1 Introduction

The dynamic analyses of different biological models are important fields of science that have received much attention in recent years [1,2,3,4,5]. For example, AlDosari et al., in 2023, investigated the dynamic behaviors of drug release using nanoparticles in cancer cells on two-dimensional materials for drug targeting [6]. Aljaloud et al. studied the emotional behaviors of cross-flow nanofluid displacement caused by the cylinder with activation energy and second-order slip characteristics [7]. In 2022, Prasad et al. investigated the peristaltic activity in the caisson nanofluid blood flow with irreversible aspects in the vertical non-uniform channel using dynamic analysis of nonlinear coupled partial differential equations [8]. Aljaloud et al. investigated phase change and heat transfer in water/copper oxide nanofluid enclosed in a cylindrical tank with a porous medium: using molecular dynamics analysis [9]. In addition, Banawas et al. investigated the coating processes of reinforced calcium phosphate cement for dental pulp with changes in primary properties using molecular dynamics analysis [10]. Studying the dynamic analysis of the gene models is also significant, since it can help prevent possible risks, such as incurable diseases.

The process of human evolution is based on inheritance by genes in DNA and the transmission of information through the functioning of nervous systems [11, 12]. A gene is a sequence of nucleotides in DNA [13]. In general, genes are parts of DNA, DNA is part of a chromosome, and a chromosome is part of a cell (Fig. 1 shows the constituent units of genes) [14]. Gene is the basic physical and functional unit of heredity, which can be inherited from parents or mutated (less than 1% of the total). Some genetic traits, such as eye color or the number of organs, are visible, and others are not, such as blood type, the risk of certain diseases, or thousands of biochemical processes [15]. Traits that are not visible may affect people after several generations.

Fig. 1
figure 1

Each cell contains chromosomes, each chromosome contains a string of DNAs, and each DNA contains different A, T, G, and C genes

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According to the previous paragraphs and dynamic analysis, identifying the dynamic behavior of gene regulatory networks is of great importance in revealing the mechanism of the cell. Various mathematical models have been introduced to identify and describe the gene mechanism, which is generally divided into two types of maps (discrete model) and differential equations (continuous model) [11, 16, 17]. Some gene models are based on integer-order differential equations. For example, Zhi-Hong Guan et al. have studied the cluster synchronization of coupled genetic regulatory networks with time-varying delays through periodic adaptive alternating control at different nodes [18]. Fengyan Wu et al. have investigated a stable two-dimensional model of gene dynamics in terms of competence development in Bacillus subtilis under Lévy noise effects and Brownian motions [19]. Qiang Lai et al. showed that gene regulatory networks have monotonic, bistable, periodic, and chaotic behaviors [20].

In contrast to continuous-time genetic models that have been widely studied, less attention has been paid to map-based discrete-time models. Distinctive features of discrete-time gene maps are their high potential for modeling complex behavior, saving memory, and simplifying large network computations. For example, Zoran Levnajić et al. investigated the collective dynamics of coupled 2D chaotic maps on the E. coli gene regulatory network [21]. Dandan Yue et al. studied the dynamics of a discrete genetic model by bifurcation diagram [22]. Ming Liu et al. examined the stability and bifurcation of a novel discrete chaotic map based on a gene regulatory network [23]. S.L.T. de Souza et al. investigated the dynamics of the map of difference equations derived from chemical reactions for gene expression and regulation by the Lyapunov and bifurcation diagrams [24].

Discrete fractional-order derivatives are used in map-based dynamic analysis to describe the effect of memory on processing, because discrete fractional-order derivatives are relatively more accurate than integer-order derivatives [25, 26]. Discrete fractional-order derivatives save previous past behaviors in their memory [27, 28]. In past studies, fractional-order derivatives have been a powerful tool in modeling biological systems [29,30,31]. For example, Karthikeyan Rajagopal et al. investigated the dynamic properties of a 1D fractional-order neuron map using the bifurcation diagram and the Lyapunov exponent diagram [32]. Bo Yan et al. studied the synchronization of multiple neural networks of Hindmarsh–Rose neurons with fractional order [33]. In addition, numerical simulations of the Hénon–Lozi map and their fractional-order forms have been investigated by Adel Ouannas et al. [34].

