
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(ap−bcab,ba) if p>bca. Further, local dynamical characteristics with topological classifications for the fixed points P0y(0,pc) and P+xy(ap−bcab,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(ap−bcab,ba). Moreover, it is also explored that for the fixed point P+xy(ap−bcab,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
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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(ap−bcab,ba) if p>bca. Further, local dynamical characteristics with topological classifications for the fixed points P0y(0,pc) and P+xy(ap−bcab,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(ap−bcab,ba). Moreover, it is also explored that for the fixed point P+xy(ap−bcab,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 10−8 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=gain−loss=axy−bx. | (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=−axy−cy+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=axy−bx, dydt=−axy−cy+p. | (1.3) |
Now, by Euler's forward scheme, the continuous-time laser model (1.3) becomes
xn+1=(1−bh)xn+ahxnyn, yn+1=(1−ch)yn−ahxnyn+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(ap−bcab,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∗):=(1−bh+ahy∗ahx∗−ahy∗1−ch−ahx∗), | (2.2) |
and
Φ1=(1−bh)xn+ahxnyn, Φ2=(1−ch)yn−ahxnyn+ph. | (2.3) |
Local dynamics for the fixed points P0y(0,pc) and P+xy(ap−bcab,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)=(1−bh+aphc0−aphc1−ch). | (3.1) |
The roots of J|P0y(0,pc) evaluated at P0y(0,pc) are λ1=1−bh+aphc and λ2=1−ch. 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>bch−2cah and c<2h; | (3.2) |
(ii) P0y(0,pc) of the laser model (1.4) is a source if
p<bch−2cah and c>2h; | (3.3) |
(iii) P0y(0,pc) of the laser model (1.4) is a saddle if
p<bch−2cah and c<2h; | (3.4) |
(iv) P0y(0,pc) of the laser model (1.4) is non-hyperbolic if
p=bch−2cah or c=2h. | (3.5) |
The J|P+xy(ap−bcab,ba) evaluated at P+xy(ap−bcab,ba) is
J|P+xy(ap−bcab,ba)=(1aph−bchb−bhb−aphb). | (3.6) |
The corresponding auxiliary equation of (3.6) is
λ2−p1λ+q1=0, | (3.7) |
where
p1=2b−aphb,q1=b−aphb+h2(ap−bc). | (3.8) |
Finally, the roots of (3.7) are
λ1,2=p1±√Δ2, | (3.9) |
where
Δ=(2b−aphb)2−4(b−aphb+h2(ap−bc))=(aphb)2−4h2(ap−bc). | (3.10) |
Hereafter, local dynamics for the fixed point P+xy(ap−bcab,ba) of the laser model (1.4) can be summarized according to the sign of Δ<0 (Δ≥0), as follows.
Lemma 3.2. If Δ=(aphb)2−4h2(ap−bc)<0, then for P+xy(ap−bcab,ba) of the laser model (1.4), following statements hold:
(i) P+xy(ap−bcab,ba) is a locally asymptotically stable focus if
p<hb2ca(bh−1); | (3.11) |
(ii) P+xy(ap−bcab,ba) is an unstable focus if
p>hb2ca(bh−1); | (3.12) |
(iii) P+xy(ap−bcab,ba) is non-hyperbolic if
p=hb2ca(bh−1). | (3.13) |
