically, and this also includes to study the fractal dimension. In Section 7, chaos control is investigated by feedback control method. Brief summary of the paper is presented in Section 8.

    Lemma 2.1. Laser model (1.4) has at most two equilibria in R2+. More specifically,

    (i) Laser model (1.4) has a boundary fixed point P0y(0,pc)  a,b,c,p,h;

    (ii) Laser model (1.4) has an interior fixed point P+xy(apbcab,ba) if p>bca.

    Hereafter, for the fixed point Pxy(x,y), linearized form is explored. The linearized system of (1.4) for the fixed point Pxy(x,y) under the map (Φ1,Φ2)(xn+1,yn+1) is

    Γn+1=J|(x,y)Γn, (2.1)

    where

    J|(x,y):=(1bh+ahyahxahy1chahx), (2.2)

    and

    Φ1=(1bh)xn+ahxnyn, Φ2=(1ch)ynahxnyn+ph. (2.3)

    Local dynamics for the fixed points P0y(0,pc) and P+xy(apbcab,ba) of (1.4) is explored in this section, as follows.

    The J|P0y(0,pc) evaluated at P0y(0,pc) is

    J|P0y(0,pc)=(1bh+aphc0aphc1ch). (3.1)

    The roots of J|P0y(0,pc) evaluated at P0y(0,pc) are λ1=1bh+aphc and λ2=1ch. So, by stability theory, dynamics of (1.4) for P0y(0,pc) can be stated, as follows.

    Lemma 3.1. For P0y(0,pc), following statements hold:

    (i) P0y(0,pc) of the laser model (1.4) is a sink if

    p>bch2cah and c<2h; (3.2)

    (ii) P0y(0,pc) of the laser model (1.4) is a source if

    p<bch2cah and c>2h; (3.3)

    (iii) P0y(0,pc) of the laser model (1.4) is a saddle if

    p<bch2cah and c<2h; (3.4)

    (iv) P0y(0,pc) of the laser model (1.4) is non-hyperbolic if

    p=bch2cah or c=2h. (3.5)

    The J|P+xy(apbcab,ba) evaluated at P+xy(apbcab,ba) is

    J|P+xy(apbcab,ba)=(1aphbchbbhbaphb). (3.6)

    The corresponding auxiliary equation of (3.6) is

    λ2p1λ+q1=0, (3.7)

    where

    p1=2baphb,q1=baphb+h2(apbc). (3.8)

    Finally, the roots of (3.7) are

    λ1,2=p1±Δ2, (3.9)

    where

    Δ=(2baphb)24(baphb+h2(apbc))=(aphb)24h2(apbc). (3.10)

    Hereafter, local dynamics for the fixed point P+xy(apbcab,ba) of the laser model (1.4) can be summarized according to the sign of Δ<0 (Δ0), as follows.

    Lemma 3.2. If Δ=(aphb)24h2(apbc)<0, then for P+xy(apbcab,ba) of the laser model (1.4), following statements hold:

    (i) P+xy(apbcab,ba) is a locally asymptotically stable focus if

    p<hb2ca(bh1); (3.11)

    (ii) P+xy(apbcab,ba) is an unstable focus if

    p>hb2ca(bh1); (3.12)

    (iii) P+xy(apbcab,ba) is non-hyperbolic if

    p=hb2ca(bh1). (3.13)

    Lemma 3.3. If Δ=(aphb)24h2(apbc)0, then for P+xy(apbcab,ba) of the laser model (1.4), following statements hold:

    (i) P+xy(apbcab,ba) is a locally asymptotically stable node if

    p<b(4bh2c)ah(2bh); (3.14)

    (ii) P+xy(apbcab,ba) is an unstable node if

    p>b(4bh2c)ah(2bh); (3.15)

    (iii) P+xy(apbcab,ba) is non-hyperbolic if

    p=b(4bh2c)ah(2bh). (3.16)

    (i) If (3.5) true then one of eigenvalues of J|P0y(0,pc) evaluated at P0y(0,pc) is 1, i.e., λ1|(3.5)=1 but λ2|(3.5)=1ch1 or 1. Thus, laser model (1.4) undergoes a flip bifurcation when (a,b,c,h,p) passes through the curve

    FB|P0y(0,pc)={(a,b,c,h,p), p=bch2cah}; (4.1)

