Solitary wave solutions of Sawada-Kotera equation using two efficient analytical methods

1.   Introduction

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 $E_1$ to higher energy state $E_2$. This condition is known as spontaneous absorption. Spontaneous emission represents the state of an atom as in excited state $E_2$ 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 $E_2$ to the lower lever $E_1$ 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:

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

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]:

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

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.

2.   Equilibrium points and linearized form of the laser model (1.4)

Lemma 2.1. Laser model (1.4) has at most two equilibria in $\mathbb{R}_+^2$. More specifically,

(i) Laser model (1.4) has a boundary fixed point $P_{0y}\left(0, \frac{p}{c}\right)\ \forall\ a, b, c, p, h$;

(ii) Laser model (1.4) has an interior fixed point $P^+_{xy}\left(\frac{ap-bc}{ab}, \frac{b}{a}\right)$ if $p > \frac{bc}{a}$.

Hereafter, for the fixed point $P_{xy}\left(x^*, y^*\right)$, linearized form is explored. The linearized system of (1.4) for the fixed point $P_{xy}\left(x^*, y^*\right)$ under the map $\left(\Phi_1, \Phi_2\right)\rightarrow \left(x_{n+1}, y_{n+1}\right)$ is

where

and

3.   Local dynamics for the fixed points of laser model (1.4)

Local dynamics for the fixed points $P_{0y}\left(0, \frac{p}{c}\right)$ and $P^+_{xy}\left(\frac{ap-bc}{ab}, \frac{b}{a}\right)$ of (1.4) is explored in this section, as follows.

3.1.   Local dynamics for $P_{0y}\left(0, \frac{p}{c}\right)$

The $J|_{P_{0y}\left(0, \frac{p}{c}\right)}$ evaluated at $P_{0y}\left(0, \frac{p}{c}\right)$ is

The roots of $J|_{P_{0y}\left(0, \frac{p}{c}\right)}$ evaluated at $P_{0y}\left(0, \frac{p}{c}\right)$ are $\lambda_1 = 1-bh+\frac{aph}{c}$ and $\lambda_2 = 1-ch$. So, by stability theory, dynamics of (1.4) for $P_{0y}\left(0, \frac{p}{c}\right)$ can be stated, as follows.

Lemma 3.1. For $P_{0y}\left(0, \frac{p}{c}\right)$, following statements hold:

(i) $P_{0y}\left(0, \frac{p}{c}\right)$ of the laser model (1.4) is a sink if

(ii) $P_{0y}\left(0, \frac{p}{c}\right)$ of the laser model (1.4) is a source if

(iii) $P_{0y}\left(0, \frac{p}{c}\right)$ of the laser model (1.4) is a saddle if

(iv) $P_{0y}\left(0, \frac{p}{c}\right)$ of the laser model (1.4) is non-hyperbolic if

3.2.   Local dynamics for $P^+_{xy}(\frac{ap-bc}{ab}, \frac{b}{a})$

The $J|_{P^+_{xy}(\frac{ap-bc}{ab}, \frac{b}{a})}$ evaluated at $P^+_{xy}(\frac{ap-bc}{ab}, \frac{b}{a})$ is

The corresponding auxiliary equation of (3.6) is

where

Finally, the roots of (3.7) are

where

Hereafter, local dynamics for the fixed point $P^+_{xy}\left(\frac{ap-bc}{ab}, \frac{b}{a}\right)$ of the laser model (1.4) can be summarized according to the sign of $\Delta < 0\ (\Delta\geq0)$, as follows.

Lemma 3.2. If $\Delta = \left(\frac{aph}{b}\right)^2 -4h^2\left(ap-bc\right) < 0$, then for $P^+_{xy}\left(\frac{ap-bc}{ab}, \frac{b}{a}\right)$ of the laser model (1.4), following statements hold:

(i) $P^+_{xy}\left(\frac{ap-bc}{ab}, \frac{b}{a}\right)$ is a locally asymptotically stable focus if

(ii) $P^+_{xy}\left(\frac{ap-bc}{ab}, \frac{b}{a}\right)$ is an unstable focus if