Fractional-order derivatives have also been applied to continuous gene models. For example, in 2018, Binbin Tao et al. proposed a new fractional-order two-gene regulatory network model with delay as a suitable description for memory and inherited traits in the genetic regulatory networks [35]. They stated that it is the first time that stability dynamics and Hopf bifurcation are investigated for the delayed fractional-order model of the two-gene regulatory network. Zhe Zhang et al. presented a new stability measure of a fractional-order gene regulation network system with time delay for stability analysis using Jensen's inequality, Wirtinger's inequality, fractional-order Lyapunov method, and integral mean value theorem [36]. Chengdai Huang et al. presented a hybrid controller to control Hopf bifurcation in a fractional-order gene regulatory network model [37]. Fengli Ren et al. studied some measures of Mittag–Leffler stability and generalized Mittag–Leffler stability using the fractional Lyapunov method for a class of fractional-order gene regulatory networks [38]. Mani Mallika Arjunan et al. investigated fractional-order gene regulatory networks with time delay and impulsive effects [39].

The underlying physical and biological mechanisms of heredity have not been investigated in discrete fractional-order map-based gene models. Discrete fractional-order derivatives in map-based gene models create more accurate inheritance characteristics than integer-order models, because it includes richer functional gene mechanisms [40]. Discrete fractional-order gene models correctly represent biological characteristics in the presence of noise, while full-order models fail to do so [40]. The physical and biological importance of fractional-order derivatives in map-based gene models was stated in the previous paragraphs. Motivated by the above descriptions, this paper aims to propose the discrete fractional-order two-gene map model and examine its dynamic behavior to help biologists describe the biological behavior of genes. Therefore, at first, the dynamic behavior of the gene map is evaluated using the phase plane, bifurcation diagram, and Lyapunov exponent, and the periodic and chaotic behaviors of the system are shown. Next, the dynamic behavior of the fractional-order system of the gene map is evaluated by the branching diagram by changing the fractional order and system parameters. The synchronization behaviors of two two-gene maps with discrete fractional order have been investigated using the electrical connection.

This paper is organized as follows: in Sect. 2, the Andrecut–Kauffman map is introduced, and the bifurcation diagram, phase plane, and Lyapunov exponents evaluate its dynamic behavior. In Sect. 3, the discrete fractional-order Andrecut–Kauffman gene model is proposed, and its dynamic behaviors are studied according to the fractional order and parameters of the evaluation model. Section 4 examines the synchronization of the proposed model by electrical connection strength, and finally, Sect. 5 presents the conclusion and future research works.

2 Andrecut–Kauffman map

In the first subsection of this section, the Andrecut–Kauffman gene model is introduced, and its time series and phase plane are examined. The bifurcation and Lyapunov diagrams analyze the dynamic behavior of the Andrecut–Kauffman gene model in the following subsection.

2.1 Mathematics of two-gene map model

The Andrecut–Kauffman model is a two-dimensional map of chemical reactions related to gene expression and regulation [16, 41]. This model is a two-dimensional map of the connection of two genes as follows:

$$\begin{aligned} &\{x\left(n+1\right)=\frac{\alpha }{1+\left(1-\varepsilon \right){x}^{m}\left(n\right)+\varepsilon {y}^{m}\left(n\right)}+{\beta }_{1}x\left(n\right), \nonumber \\ &y\left(n+1\right)=\frac{\alpha }{1+\left(1-\varepsilon \right){y}^{m}\left(n\right)+\varepsilon {x}^{m}\left(n\right)}+{\beta }_{2}y\left(n\right),\end{aligned}$$
(1)

where \(x\left(n\right)\) and \(y\left(n\right)\) are discrete dynamic variables of the system, which express the concentration levels of transcription factor proteins for two genes. In the Andrecut–Kauffman model, two genes are coupled using the parameter \(\varepsilon\). The parameters of the model are in the range \(\alpha \epsilon \left[0, 100\right]\), \({\beta }_{1}\epsilon [\mathrm{0,1})\), \({\beta }_{2}\epsilon [\mathrm{0,1})\), \(\varepsilon \epsilon \left[0, 1\right]\), and \(m=1, 2, 3, 4\). In Fig. 2, part (I) of each panel shows the time series of \(x\left(n\right)\) variable, part (II) shows the time series of \(y\left(n\right)\) variable, and part (III) shows the \(x-y\) phase portrait for random initial conditions and fixed values of \(m=3\), \(\alpha =25\), \(\varepsilon =0.1\). The parameters \({\beta }_{1}\), \({\beta }_{2}\) are different in subfigures (a–d). Panel (a) depicts a chaotic state, panel (b) shows a quasiperiodic state, panel (c) shows a hyperchaotic state, and panel (d) represents a period-four state.