    (ii) If (3.13) true then roots of J|P+xy(apbcab,ba) evaluated at P+xy(apbcab,ba) satisfying |λ1,2|(3.13)=1. Thus, laser model (1.4) undergoes Neimark-Sacker bifurcation when (a,b,c,h,p) passes through the curve

    NSB|P+xy(apbcab,ba)={(a,b,c,h,p), p=hb2ca(bh1)}; (4.2)

    (iii) If (3.16) true then one of the eigenvalues of J|P+xy(apbcab,ba) evaluated at P+xy(apbcab,ba) is 1, i.e., λ1|(3.16)=1 but λ2|(3.16)=23bh+bch22bh1 or 1. So model (1.4) undergoes a flip bifurcation when (a,b,c,h,p) passes the curve

    FB|P+xy(apbcab,ba)={(a,b,c,h,p), p=b(4bch2)ah(2bh)}. (4.3)

    We study comprehensive bifurcation analysis for fixed points P0y(0,pc) and P+xy(apbcab,ba) of the discrete model (1.4) in this section. This comprises the study of flip bifurcation for P0y(0,pc), Neimark-Sacker and flip bifurcations for P+xy(apbcab,ba) of the discrete laser model (1.4).

    Recall that for P0y(0,pc), model (1.4) undergoes a flip bifurcation when (a,b,c,h,p) passes through the curve (4.1). But by calculation it can not occur, and therefore P0y(0,pc) is degenerate with higher codimension.

    Hereafter, we will give detail Neimark-Sacker bifurcation for P+xy(apbcab,ba) of (1.4) by utilizing bifurcation theory [11,12,13]. Recall that if (4.2) holds, then |λ1,2|=1. So, consider p in a small neighborhood of p, i.e., p=p+ϵ where ϵ<<1 and hence (1.4) gives

    xn+1=(1bh)xn+ahxnyn, yn+1=(1ch)ynahxnyn+h(p+ϵ). (5.1)

    The ϵ-dependence laser model (5.1) has fixed point P+xy(a(p+ϵ)bcab,ba) and the J|P+xy(a(p+ϵ)bcab,ba) evaluated at P+xy(a(p+ϵ)bcab,ba) is

    J|P+xy(a(p+ϵ)bcab,ba)=(1a(p+ϵ)hbchbhbba(p+ϵ)hb). (5.2)

    The characteristic equation of J|P+xy(a(p+ϵ)bcab,ba) evaluated at P+xy(a(p+ϵ)bcab,ba) is

    λ2p1(ϵ)λ+q1(ϵ)=0, (5.3)

    where

    p1(ϵ)=2ba(p+ϵ)hb,q1(ϵ)=ba(p+ϵ)hb+h2(a(p+ϵ)bc). (5.4)

    The zeroes of (5.3) become

    λ1,2=p1(ϵ)±ι4q1(ϵ)p21(ϵ)2, (5.5)

    where

    Δ=p21(ϵ)4q1(ϵ)=(a(p+ϵ)hb)24h2(a(p+ϵ)bc). (5.6)

    Moreover after some computation, one has

    d|λ1,2|dϵ|ϵ=0=bch2(bh1)0. (5.7)

    Additionally, it is required that λm1,21, m=1,,4, that is same to p(0)2,0,1,2, and so it is true by calculation. Using following transformation:

    un=xnx,vn=yny, (5.8)

    the fixed point P+xy(a(p+ϵ)bcab,ba) of system (5.1) transform into P00(0,0). So, (5.1) becomes

    un+1=(1bh)(un+x)+ah(un+x)(vn+y)x,vn+1=(1ch)(vn+y)ah(un+x)(vn+y)+h(p+ϵ)y, (5.9)

    where x=a(p+ϵ)bcab and y=ba. Hereafter, if ϵ=0, then normal form for (5.9) is studied. By Taylor series expansion at (un,vn)=(0,0), from (5.9) one has

    un+1=a11un+a12vn+a13unvn,vn+1=a21un+a22vn+a23unvn, (5.10)

    with

    a11=1bh+ahy,a12=ahx,a13=ah,a21=ahy,a22=1chahx,a23=ah. (5.11)

    Hereafter, one construct invertible matrix T that put linear part of (5.10) to required canonical form

    T:=(a120ηa11ζ), (5.12)

    where

    η=2baph2b,ζ=124h2(apbc)(aphb)2. (5.13)