(iii) $P^+_{xy}\left(\frac{ap-bc}{ab}, \frac{b}{a}\right)$ is non-hyperbolic if

Lemma 3.3. If $\Delta = \left(\frac{aph}{b}\right)^2 -4h^2\left(ap-bc\right)\ge0$, then for $P^+_{xy}\left(\frac{ap-bc}{ab}, \frac{b}{a}\right)$ of the laser model (1.4), following statements hold:

(i) $P^+_{xy}\left(\frac{ap-bc}{ab}, \frac{b}{a}\right)$ is a locally asymptotically stable node if

(ii) $P^+_{xy}\left(\frac{ap-bc}{ab}, \frac{b}{a}\right)$ is an unstable node if

(iii) $P^+_{xy}\left(\frac{ap-bc}{ab}, \frac{b}{a}\right)$ is non-hyperbolic if

4.   Existence of bifurcations for $P_{0y}\left(0, \frac{p}{c}\right)$ and $P^+_{xy}\left(\frac{ap-bc}{ab}, \frac{b}{a}\right)$ of the laser model (1.4)

(i) If (3.5) true then one of eigenvalues of $J|_{P_{0y}\left(0, \frac{p}{c}\right)}$ evaluated at $P_{0y}\left(0, \frac{p}{c}\right)$ is $-1$, $i.e.$, $\lambda_1|_{(3.5)} = -1$ but $\lambda_2 |_{(3.5)} = 1-ch\not = 1 \ \mathrm{or}\ -1$. Thus, laser model (1.4) undergoes a flip bifurcation when $(a, b, c, h, p)$ passes through the curve

(ii) If (3.13) true then roots of $J|_{P^+_{xy}\left(\frac{ap-bc}{ab}, \frac{b}{a}\right)}$ evaluated at $P^+_{xy}\left(\frac{ap-bc}{ab}, \frac{b}{a}\right)$ satisfying $\left|\lambda_{1, 2}\right|_{ (3.13)} = 1$. Thus, laser model (1.4) undergoes Neimark-Sacker bifurcation when $(a, b, c, h, p)$ passes through the curve

(iii) If (3.16) true then one of the eigenvalues of $J|_{P^+_{xy}\left(\frac{ap-bc}{ab}, \frac{b}{a}\right)}$ evaluated at $P^+_{xy}\left(\frac{ap-bc}{ab}, \frac{b}{a}\right)$ is $-1$, $i.e.$, $\lambda_1|_{(3.16)} = -1$ but $\lambda_2|_{(3.16)} = \frac{2-3bh+bch^2}{2-bh}\not = 1 \ \mathrm{or} \ -1$. So model (1.4) undergoes a flip bifurcation when $(a, b, c, h, p)$ passes the curve

5.   Bifurcation analysis for fixed points $P_{0y}\left(0, \frac{p}{c}\right)$ and $P^+_{xy}\left(\frac{ap-bc}{ab}, \frac{b}{a}\right)$ of the laser model (1.4)

We study comprehensive bifurcation analysis for fixed points $P_{0y}\left(0, \frac{p}{c}\right)$ and $P^+_{xy}\left(\frac{ap-bc}{ab}, \frac{b}{a}\right)$ of the discrete model (1.4) in this section. This comprises the study of flip bifurcation for $P_{0y}(0, \frac{p}{c})$, Neimark-Sacker and flip bifurcations for $P^+_{xy}\left(\frac{ap-bc}{ab}, \frac{b}{a}\right)$ of the discrete laser model (1.4).

5.1.   Flip bifurcation for $P_{0y}\left(0, \frac{p}{c}\right)$

Recall that for $P_{0y}\left(0, \frac{p}{c}\right)$, 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 $P_{0y}\left(0, \frac{p}{c}\right)$ is degenerate with higher codimension.