Fig. 2
figure 2

Dynamics of the model for different parameters, a \({\beta }_{1}=0.11\), \({\beta }_{2}=0.42\) (chaotic), b \({\beta }_{1}=0.18\), \({\beta }_{2}=0.42\) (quasiperiodic), c \({\beta }_{1}=0.16\), \({\beta }_{2}=0.16\) (hyperchaotic), and d \({\beta }_{1}=0.25\), \({\beta }_{2}=0.25\) (periodic). Part (I) shows the time series for variable \(x\left(n\right)\), (II) shows the time series for variable \(y\left(n\right)\), (III) shows the \(x-y\) phase portrait. The initial conditions are random with fixed parameters \(m=3\), \(\alpha =25\), \(\varepsilon =0.1\)

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2.2 Dynamics of two-gene map

To study the different dynamics of the Andrecut–Kauffman model, the bifurcation diagram and Lyapunov exponent (LE) are found in different intervals according to system parameters. After each parameter's transition state, the bifurcation diagram is obtained from the 500-time series endpoints. The LE is a tool for determining the sensitivity to minor disturbances of the initial conditions, so that if the maximum value of LE is positive, chaotic behavior is seen, if the maximum value of LE is zero, periodic behavior is seen, and if the maximum value of LE is negative, the fixed-point is observed. To calculate LE in the map, first, the Jacobian matrix is obtained as follows:

$$\begin{aligned}{f}{\prime}\left(x\right)&=\frac{\partial f}{\partial x}=\left[\frac{\partial {f}_{1}}{\partial {x}_{1}} \frac{\partial {f}_{1}}{\partial {x}_{2}} \frac{\partial {f}_{2}}{\partial {x}_{1}} \frac{\partial {f}_{2}}{\partial {x}_{2}} \right] \\ &=\left[{\beta }_{1}+\frac{\alpha *n*{x}_{t}^{n-1}\left(\varepsilon -1\right)}{{\left(1+\left(1-\varepsilon \right){x}_{t}^{n}+\varepsilon {y}_{t}^{n}\right)}^{2}}-1 \right. \\ &\quad - \frac{\alpha *\varepsilon *n*{y}_{t}^{n-1}}{{\left(1+\left(1-\varepsilon \right){x}_{t}^{n}+\varepsilon {y}_{t}^{n}\right)}^{2}} \\ &\quad -\frac{\alpha *\varepsilon *n*{x}_{t}^{n-1}}{{\left(1+\left(1-\varepsilon \right){y}_{t}^{n} +\varepsilon {x}_{t}^{n}\right)}^{2}} {\beta }_{2}\\ &\quad +\left.\frac{\alpha *n*{y}_{t}^{n-1}\left(\varepsilon -1\right)}{{\left(1+\left(1-\varepsilon \right){y}_{t}^{n}+\varepsilon {x}_{t}^{n}\right)}^{2}}-1 \right].\end{aligned}$$
(2)

This system has two eigenvalues. The value of LE is obtained using the eigenvalue as follows:

$$\lambda \left( {x_{0} } \right) = \frac{1}{i}\sum\limits_{{j = 0}}^{{i - 1}} {{\text{lnln}}\left| {f\prime \left( {x_{h} } \right)} \right|} .$$
(3)

Based on the definition, the average degree of separation of Andrecut–Kauffman genetic model paths is obtained. The dynamic behaviors of the system are shown using the bifurcation diagram and maximum LE with random initial conditions (Fig. 3). The first box shows the bifurcation diagram for two variables \(x(n)\), \(y(n)\) in part (a), (b), and maximum LE in part (c) for changing the parameter \({\beta }_{1}\) in the interval \([\mathrm{0,0.25}]\). Other system's parameter values are \(m=3\), \(\alpha =25\), \(\varepsilon =0.1\), \({\beta }_{2}=0.42\). By changing the \({\beta }_{1}\) parameter, the chaotic behavior can be seen in the \(\left[\mathrm{0,0.0113}\right]\cup \left[0.1833, 0.0560\right]\cup \left[0.896, 0.1133\right]\cup \left[0.1206, 0.1493\right]\cup \left[\mathrm{0.1593,0.2080}\right]\) ranges, periodic behavior in the \(\left[0.0113, 0.0183\right]\cup \left[0.056, 0.0896\right]\cup \left[0.1133, 0.1206\right]\cup \left[\mathrm{0.1403,0.1593}\right]\) ranges, and fix point behavior in the \(\left[\mathrm{0.208,0.25}\right]\) range. In the second box, the bifurcation diagram for two variables \(x(n)\), \(y(n)\) in part (d), (e) and the maximum LE in part (f) for changing the parameter \({\beta }_{2}\) in the range [0, 0.5] are shown. The other system parameters are \(m=3\), \(\alpha =25\), \(\varepsilon =0.1\), \({\beta }_{1}=0.2\). By changing the parameter \({\beta }_{2}\) in the desired interval, the intervals \(\left[0, 0.1313\right]\cup \left[0.1373, 0.2366\right]\cup \left[0.3073, 0.4346\right]\) belong to chaotic behaviors, the intervals \(\left[0.1313, 0.1373\right]\cup \left[0.2366, 0.3073\right]\) belong to periodic behaviors and the interval \(\left[0.4346, 0.5\right]\) belongs to fixed point behavior. The third box shows the bifurcation diagram for two variables \(x(n)\) and \(y(n)\) in parts (g), (h), and the maximum LE in part (i) for changing the parameter \(\alpha\) in the range \([\mathrm{0,150}]\). The other values of system parameters are \(m=3\), \(\varepsilon =0.1\), \({\beta }_{1}=0.2\), \({\beta }_{2}=0.42\). By changing the \(\alpha\) parameter, the fix point behavior can be seen in the \(\left[0, 22.5\right]\) range, chaotic behavior in the \(\left[22.5, 57\right]\cup \left[68.6, 84.2\right]\cup \left[91.8, 113.6\right]\cup \left[128, 150\right]\) ranges, and periodic behavior in the \(\left[57, 68.6\right]\cup \left[84.2, 91.8\right]\cup \left[113.6, 128\right]\) ranges. The fourth box shows the bifurcation diagram for two variables \(x(n)\), \(y(n)\) in part (j), (k) and the maximum LE in part (l) for changing the parameter \(\varepsilon\) in the range \([\mathrm{0,0.15}]\). The other system parameters are \(m=3\), \(\alpha =25\), \({\beta }_{1}=0.2\), \({\beta }_{2}=0.42\). In the interval \(\left[0, 0.0136\right]\cup \left[0.0686, 0.1356\right]\) chaotic behaviors, in the interval \(\left[0.0136, 0.0686\right]\) periodic behaviors, and in the interval \(\left[0.1356, 0.15\right]\) fixed point behaviors are observed by changing the parameter \(\varepsilon\).