Lemma 3.3. If Δ=(aphb)2−4h2(ap−bc)≥0, then for P+xy(ap−bcab,ba) of the laser model (1.4), following statements hold:
(i) P+xy(ap−bcab,ba) is a locally asymptotically stable node if
p<b(4−bh2c)ah(2−bh); | (3.14) |
(ii) P+xy(ap−bcab,ba) is an unstable node if
p>b(4−bh2c)ah(2−bh); | (3.15) |
(iii) P+xy(ap−bcab,ba) is non-hyperbolic if
p=b(4−bh2c)ah(2−bh). | (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)=1−ch≠1 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=bch−2cah}; | (4.1) |
(ii) If (3.13) true then roots of J|P+xy(ap−bcab,ba) evaluated at P+xy(ap−bcab,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(ap−bcab,ba)={(a,b,c,h,p), p=hb2ca(bh−1)}; | (4.2) |
(iii) If (3.16) true then one of the eigenvalues of J|P+xy(ap−bcab,ba) evaluated at P+xy(ap−bcab,ba) is −1, i.e., λ1|(3.16)=−1 but λ2|(3.16)=2−3bh+bch22−bh≠1 or −1. So model (1.4) undergoes a flip bifurcation when (a,b,c,h,p) passes the curve
FB|P+xy(ap−bcab,ba)={(a,b,c,h,p), p=b(4−bch2)ah(2−bh)}. | (4.3) |
We study comprehensive bifurcation analysis for fixed points P0y(0,pc) and P+xy(ap−bcab,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(ap−bcab,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(ap−bcab,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=(1−bh)xn+ahxnyn, yn+1=(1−ch)yn−ahxnyn+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)h−bchb−hbb−a(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
λ2−p1(ϵ)λ+q1(ϵ)=0, | (5.3) |
where
p1(ϵ)=2b−a(p∗+ϵ)hb,q1(ϵ)=b−a(p∗+ϵ)hb+h2(a(p∗+ϵ)−bc). | (5.4) |
The zeroes of (5.3) become
λ1,2=p1(ϵ)±ι√4q1(ϵ)−p21(ϵ)2, | (5.5) |
where
Δ=p21(ϵ)−4q1(ϵ)=(a(p∗+ϵ)hb)2−4h2(a(p∗+ϵ)−bc). | (5.6) |
Moreover after some computation, one has
d|λ1,2|dϵ|ϵ=0=bch2(bh−1)≠0. | (5.7) |
Additionally, it is required that λm1,2≠1, m=1,⋯,4, that is same to p(0)≠−2,0,1,2, and so it is true by calculation. Using following transformation:
un=xn−x∗,vn=yn−y∗, | (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=(1−bh)(un+x∗)+ah(un+x∗)(vn+y∗)−x∗,vn+1=(1−ch)(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=1−bh+ahy∗,a12=ahx∗,a13=ah,a21=−ahy∗,a22=1−ch−ahx∗,a23=−ah. | (5.11) |
Hereafter, one construct invertible matrix T that put linear part of (5.10) to required canonical form
T:=(a120η−a11−ζ), | (5.12) |
where
η=2b−aph2b,ζ=12√4h2(ap−bc)−(aphb)2. | (5.13) |
Hence, system (5.10) then implies
Xn+1=ηXn−ζYn+ˉP,Yn+1=ζXn+ηYn+ˉQ, | (5.14) |
with
ˉP(Xn,Yn)=l11X2t+l12XtYt,ˉQ(Xn,Yn)=l21X2t+l22XtYt, | (5.15) |
and
l11=(η−a11)a13,l12=−ζa13,l21=η−a11ζ(a13(η−a11)−a12a23),l22=(η−a11)(a12a23−a13), | (5.16) |
by
(unvn):=(a120η−a11−ζ)(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 Ω≠0 (see [14,15,16]),
Ω=−ℜ((1−2ˉλ)ˉλ21−λξ11ξ20)−12‖ξ11‖2−‖ξ02‖2+ℜ(ˉλξ21), | (5.19) |
where
ξ02=18[ˉPXnXn−ˉPYnYn+2ˉQXnYn+ι(ˉQXnXn−ˉQYnYn+2ˉPXnYn)]|P00(0,0),ξ11=14[ˉPXnXn+ˉPYnYn+ι(ˉQXnXn+ˉQYnYn)]|P00(0,0),ξ20=18[ˉPXnXn−ˉPYnYn+2ˉQXnYn+ι(ˉQXnXn−ˉQYnYn−2ˉPXnYn)]|P00(0,0),ξ21=116[ˉPXnXnXn+ˉPXnYnYn+ˉQXnXnYn+ˉQYnYnYn+ι(ˉQXnXnXn+ˉQXnYnYn−ˉPXnXnYn−PYnYnYn)]|P00(0,0). | (5.20) |