5.2.   Neimark-Sacker bifurcation for $P^+_{xy}\left(\frac{ap-bc}{ab}, \frac{b}{a}\right)$

Hereafter, we will give detail Neimark-Sacker bifurcation for $P^+_{xy}\left(\frac{ap-bc}{ab}, \frac{b}{a}\right)$ of (1.4) by utilizing bifurcation theory ^[11,12,13]. Recall that if (4.2) holds, then $|\lambda_{1, 2}| = 1$. So, consider $p$ in a small neighborhood of $p^*$, $i.e.$, $p = p^*+\epsilon$ where $\epsilon < < 1$ and hence (1.4) gives

The $\epsilon$-dependence laser model (5.1) has fixed point $P^+_{xy}\left(\frac{a({p^*}+\epsilon)-bc}{ab}, \frac{b}{a}\right)$ and the $J|_{P^+_{xy}\left(\frac{a({p^*}+\epsilon)-bc}{ab}, \frac{b}{a}\right)}$ evaluated at $P^+_{xy}\left(\frac{a({p^*}+\epsilon)-bc}{ab}, \frac{b}{a}\right)$ is

The characteristic equation of $J|_{P^+_{xy}\left(\frac{a({p^*}+\epsilon)-bc}{ab}, \frac{b}{a}\right)}$ evaluated at $P^+_{xy}\left(\frac{a({p^*}+\epsilon)-bc}{ab}, \frac{b}{a}\right)$ is

where

The zeroes of (5.3) become

where

Moreover after some computation, one has

Additionally, it is required that $\lambda_{1, 2}^m\neq 1, \ m = 1, \cdots, 4$, that is same to $p(0)\neq -2, 0, 1, 2$, and so it is true by calculation. Using following transformation:

the fixed point $P^+_{xy}\left(\frac{a({p^*}+\epsilon)-bc}{ab}, \frac{b}{a}\right)$ of system (5.1) transform into $P_{00}(0, 0)$. So, (5.1) becomes

where ${x^*} = \frac{a({p^*}+\epsilon)-bc}{ab}$ and ${y^*} = \frac{b}{a}$. Hereafter, if $\epsilon = 0$, then normal form for (5.9) is studied. By Taylor series expansion at $\left(u_n, v_n\right) = (0, 0)$, from (5.9) one has

with

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

where

Hence, system (5.10) then implies

with

and

by

From (5.15), one gets

Now, in order for system (5.14) undergoes Neimark-Sacker bifurcation, it is necessary that $\Omega\not = 0$ (see ^[14,15,16]),

where

After manipulation, one gets

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 $P^+_{xy}\left(\frac{ap-bc}{ab}, \frac{b}{a}\right)$ as parameters $(a, b, c, h, p)$ goes through the curve (4.2). Additionally, attracting (repelling) closed curve bifurcates from $P^+_{xy}\left(\frac{ap-bc}{ab}, \frac{b}{a}\right)$ if $\Omega < 0 \ (\ \Omega > 0$).

5.3.   Flip bifurcation for $P^+_{xy}(\frac{ap-bc}{ab}, \frac{b}{a})$

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 $p^*$, $i.e$, $p = p^*+\epsilon$ where $\epsilon < < 1$, then laser model (1.4) becomes the form (5.1), that further takes the following form:

where

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

where

by the transformation

Hereafter, for (5.24) center manifold $M^c|_{P_{00}(0, 0)}$ at $P_{00}(0, 0)$ is explored where

After manipulations, we obtain

Finally, the map (5.24) restricts to $M^c|_{P_{00}(0, 0)}$ is

where

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

By computation, we get

and

From this analysis, we have the following theorem.

Theorem 5.2. For $P^+_{xy}\left(\frac{ap-bc}{ab}, \frac{b}{a}\right)$, map (5.1) undergoes a flip bifurcation if $\epsilon$ varies in a small neighborhood of $P_{00}(0, 0)$. Moreover, if $\Gamma_2 > 0\ (\Gamma_2 < 0)$, then period-$2$ orbits bifurcate from $P^+_{xy}\left(\frac{ap-bc}{ab}, \frac{b}{a}\right)$ are stable $(\mathrm{unstable})$.

6.   Numerical simulations

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, $P^+_{xy}\left(\frac{ap-bc}{ab}, \frac{b}{a}\right)$ 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 $P^+_{xy}\left(\frac{d}{c}, \frac{r\left(c-d\right)}{bc}\right)$ = $P^+_{xy}\left(0.07575757575757576, 3.2\right)$ 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 $P^+_{xy}(\frac{ap-bc}{ab}, \frac{b}{a})$ 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 $J_{P^+_{xy}\left(\frac{ap-bc}{ab}, \frac{b}{a}\right)}$ evaluated at $P^+_{xy}\left(\frac{ap-bc}{ab}, \frac{b}{a}\right)$ are

Moreover, from (5.7) one gets $\frac{bc h}{2(b h-1)} = 0.033939393939393936 > 0$. After some manipulation, from (5.21) one gets

Using (6.1) and (6.2) in (5.19), we obtain $\Omega = -0.0016523767625482892 < 0$. 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 $\Omega < 0$ (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.