Fig. 3
figure 3

Bifurcation diagrams for variable \(x\left(n\right)\), \(y\left(n\right)\), and maximum LE diagrams by varying different system parameters. a–c Bifurcation parameter is \({\beta }_{1}\epsilon [\mathrm{0,0.25}]\) and other parameter values are \(m=3\), \({\beta }_{2}=0.42\), \(\alpha =25\), \(\varepsilon =0.1\); d–f bifurcation parameter is \({\beta }_{2}\epsilon [\mathrm{0,0.5}]\) and other parameter values are \(m=3\), \({\beta }_{1}=0.2\), \(\alpha =25\), \(\varepsilon =0.1\); g–i bifurcation parameter is \(\alpha \epsilon [\mathrm{0,0.25}]\), and other parameter values are \(m=3\), \({\beta }_{1}=0.2\), \({\beta }_{2}=0.42\), \(\varepsilon =0.1\); j–l bifurcation parameter is \(\varepsilon \epsilon [\mathrm{0,0.25}]\) and other parameter values are \(m=3\), \({\beta }_{1}=0.2\), \({\beta }_{2}=0.42\), \(\alpha =25\)

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Part (a) of Fig. 4 shows the different behaviors of the Andrecut–Kauffman model for the simultaneous changes of two parameters \({\beta }_{1}\in \left[\mathrm{0,0.25}\right]\) and \({\beta }_{2}\in \left[\mathrm{0,0.5}\right]\) with random conditions and parameter values \(\varepsilon =0.1\), \(\alpha =25\), \(m=3\). In addition, part (b) of Fig. 4 shows the different behaviors of the model with the simultaneous change of two parameters \(\alpha \in \left[0, 150\right]\) and \(\varepsilon \in \left[\mathrm{0,0.15}\right]\) with the values of parameters \(m=3\), \({\beta }_{1}=0.2\), \({\beta }_{2}=0.42\). The model's behavior is identified based on the numerical value of Lyapunov. So that the red color shows the chaotic behavior, the green color shows the periodic behavior, and the blue color shows the fixed-point behavior.

Fig. 4
figure 4

Different behaviors of the gene model in 2D parameter planes. a For two parameters \({\beta }_{1}\) and \({\beta }_{2}\) and parameter values \(m=3\), \(\alpha =25\), \(\varepsilon =0.1\). b For two parameters \(\alpha\) and \(\varepsilon\) and parameter values \(m=3\), \({\beta }_{1}=0.2\), \({\beta }_{2}=0.42\). The initial conditions are random

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3 Fractional-order Andrecut–Kauffman map

In this section, using the bifurcation diagram, we describe the two-gene map of discrete fractional order and examine the effects of fractional-order parameters and system parameters on the model.