    Hence, system (5.10) then implies

    Xn+1=ηXnζYn+ˉP,Yn+1=ζXn+ηYn+ˉQ, (5.14)

    with

    ˉP(Xn,Yn)=l11X2t+l12XtYt,ˉQ(Xn,Yn)=l21X2t+l22XtYt, (5.15)

    and

    l11=(ηa11)a13,l12=ζa13,l21=ηa11ζ(a13(ηa11)a12a23),l22=(ηa11)(a12a23a13), (5.16)

    by

    (unvn):=(a120ηa11ζ)(XtYt). (5.17)

    From (5.15), one gets

    ˉPXtXt|P00(0,0)=2l11,ˉPXtYt|P00(0,0)=l12,ˉPYtYt|P00(0,0)=ˉPXnXnXn|P00(0,0)=ˉPXnXnYn|P00(0,0)=ˉPXnYnYn|P00(0,0)=ˉPYnYnYn|P00(0,0)=0,ˉQXtXt|P00(0,0)=2l21,ˉQXtYt|P00(0,0)=l22,ˉQYnYn|P00(0,0)=ˉQXnXnXn|P00(0,0)=ˉQXnXnYn|P00(0,0)=ˉQXnYnYn|P00(0,0)=ˉQYnYnYn|P00(0,0)=0. (5.18)

    Now, in order for system (5.14) undergoes Neimark-Sacker bifurcation, it is necessary that Ω0 (see [14,15,16]),

    Ω=((12ˉλ)ˉλ21λξ11ξ20)12ξ112ξ022+(ˉλξ21), (5.19)

    where

    ξ02=18[ˉPXnXnˉPYnYn+2ˉQXnYn+ι(ˉQXnXnˉQYnYn+2ˉPXnYn)]|P00(0,0),ξ11=14[ˉPXnXn+ˉPYnYn+ι(ˉQXnXn+ˉQYnYn)]|P00(0,0),ξ20=18[ˉPXnXnˉPYnYn+2ˉQXnYn+ι(ˉQXnXnˉQYnYn2ˉPXnYn)]|P00(0,0),ξ21=116[ˉPXnXnXn+ˉPXnYnYn+ˉQXnXnYn+ˉQYnYnYn+ι(ˉQXnXnXn+ˉQXnYnYnˉPXnXnYnPYnYnYn)]|P00(0,0). (5.20)

    After manipulation, one gets

    ξ02=14[l11l22+ι(l21+l12)],ξ11=12[l11+ιl21],ξ20=14[l11+l22+ι(l21l12)],ξ21=0. (5.21)

    So from this analysis and condition(s) for Neimark-Sacker bifurcation discussed in [17,18], we have the following result about Neimark-Sacker bifurcation of the model (1.4).

    Theorem 5.1. If (3.13) holds, then model (1.4) undergoes Neimark-Sacker bifurcation for the fixed point P+xy(apbcab,ba) as parameters (a,b,c,h,p) goes through the curve (4.2). Additionally, attracting (repelling) closed curve bifurcates from P+xy(apbcab,ba) if Ω<0 ( Ω>0).

    Recall that if parameters (a,b,c,h,p) crosses (4.3), then laser model (1.4) undergoes a flip bifurcation. Now if p in a neighborhood of p, i.e, p=p+ϵ where ϵ<<1, then laser model (1.4) becomes the form (5.1), that further takes the following form:

    un+1=^a11un+^a12vn+^a13unvn,vn+1=^a21un+^a22vn+^a23unvn+γ01ϵ, (5.22)

    where

    ^a11=1bh+ahy,^a12=ahx,^a13=ah,^a21=ahy,^a22=1chahx,^a23=ah,γ01=h, (5.23)

    by using transformation given in (5.8). Now (5.22) becomes

    (Xn+1Yn+1)=(100λ2)(XnYn)+(ˆPˆQ), (5.24)

    where

    ˆP=^a13(λ2^a11)^a12^a23^a12(1+λ2)unvnγ01(1+λ2)ϵ,ˆQ=^a13(1+^a11)+^a12^a23^a12(1+λ2)unvn+γ01(1+λ2)ϵ,unvn=^a12(1+^a11)X2n+(^a12(λ2^a11)^a12(1+^a11))XnYn+^a12(λ2^a11)Y2n, (5.25)

    by the transformation

    (unvn):=(^a12^a121^a11λ2^a11)(XnYn). (5.26)

    Hereafter, for (5.24) center manifold Mc|P00(0,0) at P00(0,0) is explored where

    Mc|P00(0,0)={(Xn,Yn):Yn=c0ϵ+c1X2n+c2Xnϵ+c3ϵ3+O((|Xn|+|ϵ|)3)}. (5.27)