After manipulation, one gets
ξ02=14[l11−l22+ι(l21+l12)],ξ11=12[l11+ιl21],ξ20=14[l11+l22+ι(l21−l12)],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(ap−bcab,ba) as parameters (a,b,c,h,p) goes through the curve (4.2). Additionally, attracting (repelling) closed curve bifurcates from P+xy(ap−bcab,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=1−bh+ahy∗,^a12=ahx∗,^a13=ah,^a21=−ahy∗,^a22=1−ch−ahx∗,^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)unvn−γ01(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^a12−1−^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=γ011−λ22,c1=−(1+^a11)((1+^a11)^a13+^a12^a23)1−λ22,c2=γ011−λ22(((1+^a11)^a13+^a12^a23)(λ2−2^a12−1)),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(λ2−2^a11−1)1+λ2[(λ2−^a11)^a13−^a12^a23],h3=[(λ2−^a11)^a13−^a12^a23][c2(λ2−2^a11−1)+2coc1(λ2−^a11)]1+λ2,h4=2coc2(λ2−^a11)1+λ2[(λ2−^a11)^a13−^a12^a23],h5=c1(λ2−2^a11−1)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=(∂2f∂xn∂ϵ+12∂f∂ϵ∂2f∂x2n)|P00(0,0),Γ2=(16∂3f∂x3n+(12∂2f∂x2n)2)|P00(0,0). | (5.31) |
By computation, we get
Γ1=ah(2−bh)2(bch2−4)(4−4bh+bch2)(b2c2h2−4bc+2b2ch2)(bch−2b2−bh+ap−bcb), | (5.32) |
and
Γ2=−2a(2ah−ah2(ap−bc)b)(bch2−42−bh)(bch2−4)(bch−2b2−bh+ap−bcb)(4−4bh+bch2)(b2c2h2−4bc+2b2ch2)+4(2−bh)2a2h4(4−4bh+bch2)2(bch−2b2−bh+ap−bcb)2. | (5.33) |
From this analysis, we have the following theorem.
Theorem 5.2. For P+xy(ap−bcab,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(ap−bcab,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(ap−bcab,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(c−d)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 1b–1l). But if p=0.217212, then P+xy(ap−bcab,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(ap−bcab,ba) evaluated at P+xy(ap−bcab,ba) are
λ1,2=0.961986±0.273103ι. | (6.1) |
Moreover, from (5.7) one gets bch2(bh−1)=0.033939393939393936>0. After some manipulation, from (5.21) one gets
ξ02=0.000225798+0.0381484ι,ξ11=−0.0106439−0.000172025ι,ξ20=−0.0108697−0.0383205ι,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 2b–2l. Moreover, M.L.E and bifurcation diagrams are drawn in Figure 3. Finally, bifurcation diagrams in 3D are presented in Figure 4.
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 |
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 0.02≤p≤4.9. If a=1.13,b=1.012, c=1.5, then from (3.16) one gets p=1.72336. So, P+xy(ap−bcab,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 6a–6h. Finally, more plots of laser model (1.4) that related with Figure 5a and 5b are drawn in Figure 7a–7f, that shows model (1.4) yields a complex dynamics having orbits of period −2, −3, −4, −5, −8 and −9.
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=1λl≥0 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.