3.1 Description of discrete fractional-order two-gene map

To convert the Andrecut–Kauffman model into a discrete fractional-order model, we first need to state the basic definition of discrete fractional order and implement it on the two-gene map model. Fractional order of Caputo type delta with symbol \(C{\Delta }_{a}^{q}X\left(t\right)\) with order \(q\) for a function \(X\left(t\right):{N}_{a}\to R\), where \({N}_{a}=\left\{a, a+1, a+2, \dots \right\}\) is defined as follows:

$$C\Delta _{a}^{q} X\left( t \right) = \Delta _{a}^{{ - \left( {n - q} \right)}} \Delta ^{n} X\left( t \right) = \frac{1}{{\Gamma \left( {n - q} \right)}}\sum\limits_{{s = a}}^{{t - \left( {n - q} \right)}} {\left( {t - s - 1} \right)^{{\left( {n - q - 1} \right)}} \Delta _{s}^{{ - n}} } ,$$
(4)

where \(q\notin N\) means the order of the derivative, \(t\in {N}_{a+n-q}\), and \(n=\lceil q\rceil +1\). The fractional sum \(q\) for \({\Delta }_{a}^{-q}X\left(t\right)\) in Eq. 4 is defined as follows:

$$\Delta _{a}^{{ - q}} X\left( t \right) = \frac{1}{{\Gamma \left( q \right)}}\sum\limits_{{s = a}}^{{t - q}} {\left( {t - s - 1} \right)^{{\left( {q - 1} \right)}} X\left( s \right)} ,$$
(5)

where \(t\in {N}_{a+n-q}\) and \(q>0\). In addition, \({t}^{\left(q\right)}\) means the falling function, and it is defined based on the gamma function (\(\Gamma\)) as follows:

$$t^{\left( q \right)} = \frac{{\Gamma \left( {t + 1} \right)}}{{\Gamma \left( {r + 1 - 1} \right)}} = t\left( {t - 1} \right) \ldots \left( {t - q + 1} \right).$$
(6)

For the numerical solution in the discrete map, its fractional order can be calculated through the following method by the fractional difference equation:

$$\begin{aligned} &\{C{\Delta }_{a}^{q}X\left(t\right)=f\left(t+q-1, x\left(t+q-1\right)\right),\\ &{\Delta }^{k}X\left(t\right)={x}_{k}, n=\lceil q\rceil +1, k=0, 1, 2, \dots , n-1, \end{aligned}$$
(7)

where the equivalent discrete integral is obtained as follows:

$$\begin{aligned} x\left( t \right) &= x_{0} \left( t \right) + \frac{1}{\Gamma \left( q \right)}\mathop \sum \limits_{s = a + n - q}^{t - q} \left( {t - s - 1} \right)^{{\left( {q - 1} \right)}} \\ &\quad \times f\left( {s + q - 1, x\left( {s + q - 1} \right)} \right),t \in N_{a + m} , \end{aligned}$$
(8)

where \({x}_{0}\left(t\right)={\sum }_{k=0}^{n-1}\frac{{\left(t-a\right)}^{\left(k\right)}}{\Gamma \left(k+1\right)}{\Delta }^{k}u\left(a\right)\). According to the definitions for the fractional order, the first-order difference of the Andrecut–Kauffman model of Eq. 1 can be rewritten as follows:

$$\begin{aligned}\{\Delta x\left(n\right)&=x\left(n+1\right)-x\left(n\right)\\ &=\frac{\alpha }{1+\left(1-\varepsilon \right){x}^{m}\left(n\right)+\varepsilon {y}^{m}\left(n\right)}+{\beta }_{1}x\left(n\right)-x\left(n\right), \\ \Delta y\left(n\right)&=y\left(n+1\right)-y\left(n\right)\\ &=\frac{\alpha }{1+\left(1-\varepsilon \right){y}^{m}\left(n\right)+\varepsilon {x}^{m}\left(n\right)}+{\beta }_{2}y\left(n\right)-y\left(n\right).\end{aligned}$$
(9)

The fractional-order discrete map is calculated as follows:

$$\begin{aligned}&\{C{\Delta }_{a}^{q}x\left(t\right)=\frac{\alpha }{1+\left(1-\varepsilon \right){x}^{m}\left(t-1+q\right)+\varepsilon {y}^{m}\left(t-1+q\right)}\\ &\quad +{\beta }_{1}x\left(t-1+q\right)-x\left(t-1+q\right),\\ & C{\Delta }_{a}^{q}y\left(t\right)=\frac{\alpha }{1+\left(1-\varepsilon \right){y}^{m}\left(t-1+q\right)+\varepsilon {x}^{m}\left(t-1+q\right)}\\ &\quad +{\beta }_{2}y\left(t-1+q\right)-y\left(t-1+q\right).\end{aligned}$$
(10)