    After manipulations, we obtain

    c0=γ011λ22,c1=(1+^a11)((1+^a11)^a13+^a12^a23)1λ22,c2=γ011λ22(((1+^a11)^a13+^a12^a23)(λ22^a121)),c3=0. (5.28)

    Finally, the map (5.24) restricts to Mc|P00(0,0) is

    f(xn)=xn+h1x2n+h2xnϵ+h3x2nϵ+h4xnϵ2+h5x3n+O((|Xn|+|ϵ|)4), (5.29)

    where

    h1=(1+^a11)1+λ2[(λ2^a11)^a13^a12^a23],h2=co(λ22^a111)1+λ2[(λ2^a11)^a13^a12^a23],h3=[(λ2^a11)^a13^a12^a23][c2(λ22^a111)+2coc1(λ2^a11)]1+λ2,h4=2coc2(λ2^a11)1+λ2[(λ2^a11)^a13^a12^a23],h5=c1(λ22^a111)1+λ2[(λ2^a11)^a13^a12^a23]. (5.30)

    Thus discriminatory quantities are non-zero in order for (5.29) undergoes flip bifurcation [19,20,21]

    Γ1=(2fxnϵ+12fϵ2fx2n)|P00(0,0),Γ2=(163fx3n+(122fx2n)2)|P00(0,0). (5.31)

    By computation, we get

    Γ1=ah(2bh)2(bch24)(44bh+bch2)(b2c2h24bc+2b2ch2)(bch2b2bh+apbcb), (5.32)

    and

    Γ2=2a(2ahah2(apbc)b)(bch242bh)(bch24)(bch2b2bh+apbcb)(44bh+bch2)(b2c2h24bc+2b2ch2)+4(2bh)2a2h4(44bh+bch2)2(bch2b2bh+apbcb)2. (5.33)

    From this analysis, we have the following theorem.

    Theorem 5.2. For P+xy(apbcab,ba), map (5.1) undergoes a flip bifurcation if ϵ varies in a small neighborhood of P00(0,0). Moreover, if Γ2>0 (Γ2<0), then period-2 orbits bifurcate from P+xy(apbcab,ba) are stable (unstable).

    In this section, obtained theoretical results are illustrated numerically. Let a=0.5, b=1.6, c=0.03, h=1.12, then from (3.13) one gets p=0.217212. In this case, P+xy(apbcab,ba) of (1.4) is a stable focus if p<0.217212, loss its stability if p=0.217212 and meanwhile it is an unstable focus if p>0.217212. For this, if p=0.11<0.217212, then from Figure 1a it is clear that P+xy(dc,r(cd)bc) = P+xy(0.07575757575757576,3.2) of laser model (1.4) is a stable focus. Correspondingly, for the remaining values of p, if p<0.217212, then, respective equilibrium point is also a stable focus (Figure 1b1l). But if p=0.217212, then P+xy(apbcab,ba) change the behavior and thus an unstable focus if p>0.217212, and as a consequence, stable curve appears, which indicates that laser model (1.4) undergoes supercritical Neimark-Sacker bifurcation. To prove this, if p=0.217222>0.217212, then roots of JP+xy(apbcab,ba) evaluated at P+xy(apbcab,ba) are

    λ1,2=0.961986±0.273103ι. (6.1)
    Figure 1.  Phase portrait for the laser model (1.4).

    Moreover, from (5.7) one gets bch2(bh1)=0.033939393939393936>0. After some manipulation, from (5.21) one gets

    ξ02=0.000225798+0.0381484ι,ξ11=0.01064390.000172025ι,ξ20=0.01086970.0383205ι,ξ21=0. (6.2)

    Using (6.1) and (6.2) in (5.19), we obtain Ω=0.0016523767625482892<0. Hence, laser model (1.4) undergoes supercritical a Neimark-Sacker bifurcation if p=0.217222>0.217212 and meanwhile stable curve appear (Figure 2a). Similarly, if p>0.217212, then Ω<0 (See Table 1) and their corresponding closed curves are drawn in Figure 2b2l. Moreover, M.L.E and bifurcation diagrams are drawn in Figure 3. Finally, bifurcation diagrams in 3D are presented in Figure 4.