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 $0.02\le p\le 4.9$. If $a = 1.13, b = 1.012, \ c = 1.5$, then from (3.16) one gets $p = 1.72336$. So, $P^+_{xy}\left(\frac{ap-bc}{ab}, \frac{b}{a}\right)$$= P^+_{xy}\left(0.375491, 0.895575\right)$ 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 $-9$.

6.1.   Fractal dimension

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

where Lyapunov exponents are $\lambda_i\ (i = 1, \cdots, n)$ and $l$ is the largest integer for which $\sum_{j = 1}^{l}\lambda_l\ge0$ and $\sum_{j = 1}^{l+1}\lambda_l < 0$. For the model (1.4), (6.3) becomes

Fixing $a = 1.13, \ b = 1.012, \ c = 1.5, \ h = 1.2$, then two Lyapunov exponents are commuted numerically, and one gets $\lambda_1 = 0.647622569300813$ ($\lambda_1 = 0.5786643421204638$) and $\lambda_2 = -1.0393814625814453$ ($\lambda_2 = -1.1111149350058382$) if $p = 1.785$ ($p = 1.89$). The fractal dimension for (1.4) becomes

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.

7.   Chaos control

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

The control force depicted in (7.1) given by

where $l_1$ and $l_2$ denotes feedback gains, and $x^* = \frac{ap-bc}{ab}$ and $y^* = \frac{b}{a}$. The $JC|_{P_{xy}(x^*, y^*)}$ of (7.1) is

where

Moreover, characteristic equation of $JC|_{P_{xy}(x^*, y^*)}$ evaluated at $P_{xy}(x^*, y^*)$ is

where

If $\lambda_{1, 2}$ are roots of (7.5), then

and

Now lines of marginal stability determines from the solution of $\lambda_1 = \pm 1$ and $\lambda_1\lambda_2 = 1$, which confirm that $|\lambda_{1, 2}| < 1$. If $\lambda_1\lambda_2 = 1$, then from (7.8), one gets

If $\lambda_1 = 1$, then from (7.7) and (7.8) one gets

Finally if $\lambda_1 = -1$, then from (7.7) and (7.8) one gets

Hence $L_1, \ L_2$ and $L_3$ in $(l_1, l_2)$-plane gives a triangular region which give $|\lambda_{1, 2}| < 1$ (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 $P^+_{xy}(\frac{ap-bc}{ab}, \frac{b}{a})$ the chaotic trajectories is stabilized.

8.   Conclusions

We have explored the local dynamical properties, bifurcations and chaos in the laser model (1.4) in the interior of $\mathbb{R}_+^2$. We have explored that for all parametric values, model (1.4) has a boundary equilibrium $P_{0y}\left(0, \frac{p}{c}\right)$ and if $p > \frac{bc}{a}$ then it has a unique positive equilibrium point $P^+_{xy}\left(\frac{ap-bc}{ab}, \frac{b}{a}\right)$. By linear stability theory, local dynamics with different topological classifications for fixed points $P_{0y}\left(0, \frac{p}{c}\right)$ and $P^+_{xy}\left(\frac{ap-bc}{ab}, \frac{b}{a}\right)$ are studied, and main finding are presented in Table 1. We have also explored existence of possible bifurcations for fixed points $P_{0y}\left(0, \frac{p}{c}\right)$ and $P^+_{xy}\left(\frac{ap-bc}{ab}, \frac{b}{a}\right)$. By computation, it is proved that for $P_{0y}\left(0, \frac{p}{c}\right)$ discrete model (1.4) cannot undergoes a flip bifurcation when parameters $(a, b, c, h, p)$ goes through the curve (4.1). But for $P^+_{xy}\left(\frac{ap-bc}{ab}, \frac{b}{a}\right)$, 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 $P^+_{xy}\left(\frac{ap-bc}{ab}, \frac{b}{a}\right)$ 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.

Acknowledgments

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

Conflict of interest

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