Based on Eqs. 9 and 10, we can write the Andrecut–Kauffman model in the following fractional order:

$$\begin{aligned} \{ x\left( t \right) &= x\left( a \right) + \frac{1}{{\Gamma \left( q \right)}}\sum _{{s = a + 1}}^{{t - q}} \left( {t - s - 1} \right)^{{\left( {q - 1} \right)}} \hfill \\ &\quad \Bigg( \frac{\alpha }{{1 + \left( {1 - \varepsilon } \right)x^{m} \left( {t - 1 + q} \right) + \varepsilon y^{m} \left( {t - 1 + q} \right)}}\\ &\qquad + \beta _{1} x\left( {t - 1 + q} \right) - x\left( {t - 1 + q} \right) \Bigg), \hfill \\ y\left( t \right) &= y\left( a \right) + \frac{1}{{\Gamma \left( q \right)}}\sum _{{s = 1 - q}}^{{t - q}} \left( {t - s - 1} \right)^{{\left( {q - 1} \right)}} \hfill \\ & \quad \Bigg( \frac{\alpha }{{1 + \left( {1 - \varepsilon } \right)y^{m} \left( {t - 1 + q} \right) + \varepsilon x^{m} \left( {t - 1 + q} \right)}}\\ &\qquad + \beta _{2} y\left( {t - 1 + q} \right) - y\left( {t - 1 + q} \right) \Bigg). \end{aligned}$$
(11)

As a result, the numerical formula of the fractional two-gene map discrete model depending on \(n\) is defined as follows:

$$\{x\left(n\right)=x\left(a\right)+\frac{1}{\Gamma \left(q\right)}\sum_{i=1}^{n}\frac{\Gamma \left(n-i+q\right)}{\Gamma \left(n-i+1\right)}\left(\frac{\alpha }{1+\left(1-\varepsilon \right){x}^{m}\left(i-1\right)+\varepsilon {y}^{m}\left(i-1\right)}+{\beta }_{1}x\left(i-1\right)-x\left(i-1\right)\right), y\left(n\right)=y\left(a\right)+\frac{1}{\Gamma \left(q\right)}\sum_{i=1}^{n}\frac{\Gamma \left(n-i+q\right)}{\Gamma \left(n-i+1\right)}\left(\frac{\alpha }{1+\left(1-\varepsilon \right){y}^{m}\left(i-1\right)+\varepsilon {x}^{m}\left(i-1\right)}+{\beta }_{2}y\left(i-1\right)-y\left(i-1\right)\right).$$
(12)

where the lower limit of \(a\) is fixed as zero, to expand the range of \(n\) in numerical simulation, the following equation is used for Eq. 12 [42]:

$$\frac{\Gamma \left(n-i+q\right)}{\Gamma \left(n-i+1\right)}={e}^{lnln \Gamma \left(n-i+q\right)-\Gamma \left(n-i+1\right) }.$$
(13)

The difference between the integer-order map in Eq. 10 and the fractional-order map in Eq. 12 is a discrete kernel function, as well as the dependence of \(x\left(n\right)\) and \(y\left(n\right)\) to the past information \(x\left(0\right), \dots , x\left(n-1\right)\) and \(y\left(0\right), \dots , y\left(n-1\right)\). Consequently, the current state depends on all past states, expressing the memory effects of discrete maps.

3.2 Effect of the fractional order

The fractional-order parameter is an essential factor affecting the overall dynamics. Figure 5 shows the bifurcation diagram of the model according to the fractional order for different parameter values. In parts (a) and (b), the bifurcation diagram of the model is plotted according to the change of the derivative order (\(q\)) in the interval \([\mathrm{0.85,1}]\) for different values of \(\alpha =0, 37.5, 75, \mathrm{112.5,150}\) with constant parameters \(m=3\), \(\varepsilon =0.1\), \({\beta }_{1}=0.2\), \({\beta }_{2}=0.42\). As the value of \(\alpha\) increases, chaotic behavior is created and increased in the system, called "anti-uniformity" [43]. In addition, increasing \(\alpha\) increases the chaotic interval in \(q\). In parts (c) and (d), the bifurcation diagrams according to the fractional order (\(q\)) in different values of \({\beta }_{1}=0.12, 0.1525, 0.185, 0.2175, 0.25\) with constant parameters as \(m=3\), \(\alpha =25\), \(\varepsilon =0.1\), \({\beta }_{2}=0.42\) is shown. As can be seen, with the increase of \({\beta }_{1}\), the dynamic behavior changes from chaotic to periodic and finally tends to the fixed point. In addition, in parts (e) and (f) of Fig. 5, the bifurcation diagrams are shown for different values of \({\beta }_{2}=0.1, 0.2, 0.3, 0.4, 0.5\) with constant parameters as \(m=3\), \(\alpha =25\), \(\varepsilon =0.1\), \({\beta }_{1}=0.2\). As \({\beta }_{2}\) increases, the chaotic behavior decreases and changes to periodic behavior, and finally to a fixed point. In the end, the bifurcation diagrams are plotted in parts (g) and (h) of Fig. 5 for different values of \(\varepsilon =0.15, 0.2375, 0.325, 0.4125, 0.5\) with parameter values of \(m=3\), \(\alpha =25\), \({\beta }_{1}=0.2\), \({\beta }_{2}=0.42\). It can be observed that the model does not bifurcate in this range of \(\varepsilon\). Table 1 describes the range of different behaviors of the model as obtained from the bifurcation diagrams in Fig. 5.