    Figure 2.  Closed curves of the laser model (1.4).
    Table 1.  Numerical values of Ω for p>0.217212.
    Bifurcation values if p>0.217212 Values of Ω
    0.217222 0.0016523767625482892<0
    0.2172124 0.0016523096396154336<0
    0.217412 0.001653861096012872<0
    0.217612 0.0016549900334033758<0
    0.220 0.0016707992836875582<0
    0.217812 0.0016563267520259238<0
    0.2179912 0.0016575224832815231<0
    0.21889 0.0016634917766999676<0
    0.219 0.0016642191214841734<0
    0.2189 0.0016635579278594646<0
    0.218899 0.001663551313003689<0
    0.21889 0.0016634917766999676<0

     | Show Table
    DownLoad: CSV
    Figure 3.  Bifurcation diagram and their corresponding M.L.E of laser model (1.4) for P+xy(apbcab,ba). (a, b) Bifurcation diagram of (1.4) if 0.000089p0.47 and initial condition (0.00549,4.037654). (c) M.L.E corresponding to (a) and (b).
    Figure 4.  3D bifurcation diagrams.

    Hereafter, we will give simulation in order to validate obtained results in Section 5.3 by fixing a=1.13,b=1.012, c=1.5 and 0.02p4.9. If a=1.13,b=1.012, c=1.5, then from (3.16) one gets p=1.72336. So, P+xy(apbcab,ba)=P+xy(0.375491,0.895575) is a stable node if p<1.72336, non-hyperbolic if p=1.72336, an unstable node if p>1.72336, and thus flip bifurcation exists if p>1.72336. Figure 5a and 5b indicates that fixed point is a stable if p<1.72336 and loss stability at p=1.72336. Now corresponding to Figure 5a and 5b, M.L.E are drawn in Figure 5c. Additionally 3D flip bifurcation diagrams equivalent to Figure 5a and 5b are presented in Figure 6a6h. Finally, more plots of laser model (1.4) that related with Figure 5a and 5b are drawn in Figure 7a7f, that shows model (1.4) yields a complex dynamics having orbits of period 2, 3, 4, 5, 8 and 9.

    Figure 5.  Bifurcation diagram and their corresponding M.L.E of (1.4) for P+xy(apbcab,ba). (a, b) Bifurcation diagram of the model if 0.02p4.9 and (0.712,0.4). (c) M.L.E corresponding to (a) and (b).
    Figure 6.  3D flip bifurcation diagrams of the model (1.4).
    Figure 7.  Complex dynamics behaviors of (1.4).

    The strange attractors designated by fractal dimension for discrete system takes the following form [22,23]:

    dL=l+lj=1λl|λl|, (6.3)

    where Lyapunov exponents are λi (i=1,,n) and l is the largest integer for which lj=1λl0 and l+1j=1λl<0. For the model (1.4), (6.3) becomes

    dL=1+λ1|λ2|, λ1>0>λ2. (6.4)

    Fixing a=1.13, b=1.012, c=1.5, h=1.2, then two Lyapunov exponents are commuted numerically, and one gets λ1=0.647622569300813 (λ1=0.5786643421204638) and λ2=1.0393814625814453 (λ2=1.1111149350058382) if p=1.785 (p=1.89). The fractal dimension for (1.4) becomes

    dL=1+0.6476225693008131.0393814625814453=1.6230858781196482 for  p=1.785,dL=1+0.57866434212046381.1111149350058382=1.5208028080280802 for  p=1.89. (6.5)

    Finally, for above chosen values, strange attractors are also plotted in Figure 8a and 8b, which shows that (1.4) gives complex dynamics if p increases.

    Figure 8.  Strange attractor of (1.4) if p=1.785 (p=1.89) with (0.040,1.04).

    We explore chaos control by utilizing state feedback control method [24,25] in this section. On adding un as a control force to (1.4), one has

    xn+1=(1bh)xn+ahxnyn+un,yn+1=(1ch)ynahxnyn+ph. (7.1)

    The control force depicted in (7.1) given by

    un=l1(xnx)l2(yny), (7.2)

    where l1 and l2 denotes feedback gains, and x=apbcab and y=ba. The JC|Pxy(x,y) of (7.1) is

    JC|Pxy(x,y)=(n11l1n12l2n21n22), (7.3)

    where

    n11=1,n12=aphbchb,n21=bh,n22=baphb. (7.4)

    Moreover, characteristic equation of JC|Pxy(x,y) evaluated at Pxy(x,y) is

    λ2tr(JC|Pxy(x,y))λ+det(JC|Pxy(x,y))=0, (7.5)

    where

    tr(JC|Pxy(x,y))=n11+n22l1,det(JC|Pxy(x,y))=n22(n11l1)n21(n12l2). (7.6)