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=(1−bh)xn+ahxnyn+un,yn+1=(1−ch)yn−ahxnyn+ph. | (7.1) |
The control force depicted in (7.1) given by
un=−l1(xn−x∗)−l2(yn−y∗), | (7.2) |
where l1 and l2 denotes feedback gains, and x∗=ap−bcab and y∗=ba. The JC|Pxy(x∗,y∗) of (7.1) is
JC|Pxy(x∗,y∗)=(n11−l1n12−l2n21n22), | (7.3) |
where
n11=1,n12=aph−bchb,n21=−bh,n22=b−aphb. | (7.4) |
Moreover, characteristic equation of JC|Pxy(x∗,y∗) evaluated at Pxy(x∗,y∗) is
λ2−tr(JC|Pxy(x∗,y∗))λ+det(JC|Pxy(x∗,y∗))=0, | (7.5) |
where
tr(JC|Pxy(x∗,y∗))=n11+n22−l1,det(JC|Pxy(x∗,y∗))=n22(n11−l1)−n21(n12−l2). | (7.6) |
If λ1,2 are roots of (7.5), then
λ1+λ2=n11+n22−l1, | (7.7) |
and
λ1λ2=n22(n11−l1)−n21(n12−l2). | (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:(1−aphb)(1−l1)+aph2−bch2−bhl2−1=0. | (7.9) |
If λ1=1, then from (7.7) and (7.8) one gets
L2:aphbl1+aph2−bch2−bhl2=0. | (7.10) |
Finally if λ1=−1, then from (7.7) and (7.8) one gets
L3:(2−l1)(2b−aph)+bh(aph−bch−bl2)=0. | (7.11) |
Hence L1, L2 and L3 in (l1,l2)-plane gives a triangular region which give |λ1,2|<1 (Figure 9a).
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(ap−bcab,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(ap−bcab,ba). By linear stability theory, local dynamics with different topological classifications for fixed points P0y(0,pc) and P+xy(ap−bcab,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(ap−bcab,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(ap−bcab,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(ap−bcab,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.
Fixed points | Corresponding behavior |
P0y(0,pc) | sink if p>bch−2cah and c<2h; |
source if p<bch−2cah and c>2h; | |
saddle if p<bch−2cah and c<2h; | |
non-hyperbolic if p=bch−2cah or c=2h. | |
P+xy(ap−bcab,ba) | locally asymptotically stable focus if |
p<hb2ca(bh−1); | |
an unstable focus if | |
p>hb2ca(bh−1); | |
non-hyperbolic if | |
p=hb2ca(bh−1); | |
locally asymptotically stable node if | |
p<b(4−bh2c)ah(2−bh); | |
an unstable node if | |
p>b(4−bh2c)ah(2−bh); | |
non-hyperbolic if | |
p=b(4−bh2c)ah(2−bh). |
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.
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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 |
Fixed points | Corresponding behavior |
P0y(0,pc) | sink if p>bch−2cah and c<2h; |
source if p<bch−2cah and c>2h; | |
saddle if p<bch−2cah and c<2h; | |
non-hyperbolic if p=bch−2cah or c=2h. | |
P+xy(ap−bcab,ba) | locally asymptotically stable focus if |
p<hb2ca(bh−1); | |
an unstable focus if | |
p>hb2ca(bh−1); | |
non-hyperbolic if | |
p=hb2ca(bh−1); | |
locally asymptotically stable node if | |
p<b(4−bh2c)ah(2−bh); | |
an unstable node if | |
p>b(4−bh2c)ah(2−bh); | |
non-hyperbolic if | |
p=b(4−bh2c)ah(2−bh). |
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 |
Fixed points | Corresponding behavior |
P0y(0,pc) | sink if p>bch−2cah and c<2h; |
source if p<bch−2cah and c>2h; | |
saddle if p<bch−2cah and c<2h; | |
non-hyperbolic if p=bch−2cah or c=2h. | |
P+xy(ap−bcab,ba) | locally asymptotically stable focus if |
p<hb2ca(bh−1); | |
an unstable focus if | |
p>hb2ca(bh−1); | |
non-hyperbolic if | |
p=hb2ca(bh−1); | |
locally asymptotically stable node if | |
p<b(4−bh2c)ah(2−bh); | |
an unstable node if | |
p>b(4−bh2c)ah(2−bh); | |
non-hyperbolic if | |
p=b(4−bh2c)ah(2−bh). |