Fig. 5
figure 5

Bifurcation diagrams of \(x\left(n\right)\) and \(y\left(n\right)\) variables according to the fractional order (\(q\)); a, b \(m=3\), \(\varepsilon =0.1\), \({\beta }_{1}=0.2\), \({\beta }_{2}=0.42\); c, d \(m=3\), \(\varepsilon =0.1\), \(\alpha =25\), \({\beta }_{2}=0.42\); e, f \(m=3\), \(\varepsilon =0.1\), \(\alpha =25\), \({\beta }_{1}=0.2\); g, h \(m=3\), \(\alpha =25\), \({\beta }_{1}=0.2\), \({\beta }_{2}=0.42\). The random initial conditions are selected

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Table 1 It shows the model's dynamic behavior according to the fractional order for the values of different parameters based on Fig. 5
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3.3 Effect of the system parameter

Another critical factor is the system parameters, which affect the system's dynamic behavior. Figure 6 shows the bifurcation diagram of the model for the variables \(x\left(n\right)\), and \(y\left(n\right)\) according to the system parameters in different fractional-order values. The bifurcation diagram in parts (a) and (b) is drawn according to the change of parameter \(\alpha\) in the range \([\mathrm{0,150}]\) for different values of fractional order \(q=0.85, 0.8875, 0.925, 0.9625, 1\) with fixed values of parameters. In parts (c) and (d), the bifurcation diagram is shown according to the change in parameter \({\beta }_{1}\) in the interval \([\mathrm{0,0.25}]\) with fixed values of parameters \(m=3\), \(\varepsilon =0.1\), \(\alpha =25\), \({\beta }_{2}=0.42\). The bifurcation diagram in parts (e) and (f) of Fig. 6 is according to the change of parameter \({\beta }_{2}\) in the interval \([\mathrm{0,0.25}]\) and parts (g) and (h), it is according to the change of the parameter \(\varepsilon\) in the range \([\mathrm{0.15,0}]\) fixed values of the parameters as \(m=3\), \(\alpha =25\), \({\beta }_{1}=0.2\), \({\beta }_{2}=0.42\). The summary of changes in the dynamic behavior of the proposed model for the system parameters in different values of the fractional order based on Fig. 6 is presented in Table 2. In all the bifurcation diagrams of Fig. 6, as the value of \(q\) increases toward integer order, the dynamic behavior changes, and the chaotic state is observed in the broader range of the parameter. In addition, reducing the value of parameter \(\alpha\) reduces chaotic behavior and creates a fixed point, and reducing the value of parameters \({\beta }_{1}\), \({\beta }_{2}\) and \(\varepsilon\) causes an increase in chaotic behavior.

Fig. 6
figure 6

Bifurcation diagrams of model parameters for variables x(n) and y(n) according to system parameter for different fractional order \(q=0.85, 0.8875, 0.925, 0.9625, 1\); a, b according to \(\alpha\) with \(m=3\), \(\varepsilon =0.1\), \({\beta }_{1}=0.2\), \({\beta }_{2}=0.42\); c, d according to \({\beta }_{1}\) with \(m=3\), \(\varepsilon =0.1\), \(\alpha =25\), \({\beta }_{2}=0.42\); e, f according to \({\beta }_{2}\) with \(m=3\), \(\varepsilon =0.1\), \(\alpha =25\), \({\beta }_{1}=0.2\); g, h according to \(\varepsilon\) with \(m=3\), \(\alpha =25\), \({\beta }_{1}=0.2\), \({\beta }_{2}=0.42\)

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Table 2 Summary of changes in the dynamic behavior of the proposed model for the system parameters in different values of the fractional order is expressed based on Fig. 6
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4 Synchronization of two coupled fractional-order models

Another important aspect of dynamic analysis is synchronizing systems in a chaotic state. The two-gene maps of discrete fractional order through electrical coupling strength in a two-way coupled topology can be formulated as follows:

$$\begin{aligned}\{{\Delta }_{c}^{q}{x}_{1}\left(n+1\right)&=\frac{\alpha }{1+\left(1-\varepsilon \right){x}_{1}^{m}\left(n\right)+\varepsilon {y}_{1}^{m}\left(n\right)}\\ &\quad +{\beta }_{1}{x}_{1}\left(n\right)-{x}_{1}\left(n\right)+{d}_{x}\left({x}_{2}\left(n\right)-{x}_{1}\left(n\right)\right),\\ {\Delta }_{c}^{q}{y}_{1}\left(n+1\right)&=\frac{\alpha }{1+\left(1-\varepsilon \right){y}_{1}^{m}\left(n\right)+\varepsilon {x}_{1}^{m}\left(n\right)}\\ &\quad +{\beta }_{2}{y}_{1}\left(n\right)-{y}_{1}\left(n\right)+{d}_{y}\left({y}_{2}\left(n\right)-{y}_{1}\left(n\right)\right),\\ {\Delta }_{c}^{q}{x}_{2}\left(n+1\right)&=\frac{\alpha }{1+\left(1-\varepsilon \right){x}_{2}^{m}\left(n\right)+\varepsilon {y}_{2}^{m}\left(n\right)}\\ &\quad +{\beta }_{1}{x}_{2}\left(n\right)-{x}_{2}\left(n\right)+{d}_{x}\left({x}_{1}\left(n\right)-{x}_{2}\left(n\right)\right),\\ {\Delta }_{c}^{q}{y}_{2}\left(n+1\right)&=\frac{\alpha }{1+\left(1-\varepsilon \right){y}_{2}^{m}\left(n\right)+\varepsilon {x}_{2}^{m}\left(n\right)}\\ &\quad +{\beta}_{2}{y}_{2}\left(n\right)-{y}_{2}\left(n\right)+{d}_{y}\left({y}_{1}\left(n\right)-{y}_{2}\left(n\right)\right),\end{aligned}$$
(14)

where \({d}_{x}\) is related to the electrical connection strength of the variable pair \(x\left(n\right)\) and \({d}_{y}\) is related to the electrical connection strength of the variable pair \(y\left(n\right)\), also subscript one is related to the first gene, and subscript two is related to the second gene. The model parameters are set at \(m=3\), \(\alpha =25\), \(\varepsilon =0.1\), \({\beta }_{1}=0.2\), \({\beta }_{2}=0.42\). For both gene models, the initial conditions are randomly chosen between \(-1\) and \(1\). To determine the synchronization level of two genes, by considering two two-gene maps of discrete fractional order, the synchronization error is calculated as a criterion for evaluating the synchronization of genes as follows:

$$Error=\frac{1}{N}\sum_{n=1}^{N}\sqrt{{\left({x}_{1}\left(n\right)-{x}_{2}\left(n\right)\right)}^{2}+{\left({y}_{1}\left(n\right)-{y}_{2}\left(n\right)\right)}^{2}},$$
(15)

Here \(N\) is the number of time series data samples. In the following, for three values of the fractional order \(q=0.8, 0.9, 1,\) we will check the synchronization of the model with the change of coupling strength in the interval \({d}_{x}\epsilon \left[\mathrm{0,0.05}\right]\) and \({d}_{y}\epsilon \left[\mathrm{0,0.05}\right]\) in Fig. 7. According to the evaluation, the results show that reducing the order of the derivative in the system increases the synchronization in the model. When \(q=1\), no synchronization occurs in coupled systems in the considered range, while for \(q=0.8\), the systems become completely synchronous. For \(q=0.9\), the systems become completely synchronous in some coupling strengths and anti-phase synchronous in others. The color islands in Fig. 7b show the anti-phase synchronization regions.

Fig. 7
figure 7

Synchronization error of two discrete fractional-order models with the change of coupling strength in the interval \({d}_{x}\epsilon \left[\mathrm{0,0.15}\right]\) and \({d}_{y}\epsilon \left[\mathrm{0,0.15}\right]\) for a \(q=1\); b \(q=0.9\); c \(q=0.8\)

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5 Conclusion and future work

This paper proposes the two-gene Andrecut–Kauffman map by discrete fractional order. The dynamic analysis of the model was presented through bifurcation diagrams and the synchronization between two genes. Adding the discrete fraction calculation to the gene map in the biological mode simulates the hereditary behaviors on the model. In the physical concept, all the previous states determine the system states in fractional order. At first, the Andrecut–Kauffman gene map was examined using the phase plane, bifurcation diagram, and Lyapunov exponent. Different behaviors of the gene map, such as chaotic and periodic states, were displayed in the system. The bifurcation diagrams showed that increasing the order of the derivatives causes chaotic behavior in the system. Moreover, there is a limit for the fractional order below which no chaos is observed. Finally, the synchronization of two two-gene maps with discrete fractional order was investigated with an electrical connection. The results showed that increasing the fractional order decreases the model's synchronization level.