    If λ1,2 are roots of (7.5), then

    λ1+λ2=n11+n22l1, (7.7)

    and

    λ1λ2=n22(n11l1)n21(n12l2). (7.8)

    Now lines of marginal stability determines from the solution of λ1=±1 and λ1λ2=1, which confirm that |λ1,2|<1. If λ1λ2=1, then from (7.8), one gets

    L1:(1aphb)(1l1)+aph2bch2bhl21=0. (7.9)

    If λ1=1, then from (7.7) and (7.8) one gets

    L2:aphbl1+aph2bch2bhl2=0. (7.10)

    Finally if λ1=1, then from (7.7) and (7.8) one gets

    L3:(2l1)(2baph)+bh(aphbchbl2)=0. (7.11)

    Hence L1, L2 and L3 in (l1,l2)-plane gives a triangular region which give |λ1,2|<1 (Figure 9a).

    Figure 9.  Control of chaotic trajectories of (7.1) for a=1.13, b=1.012, c=2.4, h=1.2,p=1.32 with (0.01,0.723), (a) Stability region in (l1,l2)-plan. (b, c) Behavior for xn and yn, respectively.

    To justify how this method works and also control chaos at unstable state, we presented simulation. Figure 9b and 9c shows that for P+xy(apbcab,ba) the chaotic trajectories is stabilized.

    We have explored the local dynamical properties, bifurcations and chaos in the laser model (1.4) in the interior of R2+. We have explored that for all parametric values, model (1.4) has a boundary equilibrium P0y(0,pc) and if p>bca then it has a unique positive equilibrium point P+xy(apbcab,ba). By linear stability theory, local dynamics with different topological classifications for fixed points P0y(0,pc) and P+xy(apbcab,ba) are studied, and main finding are presented in Table 1. We have also explored existence of possible bifurcations for fixed points P0y(0,pc) and P+xy(apbcab,ba). By computation, it is proved that for P0y(0,pc) discrete model (1.4) cannot undergoes a flip bifurcation when parameters (a,b,c,h,p) goes through the curve (4.1). But for P+xy(apbcab,ba), laser model (1.4) undergoes both Neimark-Saker and flip bifurcations when (a,b,c,h,p) respectively goes through the parametric curves (4.2) and (4.3). It is important here to mention that the existence of Neimark-Sacker bifurcation for the unique positive equilibrium point P+xy(apbcab,ba) of discrete laser model (1.4) means that there exists a periodic or quasi-periodic oscillations between laser photon and number of atom in the excited state. Some numerical simulations have been implemented to validate not only obtain results but also exhibit complex dynamics of period 2, 3, 4, 5, 8 and 9. The M.L.E as well as fractal dimension has computed numerically to validate chaotic behaviors of the laser model. Finally, feedback control method is applied to stabilized chaos exists in laser model (1.4). The chaos control and bifurcation analysis of a discrete fractional-order laser model are our next aim to study.

    Table 2.  Fixed points P0y(0,pc) and P+xy(apbcab,ba) along their behavior of model (1.4).
    Fixed pointsCorresponding behavior
    P0y(0,pc) sink if p>bch2cah and c<2h;
    source if p<bch2cah and c>2h;
    saddle if p<bch2cah and c<2h;
    non-hyperbolic if p=bch2cah or c=2h.
    P+xy(apbcab,ba) locally asymptotically stable focus if
    p<hb2ca(bh1);
    an unstable focus if
    p>hb2ca(bh1);
    non-hyperbolic if
    p=hb2ca(bh1);
    locally asymptotically stable node if
    p<b(4bh2c)ah(2bh);
    an unstable node if
    p>b(4bh2c)ah(2bh);
    non-hyperbolic if
    p=b(4bh2c)ah(2bh).

     | Show Table
    DownLoad: CSV

    This work was partially supported by the Higher Education Commission (HEC) of Pakistan.

    The authors declare that they have no conflicts of interest regarding the publication of this paper.



    [1] J. C. Ion, Laser processing of engineering materials: principles, procedure and industrial application, Elsevier, 2005.
    [2] Z. X. Guo, S. Kumar, Discrete-ordinates solution of short-pulsed laser transport in two-dimensional turbid media, Appl. Opt., 40 (2001), 3156–3163. https://doi.org/10.1364/AO.40.003156 doi: 10.1364/AO.40.003156
    [3] X. Y. Jiang, C. M. Soukoulis, Time dependent theory for random lasers, Phys. Rev. Lett., 85 (2000), 70. https://doi.org/10.1103/PhysRevLett.85.70 doi: 10.1103/PhysRevLett.85.70
    [4] A. R. Jha, Infrared technology: applications to electro-optics, photonic devices and sensors, New York: Wiley, 2000.
    [5] B. A. Lengyel, Introduction to laser physics, New York: Wiley, 1966.
    [6] J. Ohtsubo, Semiconductor lasers: stability, instability and chaos, Berlin, Heidelberg: Springer, 2013. https://doi.org/10.1007/978-3-642-30147-6
    [7] M. J. Weber, Science and technology of laser glass, J. Non-Cryst. Solids, 123 (1990), 208–222. https://doi.org/10.1016/0022-3093(90)90786-L doi: 10.1016/0022-3093(90)90786-L
    [8] P. W. Milonni, J. H. Eberly, Lasers physics, New York: Wiley, 2010. https://doi.org/10.1002/9780470409718
    [9] H. Haken, Synergetics, Berlin, Heidelberg: Springer, 1983. https://doi.org/10.1007/978-3-642-88338-5
    [10] S. H. Strogatz, Nonlinear dynamics and chaos with student solutions manual: with applications to physics, biology, chemistry, and engineering, Boca Raton: CRC Press, 2018. https://doi.org/10.1201/9780429399640
    [11] J. Guckenheimer, P. Holmes, Nonlinear oscillations, dynamical systems and bifurcation of vector fields, New York: Springer, 1983. https://doi.org/10.1007/978-1-4612-1140-2
    [12] Y. A. Kuznetsov, Elements of applied bifurcation theorey, New York: Springer, 2004. https://doi.org/10.1007/978-1-4757-3978-7
    [13] Z. Y. Hu, Z. D. Teng, L. Zhang, Stability and bifurcation analysis of a discrete predator-prey model with nonmonotonic functional response, Nonlinear Anal. Real World Appl., 12 (2011), 2356–2377. https://doi.org/10.1016/j.nonrwa.2011.02.009 doi: 10.1016/j.nonrwa.2011.02.009
    [14] A. Q. Khan, J. Y. Ma, D. M. Xiao, Bifurcations of a two-dimensional discrete time plant-herbivore system, Commun. Nonlinear Sci. Numer. Simul., 39 (2016), 185–198. https://doi.org/10.1016/j.cnsns.2016.02.037 doi: 10.1016/j.cnsns.2016.02.037
    [15] A. Q. Khan, J. Y. Ma, D. M. Xiao, Global dynamics and bifurcation analysis of a host-parasitoid model with strong Allee effect, J. Biol. Dyn., 11 (2017), 121–146. https://doi.org/10.1080/17513758.2016.1254287 doi: 10.1080/17513758.2016.1254287
    [16] Z. J. Jing, J. P. Yang, Bifurcation and chaos in discrete-time predator-prey system, Chaos Solitons Fract., 27 (2006), 259–277. https://doi.org/10.1016/j.chaos.2005.03.040 doi: 10.1016/j.chaos.2005.03.040
    [17] C. H. Zhang, X. P. Yan, G. H. Cui, Hopf bifurcations in a predator-prey system with a discrete delay and a distributed delay, Nonlinear Anal. Real World Appl., 11 (2010), 4141–4153. https://doi.org/10.1016/j.nonrwa.2010.05.001 doi: 10.1016/j.nonrwa.2010.05.001
    [18] M. Sen, M. Banerjee, A. Morozov, Bifurcation analysis of a ratio-dependent prey-predator model with the Allee effect, Ecol. Complex., 11 (2012), 12–27. https://doi.org/10.1016/j.ecocom.2012.01.002 doi: 10.1016/j.ecocom.2012.01.002
    [19] C. D. Huang, J. Wang, X. P. Chen, J. D. Cao, Bifurcations in a fractional-order BAM neural network with four different delays, Neural Netw., 141 (2021), 344–354. https://doi.org/10.1016/j.neunet.2021.04.005 doi: 10.1016/j.neunet.2021.04.005
    [20] E. Kaslik, I. R. Radulescu, Stability and bifurcations in fractional-order gene regulatory networks, Appl. Math. Comput., 421 (2022), 126916. https://doi.org/10.1016/j.amc.2022.126916 doi: 10.1016/j.amc.2022.126916
    [21] J. Alidousti, Stability and bifurcation analysis for a fractional prey-predator scavenger model, Appl. Math. Model., 81 (2020), 342–355. https://doi.org/10.1016/j.apm.2019.11.025 doi: 10.1016/j.apm.2019.11.025
    [22] J. H. E. Cartwright, Nonlinear stiffness, Lyapunov exponents, and attractor dimension, Phys. Lett. A, 264 (1999), 298–302. https://doi.org/10.1016/S0375-9601(99)00793-8
    [23] J. L. Kaplan, J. A. Yorke, Preturbulence: a regime observed in a fluid flow model of Lorenz, Commun. Math. Phys., 67 (1979), 93–108. https://doi.org/10.1007/BF01221359 doi: 10.1007/BF01221359
    [24] S. N. Elaydi, An introduction to difference equations, New York: Springer, 1996. https://doi.org/10.1007/978-1-4757-9168-6
    [25] S. Lynch, Dynamical systems with applications using Mathematica, Boston: Birkhäuser, 2007.
  • This article has been cited by:

    1. Khush Bukht Mehdi, Zubia Mehdi, Shamaila Samreen, Imran Siddique, Adel A. Elmandouh, Mamdouh E. Elbrolosy, M.S. Osman, Novel exact traveling wave solutions of the space-time fractional Sharma Tasso-Olver equation via three reliable methods, 2024, 11, 26668181, 100784, 10.1016/j.padiff.2024.100784
    2. Jan Muhammad, Usman Younas, Aziz Khan, Thabet Abdeljawad, D.K. Almutairi, On the study of double dispersive equation in the Murnaghan’s rod: Dynamics of diversity wave structures, 2024, 12, 26668181, 100916, 10.1016/j.padiff.2024.100916
    3. Md. Tarikul Islam, Tobibur Rahman, Mustafa Inc, Md. Ali Akbar, Comprehensive soliton solutions of fractional stochastic Kraenkel–Manna–Merle equations in ferromagnetic materials, 2024, 56, 1572-817X, 10.1007/s11082-024-06471-y
    4. U.H.M. Zaman, Mohammad Asif Arefin, M. Ali Akbar, M. Hafiz Uddin, Diverse soliton wave profile analysis in ion-acoustic wave through an improved analytical approach, 2024, 12, 26668181, 100932, 10.1016/j.padiff.2024.100932
    5. Ejaz Hussain, Abdul Mutlib, Zhao Li, Adham E.Ragab, Syed Asif Ai Shah, Emad A. Az-Zo’bi, Nida Raees, Theoretical examination of solitary waves for Sharma–Tasso–Olver Burger equation by stability and sensitivity analysis, 2024, 75, 0044-2275, 10.1007/s00033-024-02225-8
    6. Sujoy Devnath, Mst. Munny Khatun, M. Ali Akbar, Analytical solutions and soliton behaviors in the space fractional Heisenberg ferromagnetic spin chain equation, 2024, 11, 26668181, 100783, 10.1016/j.padiff.2024.100783
    7. Muhammad Iqbal, Muhammad Bilal Riaz, Muhammad Aziz ur Rehman, Investigating optical soliton pattern and dynamical analysis of Lonngren wave equation via phase portraits, 2024, 11, 26668181, 100862, 10.1016/j.padiff.2024.100862
    8. Chongkun Xu, Mati ur Rahman, Homan Emadifar, Bifurcations, chaotic behavior, sensitivity analysis and soliton solutions of the extended Kadometsev–Petviashvili equation, 2024, 56, 0306-8919, 10.1007/s11082-023-05958-4
    9. Muhammad Bilal, Javed Iqbal, Kamal Shah, Bahaaeldin Abdalla, Thabet Abdeljawad, Ikram Ullah, Analytical solutions of the space–time fractional Kundu–Eckhaus equation by using modified extended direct algebraic method, 2024, 11, 26668181, 100832, 10.1016/j.padiff.2024.100832
    10. Messaoud Berkal, Mohammed Bakheet Almatrafi, Bifurcation and Stability of Two-Dimensional Activator–Inhibitor Model with Fractional-Order Derivative, 2023, 7, 2504-3110, 344, 10.3390/fractalfract7050344
  • Reader Comments
  • © 2023 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(2034) PDF downloads(99) Cited by(10)

Figures and Tables

Figures(9)  /  Tables(2)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog