Rich Dynamics Caused by a Fractional Diffusion Operator in Nonchaotic Rulkov Maps

Abstract

There are few works about Neimark–Sacker bifurcating analysis on discrete dynamical systems with linear diffusion and delayed coupling under periodic/Neumann-boundary conditions. In this paper, we build up the framework for Neimark–Sacker bifurcations caused by Turing instability on high-dimensional discrete-time dynamical systems with symmetrical property in the linearized system. The fractional diffusion operator in higher-dimensional discrete dynamical systems is introduced and regular/chaotic Turing patterns are discovered by the computation of the largest Lyapunov exponents.

1. Introduction

How to exhibit bifurcation analysis, set up a graph related to Lyapunov exponents, or find a positive topological entropy about higher-dimensional systems with the fractional diffusion coupling under the classical boundary conditions are naturally interesting problems. We can refer the readers to see open problem 2 in [1] (p. 14) or those in [2], etc. However, to the best of our knowledge, few researchers have paid their attention to these problems. Thus, how to define the action of fractional diffusion operators on discrete-time dynamical systems, for instance, remains unclear. In this paper, we shall try to answer this question.

We have found that Neimark–Sacker bifurcations can be caused by Turing instability. Thus, a scheme needs to be built about Neimark–Sacker bifurcations in types of discrete-time dynamical systems under linear fractional diffusion coupling with Direchlet/Neumann/periodic-boundary conditions. We explore the complexity induced by the diffusion coupling. Bifurcation analysis is set up and chaotic Turing patterns are shown by the occurrence of positive Lyapunov exponents. Our stimulating model can be viewed as an interesting case study of the following phenomenon: individuals exhibit simple dynamics, but when viewed as a whole, the dynamical behavior of diffusion coupling is very complicated, specifically things such as the existence of fluctuation, limit cycles, and even chaos. This interesting phenomenon was also called Emergence by Holland in [3]: the diffusion can set up Turing instability and then regular/chaotic patterns in a simple system, and multiple attractive patterns can emerge.

Naturally, there has been some research about some discrete-time dynamical systems by diffusion coupling with classical boundary conditions; for example, p-2 patterns in [4] for two-dimensional systems and in [1,2] for higher-dimensional systems ($n\ge 2$). To the best of our knowledge, there few researchers have been concerned with Neimark–Sacker bifurcations analysis on discrete-time dynamical systems with diffusion and delayed coupling under the given classical boundary conditions.

A framework about Hopf bifurcating analysis on high-dimensional discrete-time dynamical systems has been provided by [5] (pp. 185–187), but it is unknown how to compute formulae (5.50) or (5.51) in [5] (p. 186). In this paper, we give a clear presentation of this, and regular/chaotic oscillating Turing patterns are the main topics in the present paper.

We investigate the dynamical behavior of discrete-time dynamical systems with the fractional diffusion operator as follows:

$$\begin{array}{cc}\hfill u_{ij}(m+1)=& -{\gamma }^C{(-\nabla )}^{\beta /2}{u}_{ij}\left(m\right)+{\displaystyle \frac{a}{1+u_{ij}{(m-\tau )}^2}}+v_{ij}\left(m\right),\hfill \\ \hfill v_{ij}(m+1)=& -\mu (u_{ij}\left(m\right)-\rho )+v_{ij}\left(m\right),\hfill \end{array}$$

(1)

under the Neumann-boundary condition:

$$u_{0l}=u_{1l},u_{s+1,l}=u_{s,l},u_{k0}=x_{k1},u_{k,t+1}=u_{k,t};$$

(2)

or the periodic-boundary condition:

$$u_{0l}=u_{s,l},u_{k0}=x_{k,t},u_{k,t+1}=u_{k,1},k\in \mathbb{N}[1,s],l\in \mathbb{N}[1,t]$$

(3)

and ${}^C(-\nabla )^{\beta /2}$ is defined by the Caputo-like fractional diffusion difference operator, and if $\beta =2$, then:

$${}^C(-\nabla )^{\beta /2}=-\nabla {\psi }_{kl}=-({\psi }_{k+1,l}+{\psi }_{k-1,l}+{\psi }_{k,l-1}+{\psi }_{k,l+1}-4{\psi }_{kl}),$$

where $s,t,n$ are positive integers and $\mathbb{N}[1,n]=\{1,\dots ,n\}$. This may be a very interesting problem, including the idea about how to define the Caputo-like diffusion difference operator, how to determine its dynamical behavior in discrete-time dynamical systems, etc. Until now, there have been few works concerned with such a question. We investigate some interesting phenomena, i.e., the occurrence of fractional impact on discrete-time dynamical systems.

The single individual model was discussed in [6], which can be viewed as a model about the nonlinear delayed feedback control on classical chaotic Rulkov maps [7,8]. There may exist a delay during processing information in intra-neural communication [9], information propagation through the cortical network [10], or simulating the finite transmission velocity through the network node [11,12]. For other works related to physical connections with/without delay coupling, we refer the readers to Refs. [7,13,14,15,16,17,18,19].

Compared to those discussed in [6,20,21,22,23,24,25,26,27,28,29,30], there occurs a new class of characteristic polynomials in the present as follows:

$$(x-1)(x^{\tau +1}-\varsigma x^{\tau }+\eta )+\mu x^{\tau }$$

(4)

which is essentially different when compared with those in [6]. For instance, one can find that there are two pairs of complex conjugate eigenvalues lying on the critical unit circle for $\tau =2$, though when diffusion emerges ($\gamma >0$), there is only one pair of complex conjugate eigenvalues with a modulus equal to 1. In fact, $Turinginstability$ occurs (please see Figure 1). Moreover, in [28], it was pointed out that delay can induce a larger stable region ($\tau =0,1,2$) [6] or smaller $\tau =3$ and $\tau =1$.This implies that the delayed feedback control may not be an efficient control method, even though it is widely used.

The following is the novelty in this paper:

• Build up the framework for Neimark–Sacker bifurcations caused by Turing instability on high-dimensional discrete-time dynamical systems with symmetrical property in the linearized system;
• Design bifurcation diagrams and the graphs about Lyapunov exponents;
• Introduce the fractional diffusion operator in higher-dimensional discrete dynamical systems.

Now let us list the structure of the rest part of this paper as follows. A framework for Neimark–Sacker bifurcation is presented in Section 2 for $v=2$. Turing instability is analyzed. Eigenvalue problems are also discussed and numerical simulation is shown. Bifurcation diagrams are proposed and regular/chaotic oscillating Turing patterns can be explored by the computation of the largest Lyapunov exponents. Turing instability, including Regular/Chaotic Motions in the given system caused by diffusion and delay, is analyzed. In Section 3, the fractional diffusion operator in higher-dimensional discrete dynamical systems is introduced and Neimark–Sacker bifurcations caused by Turing instability are analyzed. Further discussion on dynamical behavior caused by the fractional diffusion operator is set up in Section 4 and the conclusions are drawn in Section 5.

2. Regular Diffusion Operator $\mathit{v}\mathbf{=}\mathbf{2}$: Turing Instability and Neimark–Sacker Bifurcations

By employing the projection technique proposed by Kuznetsov [5] (p. 182), one can rewrite the given system as:

$$\tilde{x}=Mx+M_1{x}_1+\cdots +M_{\tau }{x}_{\tau }+H(x,x_1,\dots ,x_{\tau }),x\in {\mathbb{R}}^{2st},$$

(5)

where $x=(u_{11},\cdots ,u_{st},v_{11},\cdots ,v_{st}),$ $x_{\tau }=x(•-\tau )$, $M_1=\cdots =M_{\tau -1}=0$, and

$$M=\left(\begin{array}{cc}\gamma A& I\\ \mu I& I\end{array}\right),M_{\tau }=\left(\begin{array}{cc}-{\displaystyle \frac{2a\rho }{{(1+{\rho }^2)}^2}}I& 0\\ 0& 0\end{array}\right)$$

where $H\left(x\right)={\displaystyle \frac{B(x,x)}{2}}+{\displaystyle \frac{C(x,x,x)}{6}}+{O\left(\right|\left|x\right||}^4)$ is a smooth function which can have the following form by the Taylor expansion at the fixed point:

$$E^{\ast }=(\stackrel{st}{\overbrace{\rho ,\dots ,\rho }},\stackrel{st}{\overbrace{\rho -{\displaystyle \frac{a}{1+{\rho }^2}},\dots ,\rho -{\displaystyle \frac{a}{1+{\rho }^2}}}})$$

with the components of the vectors B and C are listed as follows:

$$B_i=-{\displaystyle \frac{2a(1-3{\rho }^2)}{{(1+{\rho }^2)}^3}}{x}_i^2(•-\tau )(i=1,\dots ,st),B_i=0,(i=st+1,\dots ,2st)$$

and

$$C_i={\displaystyle \frac{12a\rho (1-2{\rho }^2)}{{(1+{\rho }^2)}^4}}{x}_i^3(•-\tau )(i=1,\dots ,st),C_i=0,(i=st+1,\dots ,2st).$$

According to those results obtained by [1,2,31,32], we have the following theorem.

Theorem 1.

${\lambda }_{k,l}=-4({\sin }^2{\displaystyle \frac{(k-1)\pi }{2s}}+{\sin }^2{\displaystyle \frac{(l-1)\pi }{2t}})$ are all the eigenvalues of the matrix N related to Neumann boundary problems Equation (2) [1] and the corresponding eigenvector is ${\psi }^{kl}=({\psi }_{11},\dots ,{\psi }_{st})$ where:

$${\psi }^{kl}=\left\{\begin{array}{cc}{\displaystyle \frac{1}{\sqrt{st}}}(1,1,\dots ,1),\hfill & k=1,l=1,\hfill \\ \sqrt{{\displaystyle \frac{2}{st}}}{\psi }^{k1},\hfill & k\in \mathbb{N}(2,s),l=1,\hfill \\ \sqrt{{\displaystyle \frac{2}{st}}}{\psi }^{1l},\hfill & k=1,l\in \mathbb{N}(2,t),\hfill \\ {\displaystyle \frac{2}{\sqrt{st}}}{\psi }^{kl}\hfill & k\in \mathbb{N}(2,s),l\in \mathbb{N}(2,t).\hfill \end{array}\right.$$

such that $P^{-1}NP=P^{T}NP=\Lambda =diag[{\lambda }_{11},\cdots ,{\lambda }_{st}]$, $P=[{\psi }^{11},\cdots {\psi }^{st}]$ and

$${\psi }_{ij}^{kl}=\cos {\displaystyle \frac{(k-1)(2i-1)\pi }{2s}}\cos {\displaystyle \frac{(l-1)(2j-1)\pi }{2t}},$$

where N related to (2) has a similar form as defined by [2] (p. 9203). In fact, $N={\left(n_{uv}\right)}_{s\times t}$ where $n_{uv}=0,$ except that $n_{11}=n_{t,t}=n_{st,st}=n_{(s-1)t+1,(s-1)t+1}=-2$, and $n_{12}=1,n_{1,t+1}=n_{t,t-1}=n_{t,2t}=1$:

$$n_{st,st-1)}=n_{st,(s-1)t}=n_{(s-1)t+1,(s-1)t+2}=n_{(s-1)t+1,(s-2)t+1}=1;$$

if $u=2:s-1$ or $v=2:t-1$, then $n_{u,u}=n_{(s-1)t+u,(s-1)t+u}=n_{(v-1)t+1,(v-1)t+1}=n_{vt,vt}=-3$, and

$$n_{u,u-1}=n_{u,u+1}=n_{u,u+t}=n_{(s-1)t+u,(s-1)t+u,(s-1)t+u-1}=n_{(s-1)t+u,(s-1)t+u,(s-1)t+u+1}$$

$$=n_{(s-1)t+u,\left(\right(s-2)t+u}=n_{(v-1)t+1,(v-1)t+2}=n_{(v-1)t+1,(v-2)t+1}=n_{(v-1)t+1,vt+1}=n_{vt,vt-1}=$$

$$=n_{vt,(v-1)t}=n_{vt,(v+1)t}=1;$$

if $u\in \mathbb{N}[2,s-1]$ and $v\in \mathbb{N}[2,t-1]$, then $n_{(v-1)t+u,(v-1)t+u}=-4,$ $n_{(v-1)t+u,(v-1)t+u-1}=n_{(v-1)t+u,(v-1)t+u+1}=n_{(v-1)t+u,(v-2)t+u}=n_{(v-1)t+u,vt+u}=1.$

Theorem 2.

${\lambda }_{k,l}=-4({\sin }^2{\displaystyle \frac{2(k-1)\pi }{s}}+{\sin }^2{\displaystyle \frac{2(l-1)\pi }{t}})$ are all the eigenvalues of the matrix N related to Equation (3) and $\{{\epsilon }^{kl}\in {\mathbb{R}}^{s\times t}\}$, a standard orthogonal basis can be formed such that $U^{-1}NU=U^{T}NU=\Lambda =diag[{\lambda }_{11},\cdots ,{\lambda }_{st}]$ and $U=[{\epsilon }^{11},\cdots {\epsilon }^{st}]$, where N has another form as follows: $N={\left(n_{uv}\right)}_{s\times t}$ where $n_{uv}=0,$ except for $n_{uu}=-4,(u=1,\cdots ,st)$, $n_{12}=n_{1,t+1}=n_{1t}=n_{1,(s-1)t+1}=n_{t,t-1}=n_{t,2t}=n_{t,1}=n_{t,st}=n_{st,st-1)}=n_{st,(s-1)t}=$

$$n_{st,t}=n_{st,(s-1)t+1}=n_{(s-1)t+1,(s-1)t+2}=n_{(s-1)t+1,(s-2)t+1}=n_{(s-1)t+1,1}=n_{(s-1)t+1,st}=1;$$

if $u\in \mathbb{N}[2,s-1]$, then:

$$n_{u,u-1}=n_{u,u+1}=n_{u,u+t}=n_{u,u+(s-1)t}=n_{(s-1)t+u,(s-1)t+u,(s-1)t+u-1}=$$

$$n_{(s-1)t+u,(s-1)t+u,(s-1)t+u+1}=n_{(s-1)t+u,\left(\right(s-2)t+u}=n_{(s-1)t+u,u}=1;$$

if $u\in \mathbb{N}[2,s-1]$ and $v\in \mathbb{N}[2,t-1]$, then:

$$n_{(v-1)\ast t+u,(v-1)t+u-1}=n_{(v-1)\ast t+u,(v-1)t+u+1}=n_{(v-1)\ast t+u,(v-2)t+u}=n_{(v-1)\ast t+i,vt+u}=1.$$

and if $v\in \mathbb{N}[2,s-1]$, then:

$$n_{(v-1)t+1,(v-1)t+2}=n_{(v-1)\ast t+1,(v-2)t+1}=n_{(v-1)t+1,vt}=n_{(v-1)t+1,vt+1}=n_{vt,vt-1}=$$

$$=n_{vt,(v-1)t}=n_{vt,(v+1)t}=n_{vt,(v-1)t+1}=1.$$

In addition, if s is odd or t is odd, one can choose ${\epsilon }^{kl}={\eta }^{kl}/\left|{\eta }^{kl}\right|$, where:

$${\eta }_{uv}^{kl}=\cos {\displaystyle \frac{2(k-1)u\pi }{s}}\cos {\displaystyle \frac{2(l-1)v\pi }{t}},$$

and ${\left|\eta \right|}^2=<\eta ,\eta >$ and $<\varphi ,\psi >={\sum }_{i=1}^{st}{\varphi }_i{\overline{\psi }}_i$ for $\varphi ,\psi \in {\mathbb{C}}^{st}$. For related works about eigenvalues problems please see Theorem 2.2 [2] (p. 9203).

2.1. Eigenvalue Problems

It is known that there is a unique fixed point $E^{\ast }=(\rho ,\rho -{\displaystyle \frac{a}{1+{\rho }^2}})$ for $\mu \ne 0$ without diffusion coupling [6]. Now let us investigate the linearized system near the synchronized fixed point $E^{\ast }$, that is:

$$\begin{array}{ccc}\hfill u_{ij}(m+1)& =& -{\displaystyle \frac{2a\rho }{{(1+{\rho }^2)}^2}}{u}_{ij}(m-\tau )+v_{ij}\left(m\right)+\gamma \nabla u_{ij}\left(m\right),\hfill \\ \hfill v_{ij}(m+1)& =& -\mu u_{(}m)+v_{ij}\left(m\right).\hfill \end{array}$$

(6)

If we introduce ${(u_{ij},v_{ij})}^T\left(m\right)={\lambda }^m\alpha$, then the characteristic matrix of (6) becomes:

$$\mathbb{Q}\left(\lambda \right)=\left(\begin{array}{cc}{\lambda }^{\tau +1}I-{\lambda }^{\tau }\gamma A+{\displaystyle \frac{2a\rho }{{(1+{\rho }^2)}^2}}I& -{\lambda }^{\tau }I\\ \mu {\lambda }^{\tau }I& (\lambda -1){\lambda }^{\tau }I\end{array}\right).$$

Hence, the characteristic equation, in view of the symmetric property A and $(\lambda -1)({\lambda }^{\tau +1}I-{\lambda }^{\tau }A+{\displaystyle \frac{2a\rho }{{(1+{\rho }^2)}^2}}I)+\mu {\lambda }^{\tau }I$, is as follows:

$$\begin{array}{ccc}\hfill \det \mathbb{Q}\left(\lambda \right)& =& \det \left(\begin{array}{cc}{\lambda }^{\tau +1}I-{\lambda }^{\tau }\gamma A+{\displaystyle \frac{2a\rho }{{(1+{\rho }^2)}^2}}I& -{\lambda }^{\tau }I\\ (\lambda -1)({\lambda }^{\tau +1}I-\gamma {\lambda }^{\tau }A+{\displaystyle \frac{2a\rho }{{(1+{\rho }^2)}^2}}I)+\mu {\lambda }^{\tau }I& 0\end{array}\right)\hfill \\ \hfill & =& {\lambda }^{st\tau }\det ((\lambda -1)({\lambda }^{\tau +1}I-\gamma {\lambda }^{\tau }A+{\displaystyle \frac{2a\rho }{{(1+{\rho }^2)}^2}}I)+\mu {\lambda }^{\tau }I)\hfill \\ \hfill & =& {\lambda }^{\tau st}L\left(\lambda \right),\hfill \end{array}$$

(7)

where

$$L\left(\lambda \right)=f\left(\lambda \right)={\mathsf{\Pi }}_{i=1}^{st}\left(({\lambda }^2-(1+\gamma {\lambda }_{kl})\lambda +(\mu +\gamma {\lambda }_{kl}){\lambda }^{\tau }+(\lambda -1){\displaystyle \frac{2a\rho }{{(1+{\rho }^2)}^2}}\right).$$

(8)

By choosing the notation $\eta$ defined as ${\displaystyle \frac{2a\rho }{{(1+{\rho }^2)}^2}}$, and setting up the direct computation or similar analysis as [6], it is easy to verify that there is a positive root of the polynomial Equation (8) or Equation (4) greater than 1 for $\mu <0$. Therefore, the equilibrium becomes unstable if $\mu <0$. Moreover, there can occur a codimension-two bifurcation with 1:1 resonance at $({\mu }^{\ast },{\eta }^{\ast })=(0,-1)$ for $k=1,l=1$, and $\lambda =1$ is a single eigenvalue at $({\mu }^{\ast },{\eta }^{\ast })=(0,-1)$ for $k\ne 1$ or $l\ne 1$.

In what follows, let us begin a study of the characteristic polynomial Equation (4) for $\tau =0,1,2$ in detail and all possible higher-codimensional bifurcations. Suppose that $\rho <0$ and $a,\mu >0$, and the diffusion coefficient $\gamma \geqq 0$. Then, $\eta <0$.

If $\tau =1$, then $L\left(\lambda \right)$ becomes a cubic polynomial ${\lambda }^3-\gamma {\lambda }_{kl}{\lambda }^2+(\mu +\eta +\gamma {\lambda }_{kl})\lambda -\eta .$ Set $p=(\mu +\eta +\gamma {\lambda }_{kl})-{\gamma }^2{\lambda }_{kl}^2/3$, $q=-2{\gamma }^3{\lambda }_{kl}^3/27+\gamma {\lambda }_{kl}(\mu +\eta +\gamma {\lambda }_{kl})/3-\eta$. Then, $\Delta =q^2+4p^3/27={(9\mu -18\eta -2)}^2/{27}^2+4{(3\mu +3\eta -1)}^3/{27}^2$. If $\Delta \ge 0$, then there exist three real eigenvalues, whereas $\Delta <0$, there can occur a pair of complex conjugate eigenvalues. One can obtain the following theorem.

Theorem 3.

Suppose that:

• $-1<\eta <0$;
• $\mu =1-{\eta }^2+\gamma {\lambda }_{kl}\eta ;$
• $-1<\eta -\gamma {\lambda }_{kl}<0.$

Then there exists a pair of complex conjugate roots of the polynomial Equation (4) with $\tau =1$ with modulus equal to 1. Moreover, the transversality condition holds true: ${\displaystyle \frac{{d\left|\lambda \right|}^2}{d\gamma }}{|}_{\left|\lambda \right|=1}=2\left(\overline{\lambda }{\displaystyle \frac{d|\lambda }{d\gamma }}+\lambda {\displaystyle \frac{d\overline{\lambda }}{d\gamma }}\right)>0$.

In the case $\tau =2$, $L\left(\lambda \right)$ is a quartic polynomial of a form as ${\lambda }^4-\gamma {\lambda }_{kl}{\lambda }^3+(\mu +\gamma {\lambda }_{kl}){\lambda }^2+\eta \lambda -\eta .$ It is easy to verify the following theorem.

Theorem 4.

If the following hypotheses become valid:

• $-1<\eta <0$;
• $\mu +\eta -1={\displaystyle \frac{\eta \gamma {\lambda }_{kl}(1+\eta +\gamma {\lambda }_{kl})}{{(1+\eta )}^2}};$
• $-3(\eta +1)<\gamma {\lambda }_{kl}<\eta +1.$

Then there exists a pair of complex conjugate roots of the polynomial Equation (4) with $\tau =2$ with modulus equal to 1. Furthermore, all roots are single and the transversality condition holds true: ${\displaystyle \frac{{d\left|\lambda \right|}^2}{d\gamma }}{|}_{\left|\lambda \right|=1}=2\left(\overline{\lambda }{\displaystyle \frac{d|\lambda }{d\gamma }}+\lambda {\displaystyle \frac{d\overline{\lambda }}{d\gamma }}\right)>0$.

Remark 1.

If compared with those in Ref. [17], there are no two pairs of complex conjugate roots with the module equal to 1 at the same time for $\eta =-1$ and the diffusion term $\gamma >0$.

2.2. Neimark–Sacker Bifurcations

Theorem 5.

Suppose that $-1<\eta <0$, $\mu >0$ and

• $\tau =1$, $\mu =1-\eta +\eta {\gamma }^{\ast }{\lambda }_{st}$ and ${\gamma }^{\ast }\in (0,(1+\eta )/{\lambda }_{st}),$
• $\tau =2$, $\mu +\eta -1={\displaystyle \frac{\eta {\gamma }^{\ast }{\lambda }_{kl}(1+\eta +{\gamma }^{\ast }{\lambda }_{st})}{{(1+\eta )}^2}},$ and ${\gamma }^{\ast }\in (0,(\eta +1)/{\lambda }_{st}),$

with the single multiplier ${\mu }_{1,2}=\exp (\pm i{\theta }_0)$ and there occurs the absence of strong resonances, i.e., $\exp \left(ik{\theta }_0\right)\ne 1,k=1,2,3,4$, system Equations (1) and (2) at the synchrous fixed point $E^{\ast }$ are locally topologically equivalent to the normal form as follows:

$$\tilde{z}=e^{i{\theta }_0}z(1+d_1{\left|z\right|}^2{)+O(\left|z\right|}^4)$$

near the critical value ${\gamma }^{\ast }$. Moreover, if $d_1<0$, the unique stable limit cycle emerges by Neimark–Sacker bifurcations. Otherwise, if $d_1>0$, the bifurcating invariant limit cycle is unstable.

Proof. 

It is known that the linear operator has a simple root with one pair of complex eigenvalues denoted by ${\mu }_{1,2}=exp(\pm i{\theta }_0)$, $0<{\theta }_0<\pi$. Moreover, these multipliers are the only eigenvalues lying on the unit circle $\left|\mu \right|=1$. Suppose that $q^{\ast }\in {\mathbb{C}}^{2st}$ be a complex eigenvector corresponding to the critical eigenvalue ${\mu }_1.$ Then:

$$(exp\left(i\tau {\theta }_0\right)M+M_{\tau })q^{\ast }=exp\left(i{\theta }_0\right)q^{\ast },(exp\left(i\tau {\theta }_0\right)M+M_{\tau }){\overline{q}}^{\ast }=exp(-i{\theta }_0){\overline{q}}^{\ast }.$$

One can rewrite system Equation (5) as another form of the following type:

$$\tilde{U}=M_{‡}U+F_{‡}\left(U\right),U\in {\mathbb{R}}^{(\tau +2)st},$$

(9)

where $U=(u_{11},\dots ,u_{st},v_{11},\dots ,v_{st},u_{11}(•-1),\dots ,u_{st}(•-1),\dots ,u_{11}(•-\tau ),\dots ,u_{st}(•-\tau ))$ and

$$M_{‡}=\left(\begin{array}{ccccccc}A& I& 0& 0& \cdots & 0& -{\displaystyle \frac{2a\rho }{{(1+{\rho }^2)}^2}}I\\ \mu I& I& 0& 0& \cdots & 0& 0\\ I& 0& 0& 0& \cdots & 0& 0\\ 0& 0& I& 0& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮\\ 0& 0& 0& 0& \cdots & I& 0\end{array}\right),F_{‡}=\left(\begin{array}{c}F(x,x_1,\dots ,x_{\tau })\\ 0\\ ⋮\\ 0\end{array}\right).$$

If we choose $q=({\left(q^{\ast }\right)}^T,exp(-i{\theta }_0)q_{11}^{\ast },\dots ,\exp (-i\tau {\theta }_0)q_{11}^{\ast },\dots ,$ $q_{st}^{\ast }exp(-i\tau {\theta }_0){)}^T$, then:

$$M_{‡}q=\exp \left(i{\theta }_0\right)q,M_{‡}\overline{q}=\exp (-i{\theta }_0)\overline{q}.$$

By introducing the adjoint eigenvector $p\in {\mathbb{C}}^N(N=(\tau +2)st)$, we have:

$$M_{‡}p=exp\left(i{\theta }_0\right)p,M_{‡}\overline{p}=exp(-i{\theta }_0)\overline{p},$$

with $<p,q>=1$ and $<p,q>={\sum }_{i=1}^N{p}_i{\overline{q}}_i$. The critical real eigenspace $T^c$ corresponding to ${\mu }_{1,2}$ is two-dimensional, spanned by $\{\Re q,\Im q\}$, i.e., $q=\Re q+i\Im q$. Obviously, $<p,\overline{q}>=0$. The real eigenspace $T^{su}$ corresponding to all eigenvalues of $M_{‡}$ other than ${\mu }_{1,2}$ is ($N-2$)-dimensional. Its validity holds true: $y\in T^{su}\subset {\mathbb{R}}^{N-2}$ if and only if $<p,y>=0$ or both $<\Re p,y>=0$ and $<\Im p,y>=0$.

Choose a standard orthogonal basis ${\{{\chi }^i\in {\mathbb{R}}^N\}}_{i=1}^{N-2}$, for instance. Then, compute the fundamental solution basis of the following homogeneous linear equations $\left(\begin{array}{c}\Re q^T\\ \Im q^T\end{array}\right)Y=0$, which can be found by the well-known Gram–Schmidt process and unitization. Next, for any vector $x\in {\mathbb{R}}^N$, we have $x=zq+\overline{zq}+{\sum }_{i=1}^{N-2}{y}_i{\chi }^i$, where $zq+\overline{zq}\in T^c,\sum y_i{\chi }^i\in T^{su}$ with $z=<p,x>$. We obtain the coordinate on $T^c$.

Now, we find that:

$$\begin{array}{ccc}\hfill z& =& <p,x>,\hfill \\ \hfill {\sum }_{i=1}^{N-2}{y}_i{\chi }^i& =& x-<p,x>q-<\overline{p},x>\overline{q}.\hfill \end{array}$$

In these coordinates, the map Equation (9) can be written as:

$$\begin{array}{ccc}\hfill \tilde{z}& =& e^{i{\theta }_0}z+<p,F(zq+\overline{zq}+y)>,\hfill \\ \hfill {\tilde{y}}_i& =& <{\chi }^i,A{\chi }^i>y_i+<{\chi }^i,F(zq+\overline{zq}+{\sum }_{i=1}^{N-2}{y}_i{\chi }^i)>.\hfill \end{array}$$

(10)

Furthermore, one can obtain that:

$$\begin{array}{ccc}\hfill \tilde{z}& =& e^{i{\theta }_0}z+{\displaystyle \frac{1}{2}}{G}_{20}{z}^2+G_{11}z\overline{z}+{\displaystyle \frac{1}{2}}{G}_{02}{\overline{z}}^2+{\displaystyle \frac{1}{2}}{G}_{21}{z}^2\overline{z}+\hfill \\ \hfill & & +{\sum }_{i=1}^{N-2}\left(z<p,B(q,y_i{\chi }^i)>+\overline{z}<p,B(\overline{q},y_i\chi )>\right)+\cdots ,\hfill \\ \hfill {\tilde{y}}_i& =& <{\chi }^i,A{\chi }^i>y_i+{\displaystyle \frac{1}{2}}{H}_{20}^i{z}^2+H_{11}^{i}z\overline{z}+{\displaystyle \frac{1}{2}}{H}_{02}^i{\overline{z}}^2+\cdots .\hfill \end{array}$$

(11)

where $G_{20},G_{11},G_{02},G_{21}\in {\mathbb{C}}^1,H_{ij}\in {\mathbb{C}}^{N-2}$, which have the following expressions:

$$G_{20}=<p,B(q,q)>,G_{11}=<p,B(q,\overline{q})>,G_{02}=<p,B(\overline{q},\overline{q})>,G_{21}=<p,C(q,q,\overline{q})>.$$

Since $y\in {\mathbb{R}}^{N-2}$, we have $H_{ij}={\overline{H}}_{ji}$.

The center manifold $W^c$ now can be represented [33] (p. 258) as:

$$W^c=\{(u,v)\in {\mathbb{R}}^2\times {\mathbb{R}}^{N-2}|v=h(z,\overline{z})={\displaystyle \frac{1}{2}}{w}_{20}{z}^2+w_{11}z\overline{z}+{\displaystyle \frac{1}{2}}{w}_{02}{\overline{z}}^2+{O\left(\right|z|}^3\left)\right),$$

where $v\in {\mathbb{R}}^{N-2}$, and

$$(e^{i{\theta }_0}I-\Lambda )w_{20}=H_{20},(e^{i{\theta }_0}I-\Lambda )w_{11}=H_{11},(e^{i{\theta }_0}I-\Lambda )w_{02}=H_{02}$$

and $\Lambda =diag\left[<{\chi }^1,A{\chi }^1>,\dots ,<{\chi }^{N-2},A{\chi }^{N-2}>\right]$, which is invertible.

Thus, in the absence of strong resonances, i.e., $\exp \left(ik{\theta }_0\right)\ne 1,k=1,2,3,4$, the restricted map can generically be rewritten as:

$$\tilde{z}=e^{i{\theta }_0}z+{\displaystyle \frac{1}{2}}{g}_{20}{z}^2+g_{11}z\overline{z}+{\displaystyle \frac{1}{2}}{g}_{02}{\overline{z}}^2+{\displaystyle \frac{1}{2}}{g}_{21}{z}^2\overline{z}+\cdots$$

A normal form ban be found by further analysis as follows: $\tilde{z}=e^{i{\theta }_0}z(1+d_1{\left|z\right|}^2{)+O(\left|z\right|}^4)$, where the direction of bifurcating closed invariant curves is pointed out by the real number $d=\Re d_1$, which could be computed by the formula as follows:

$$d=\Re ({\displaystyle \frac{e^{-i{\theta }_0}{g}_{21}}{2}})-\Re ({\displaystyle \frac{(1-2e^{i{\theta }_0})e^{-2i{\theta }_0}}{2(1-e^{i{\theta }_0})}}{g}_{20}{g}_{11})-{\displaystyle \frac{1}{2}}|g_{11}{|}^2-{\displaystyle \frac{1}{4}}{|g_{02}|}^2.$$

□

Remark 2.

A framework for Neimark–Sacker bifurcation in high-dimensional discrete-time systems was proposed by [5] (pp. 185–187). However, it is still unknown how to compute the variable y in the Formulae (5.50) or (5.51) in [5] (p. 186). In this paper, we give a clear presentation.

Below, we investigate the case for even t and s and for odd t or odd s with repeated eigenvalues. Using the same technique as earlier we obtain the following theorem.

Theorem 6.

Suppose that $-1<\eta <0$ and, with the single multiplier ${\mu }_{1,2}=(f^{\prime }\left(x^{\ast }\right)-\gamma {\lambda }_{s,t})=\exp (\pm i\theta )$:

• $\tau =1$, $\mu =1-\eta +8\eta {\gamma }^{\ast },$ and $\gamma \in (0,(1+\eta )/8)$;
• $\tau =2$, $\mu +\eta -1={\displaystyle \frac{8\eta {\gamma }^{\ast }(1+\eta +8{\gamma }^{\ast })}{{(1+\eta )}^2}}$ and ${\gamma }^{\ast }\in (0,(\eta +1)/8)$.

Moreover, since there exists the absence of strong resonances, i.e., $\exp \left(ik{\theta }_0\right)\ne 1,k=1,2,3,4$, then system Equations (1)–(3) at the unique fixed point $E^{\ast }$ is locally topologically equivalent to the normal form as follows:

$$\tilde{z}=e^{i{\theta }_0}z(1+d_1{\left|z\right|}^2{)+O(\left|z\right|}^4)$$

near the critical value ${\gamma }^{\ast }$. Moreover, if $d_1>0$, the stable limit cycle emerges by Neimark–Sacker bifurcations. Otherwise, if $d_1<0$, the bifurcating invariant limit cycle is unstable.

Remark 3.

There can exist multistability, which is a well-known phenomenon discussed in [11,12,34] in high-dimensional systems Equations (1) and (2)) or (1)–(3)).

2.3. Numerical Simulations: Regular/Chaotic Motions Caused by Diffusion and Delay

In this section, we focus on rich dynamic exhibited by system Equation (1) and (2). Assume that $\gamma >0$, $a,\mu >0$ and $\rho =1$. The initial condition is randomly chosen around the equilibrium $E^{\ast }$.

Bifurcation diagrams and the graph related the largest Lyapunov exponent are shown by Figure 2, Figure 3 and Figure 4 as the parameter $\gamma$, $\mu$ or a change their values by $n=\mathrm{10,000}$ and $s=t=34$, $a=1.2$, $\rho =1,\mu =0.8$. As the diffusion strength coefficient $\gamma$ has a continuous change, the unique fixed point solution $E^{\ast }$ becomes unstable and there occur aperiodic patterns (invariant limit cycles or Tori (Figure 3ii,iii)) and even chaos (Figure 2 and Figure 4), because of the existence of a positive Lyapunov exponent (for the computation of Lyapunov exponents, please see [26,35,36,37,38] and the references cited therein). It is also shown by Figure 4 that there exist two (blue, yellow) separate regions, which implies that there are regular (the blue region in Figure 4 with the largest Lyapunov exponent ${\sigma }_1\leqq 0$) or irregular (unbounded solutions or chaotic attractors: the yellow region in Figure 4 with the largest Lyapunov exponent ${\sigma }_1>0$) motions. It is shown by Figure 2 that the largest Lyapunov exponent can become positive and that invariant closed curves occur due to Neimark–Sacker bifurcation. Further discussions are needed as to whether invariant chaotic sets like the classical logistic map with $f\left(x\right)=\varrho x(1-x)$ [39] do or do not exist for sufficiently larger $\gamma$, a, or $\mu$. As pointed out by Ref. [15], symbolic dynamics may help us give a clear understanding about the complexity generated by a given model. Regular/chaotic oscillating Turing patterns can be discovered by Figure 2 for $s=t=64$ and $\gamma =0.114,0.118$, and 0.13, respectively. Turing instability [1,4,40] and multistability emerges [11,12,26,27,34,38,41].

3. Fractional Diffusion Difference Operator $\mathbf{1}\mathbf{<}\mathit{\beta }\mathbf{<}\mathbf{2}$: Turing Instability and Neimark–Sacker Bifurcations

Let us introduce the fractional diffusion difference operator ${}^C(-\Delta )^{\beta /2}$ with the following properties:

• ${\lambda }_{kl}^{\beta }=2^{\beta }({\sin }^{\beta }{\displaystyle \frac{(k-1)\pi }{2s}}+{\sin }^{\beta }{\displaystyle \frac{(l-1)\pi }{2t}})$ are all the eigenvalues of the operator ${}^C(-\Delta )^{\beta /2}$ and the selection of vectors $\left\{{\xi }^{kl}\right\}\in {\mathbb{R}}^{s\times t}$ is a standard orthogonal basis, such that:

  $${}^C(-\Delta )^{\beta /2}({\xi }^{11},\cdots ,{\xi }^{st})=-({\xi }^{11},\cdots ,{\xi }^{st})M,$$

  which implies that M is the matrix representation of the operator ${}^C\Delta ^{\beta /2}$ related to Neumann boundary problems Equation (2) with the symmetrical property;
• The symmetrical matrix $-M$ is the matrix representation of the operator ${}^C(-\Delta )^{\beta /2}$ related to periodic boundary problems Equation (3), and ${\lambda }_{kl}^{\beta }=2^{\beta }({\sin }^{\beta }{\displaystyle \frac{2(k-1)\pi }{s}}+{\sin }^{\beta }{\displaystyle \frac{2(l-1)\pi }{t}})$, $1<\beta \le 2$) are all the eigenvalues of the operator ${}^C(-\Delta )^{\beta /2}$ or the matrix $-M$.

Moreover, we have the following theorem.

Theorem 7.

Suppose that ${\lambda }_{k,l}^{\beta }=2^{\beta }({\sin }^{\beta }{\displaystyle \frac{(k-1)\pi }{2s}}+{\sin }^{\beta }{\displaystyle \frac{(l-1)\pi }{2t}})$ are all the eigenvalues of the fractional diffusion difference operator ${}^C(-\Delta )^{\beta /2}$ related to Neumann boundary problems (2). If one choose the corresponding eigenvector ${\psi }^{kl}=({\psi }_{11},\dots ,{\psi }_{st})$ where:

$${\psi }^{kl}=\left\{\begin{array}{cc}{\displaystyle \frac{1}{\sqrt{st}}}(1,1,\dots ,1),\hfill & k=1,l=1\hfill \\ \sqrt{{\displaystyle \frac{2}{st}}}{\psi }^{k1},\hfill & k\in \mathbb{N}(2,s),\hfill \\ \sqrt{{\displaystyle \frac{2}{st}}}{\psi }^{1l},\hfill & l\in \mathbb{N}(2,t),\hfill \\ {\displaystyle \frac{2}{\sqrt{st}}}{\psi }^{kl}\hfill & k\in \mathbb{N}(2,s),l\in \mathbb{N}(2,t).\hfill \end{array}\right.$$

Then, $P^{-1}MP=P^{T}MP=\Lambda =diag[-{\lambda }_{11}^{\beta },\cdots ,-{\lambda }_{st}^{\beta }]$, where $P=[{\psi }^{11},\cdots {\psi }^{st}]$ and ${\psi }_{ij}^{kl}=\cos {\displaystyle \frac{(k-1)(2i-1)\pi }{2s}}\cos {\displaystyle \frac{(l-1)(2j-1)\pi }{2t}}$ (for more details about $\beta =2$, please see [1]).

Theorem 8.

Suppose that ${\lambda }_{k,l}^{\beta }=2^{\beta }({\sin }^v{\displaystyle \frac{2(k-1)\pi }{s}}+{\sin }^{\beta }{\displaystyle \frac{2(l-1)\pi }{t}})$ are all the eigenvalues of the fractional diffusion difference operator ${}^C(-\Delta )^{\beta /2}$ related to periodic boundary problems Equation (3) and a selection of vectors $\left\{{\epsilon }^{kl}\right\}\in {\mathbb{R}}^{s\times t}$ is a standard orthogonal basis such that $U^{-1}MU=U^{T}MU=\Lambda =diag[-{\lambda }_{11}^{\beta },\cdots ,-{\lambda }_{st}^{\beta }]$ and $U=[{\epsilon }^{11},\cdots {\epsilon }^{st}]$. In addition, if s is odd or t is odd, one can choose ${\epsilon }^{kl}={\eta }^{kl}/\left|{\eta }^{kl}\right|$, where ${\eta }_{uv}^{kl}=\cos {\displaystyle \frac{2(k-1)u\pi }{s}}\cos {\displaystyle \frac{2(l-1)\pi }{t}}$, ${\left|\eta \right|}^2=<\eta ,\eta >$ (for more details about $\beta =2$, please see [2]).

Moreover, one can find that:

• (a) Assume that t and s are both even, $8^{\beta /2}$ is the unique largest eigenvalue ${\lambda }^{†}\in \left\{{\lambda }_{ij}^{\beta }\right\}$, and $i=s/2+1$, $j=t/2+1$.
• (b) Assume that t and s are both odd, $2^{\beta }({\sin }^{\beta }{\displaystyle \frac{(s-1)\pi }{s}}+{\sin }^v{\displaystyle \frac{(t-1)\pi }{t}})$ is the largest eigenvalue ${\lambda }^{†}\in \left\{{\lambda }_{ij}^{\beta }\right\}$, the multiplicity is 4, and the case (1) $i=(s+1)/2$, $j=(t+1)/2$, or (2) $i=(s+1)/2$, $j=(t+3)/2$, or (3) $i=(s+3)/2$, $j=(t+1)/2$, or (iv) $i=(s+3)/2$, $j=(t+3)/2$, is valid, respectively.
• (c) Assume that t is even and s is odd, $2^{\beta }({\sin }^{\beta }{\displaystyle \frac{(s-1)\pi }{s}}+1)$ is the largest eigenvalue ${\lambda }^{†}\in \left\{{\lambda }_{ij}^{\beta }\right\}$, the multiplicity is 2, and (1) $i=(s+1)/2$, $j=t/2+1$ or (2) $i=(s+3)/2$, $j=t/2+1$ become valid, respectively.
• (d) Assume that s is even and t is odd, $2^{\beta }({\sin }^{\beta }{\displaystyle \frac{(t-1)\pi }{t}}+1)$ is the largest eigenvalue ${\lambda }^{†}\in \left\{{\lambda }_{ij}^{\beta }\right\}$, the multiplicity is 2, and (1) $i=s/2+1$, $j=(t+1)/2$ or (2) $i=s/2+1$, $j=(t+3)/2$ hold true, respectively.
• (e) The single root 0 is the smallest eigenvalue among all ${\lambda }_{ij}^{\beta }$.

Rewrite system Equation (1) as:

$$\tilde{x}=M_{0}x+M_1{x}_1+\cdots +M_{\tau }{x}_{\tau }+\tilde{H}(x,x_1,\dots ,x_{\tau }),x\in {\mathbb{R}}^{2st},$$

(12)

where $x=(u_{11},\cdots ,u_{st},v_{11},\cdots ,v_{st}),$ $x_{\tau }=x(•-\tau )$, $M_1=\cdots =M_{\tau -1}=0$, and:

$$M_0=\left(\begin{array}{cc}-\gamma M& I\\ \mu I& I\end{array}\right),\dots ,M_{\tau }=\left(\begin{array}{cc}-\eta I& 0\\ 0& 0\end{array}\right),$$

where $\tilde{H}\left(x\right)={\displaystyle \frac{\tilde{B}(x,x)}{2}}+{\displaystyle \frac{\tilde{C}(x,x,x)}{6}}+{O\left(\right|\left|x\right||}^4)$ is a smooth function which can have the following form by the Taylor expansion at $E^{\ast }=(\stackrel{st}{\overbrace{\rho ,\dots ,\rho }},\stackrel{st}{\overbrace{\rho -{\displaystyle \frac{a}{1+{\rho }^2}},\dots ,\rho -{\displaystyle \frac{a}{1+{\rho }^2}}}})$ and:

$${\tilde{B}}_i=-{\displaystyle \frac{2a(1-3{\rho }^2)}{{(1+{\rho }^2)}^3}}{x}_i^2(•-\tau )(i=1,\dots ,st),{\tilde{B}}_i=0,(i=st+1,\dots ,2st)$$

and

$${\tilde{C}}_i={\displaystyle \frac{12a\rho (1-2{\rho }^2)}{{(1+{\rho }^2)}^4}}{x}_i^3(•-\tau )(i=1,\dots ,st),{\tilde{C}}_i=0,(i=st+1,\dots ,2st).$$

3.1. Eigenvalue Problems

The linearized system becomes:

$$\begin{array}{ccc}\hfill u_{ij}(m+1)& =& -\eta u_{ij}(m-\tau )+v_{ij}\left(m\right)+{\gamma }^C{\Delta }^{\beta /2}{u}_{ij}\left(m\right),\hfill \\ \hfill v_{ij}(m+1)& =& -\mu u(m)+v_{ij}\left(m\right).\hfill \end{array}$$

(13)

Then, by introducing ${(u_{ij},v_{ij})}^T\left(m\right)={\lambda }^m\alpha$, the characteristic matrix of (13) is as follows:

$$\mathbb{Q}\left(\lambda \right)=\left(\begin{array}{cc}{\lambda }^{\tau +1}I+{\lambda }^{\tau }\gamma M+\eta I& -{\lambda }^{\tau }I\\ \mu {\lambda }^{\tau }I& (\lambda -1){\lambda }^{\tau }I\end{array}\right).$$

Hence, the characteristic equation, in view of the symmetric property $(\lambda -1)({\lambda }^{\tau +1}I+{\lambda }^{\tau }M+\eta I)+\mu {\lambda }^{\tau }I$, is as follows:

$$\begin{array}{ccc}\hfill \det \mathbb{Q}\left(\lambda \right)& =& \det \left(\begin{array}{cc}{\lambda }^{\tau +1}I+{\lambda }^{\tau }\gamma M+\eta I& -{\lambda }^{\tau }I\\ (\lambda -1)({\lambda }^{\tau +1}I+\gamma {\lambda }^{\tau }M+\eta I)+\mu {\lambda }^{\tau }I& 0\end{array}\right)\hfill \\ \hfill & =& {\lambda }^{st\tau }\det ((\lambda -1)({\lambda }^{\tau +1}I-\gamma {\lambda }^{\tau }\Lambda +\eta I)+\mu {\lambda }^{\tau }I)\hfill \\ \hfill & =& {\lambda }^{\tau st}L\left(\lambda \right),\hfill \end{array}$$

(14)

where

$$L\left(\lambda \right)={\mathrm{\Pi }}_{i=1}^{st}\left(({\lambda }^2-(1+\gamma {\lambda }_{kl}^{\beta })\lambda +(\mu +\gamma {\lambda }_{kl}^{\beta }){\lambda }^{\tau }+(\lambda -1)\eta \right).$$

(15)

In the next section, we are interested in the following characteristic polynomial:

$$L_{kl}\left(\lambda \right)=({\lambda }^2-(1+\gamma {\lambda }_{kl}^{\beta })\lambda +(\mu +\gamma {\lambda }_{kl}^{\beta }){\lambda }^{\tau }+(\lambda -1)\eta$$

(16)

for $\tau =0,1,2$. Suppose that $\rho <0$ and $a,\mu >0$, and the diffusion coefficient $\gamma \geqq 0$. Then, $\eta <0$.

If $\tau =1$, $L_1\left(\lambda \right)$ becomes a cubic polynomial ${\lambda }^3-\gamma {\lambda }_{kl}^{\beta }{\lambda }^2+(\mu +\eta +\gamma {\lambda }_{kl}^{\beta })\lambda -\eta .$ Set $p=(\mu +\eta +\gamma {\lambda }_{kl}^{\beta })-{\gamma }^2{({\lambda }_{kl}^{\beta })}^2/3$, $q=-2{\gamma }^3{({\lambda }_{kl}^{\beta })}^3/27+\gamma {\lambda }_{kl}^{\beta }(\mu +\eta +\gamma {\lambda }_{kl}^{\beta })/3-\eta$. Then, $\Delta =q^2+4p^3/27={(9\mu -18\eta -2)}^2/{27}^2+4{(3\mu +3\eta -1)}^3/{27}^2$. If $\Delta \ge 0$, then there are three real eigenvalues, whereas if $\Delta <0$, there exists one pair of complex conjugate eigenvalues. Therefore, one can obtain the following theorem.

Theorem 9.

Suppose that:

• $-1<\eta <0$;
• $\mu =1-{\eta }^2+\gamma {\lambda }_{kl}^{\beta }\eta ;$
• $-1<\eta -\gamma {\lambda }_{kl}^{\beta }<0.$

Then, under the assumption that $\tau =1$, there exists one pair of complex conjugate roots of the algebraic Formula (16) with modulus equal to 1. Moreover, transversality conditions hold true: ${\displaystyle \frac{{d\left|\lambda \right|}^2}{d\gamma }}{|}_{\left|\lambda \right|=1}=2\left(\overline{\lambda }{\displaystyle \frac{d|\lambda }{d\gamma }}+\lambda {\displaystyle \frac{d\overline{\lambda }}{d\gamma }}\right)>0$.

If $\tau =2$, $L_1\left(\lambda \right)$ is a quartic polynomial of a form as ${\lambda }^4-\gamma {\lambda }_{kl}^{\beta }{\lambda }^3+(\mu +\gamma {\lambda }_{kl}^{\beta }){\lambda }^2+\eta \lambda -\eta .$

Theorem 10.

Suppose that:

• $-1<\eta <0$;
• $\mu +\eta -1={\displaystyle \frac{\eta \gamma {\lambda }_{kl}^{\beta }(1+\eta +\gamma {\lambda }_{kl}^{\beta })}{{(1+\eta )}^2}};$
• $-3(\eta +1)<\gamma {\lambda }_{kl}^{\beta }<\eta +1.$

Then, there exists one pair of complex conjugate roots of the algebraic Formula (16) with modulus equal to 1. Moreover, the transversality condition holds true: ${\displaystyle \frac{{d\left|\lambda \right|}^2}{d\gamma }}{|}_{\left|\lambda \right|=1}=2\left(\overline{\lambda }{\displaystyle \frac{d|\lambda }{d\gamma }}+\lambda {\displaystyle \frac{d\overline{\lambda }}{d\gamma }}\right)>0$.

3.2. Neimark–Sacker Bifurcations

Theorem 11.

Suppose that $-1<\eta <0$, $\mu >0$ and, with the single multiplier ${\mu }_{1,2}=\exp (\pm i{\theta }_0)$:

• $\tau =1$, $\mu =1-\eta +\eta {\gamma }^{\ast }{\lambda }_{st}^{\beta }$ and ${\gamma }^{\ast }\in (0,(1+\eta )/{\lambda }_{st}^{\beta })$;
• $\tau =2$, $\mu +\eta -1={\displaystyle \frac{\eta {\gamma }^{\ast }{\lambda }_{st}^{\beta }(1+\eta +{\gamma }^{\ast }{\lambda }_{st}^{\beta })}{{(1+\eta )}^2}},$ and ${\gamma }^{\ast }\in (0,(\eta +1)/{\lambda }_{st}^{\beta })$.

Moreover, if there is an absence of strong resonances, i.e., $\exp \left(ik{\theta }_0\right)\ne 1,k=1,2,3,4$, system Equations (1) and (2) at the unique fixed point $E^{\ast }$ is locally topologically equivalent to the normal form as follows:

$$\tilde{z}=e^{i{\theta }_0}z(1+d_1{\left|z\right|}^2{)+O(\left|z\right|}^4),$$

near the critical value ${\gamma }^{\ast }$. Moreover, if $d_1<0$, the unique stable limit cycle emerges by Neimark–Sacker bifurcations. Otherwise, if $d_1>0$, the bifurcating invariant limit cycle is unstable.

Below, we assume that both t and s are even. Using the same techniques and method as earlier we obtain the following theorem.

Theorem 12.

Suppose that $-1<\eta <0$ and, with the single multiplier ${\mu }_{1,2}=(f^{\prime }\left(x^{\ast }\right)-\gamma {\lambda }_{s,t}^{\beta })=\exp (\pm i\theta )$:

• $\tau =1$, $\mu =1-\eta +8^{\beta /2}\eta {\gamma }^{\ast },$ and $\gamma \in (0,(1+\eta )/8^{\beta /2}).$
• $\tau =2$, $\mu +\eta -1={\displaystyle \frac{8^{\beta /2}\eta {\gamma }^{\ast }(1+\eta +8^{\beta /2}{\gamma }^{\ast })}{{(1+\eta )}^2}}$ and ${\gamma }^{\ast }\in (0,(\eta +1)/8)$

Moreover, if there is an absence of strong resonances, i.e., $\exp \left(ik{\theta }_0\right)\ne 1,k=1,2,3,4$, system Equations (1)–(3) at the unique fixed point $E^{\ast }$ are locally topologically equivalent to the normal form as follows:

$$\tilde{z}=e^{i{\theta }_0}z(1+d_1{\left|z\right|}^2{)+O(\left|z\right|}^4).$$

near the critical value ${\gamma }^{\ast }$. Moreover, if $d_1>0$, the stable limit cycle emerges by Neimark–Sacker bifurcations. Otherwise, if $d_1<0$, the bifurcating invariant limit cycle is unstable.

Since the proofs of the above two theorems are similar to the techniques and method in Theorem 5, they are omitted.

4. Further Discussion

Let us depict Figure 5 and Figure 6 for some further discussion. Stability caused by the the fractional diffusion operator in system (1) with the Neumann boundary conditions (2) is shown in Figure 5. Two interesting Turing patterns in system (1) with the Neumann boundary conditions (2) are shown in Figure 6. Rich dynamical behavior can occur if the fractional fact is introduced to the well-known diffusion operator.

5. Conclusions

In this paper, some interesting results are presented in a classical delayed Rulkov model with diffusion coupling. Rich dynamics caused by diffusion and delay are investigated. Regular/chaotic Turing patterns are shown by the computation of negative/zero/positive largest Lyapunov exponents. Neimark–Sacker bifurcating analysis and the largest Lyapunov exponent diagrams are depicted. The complexity in discrete-time dynamical systems can be induced by diffusion and delay. The fractional diffusion difference operator is introduced based on the idea developed by [42] and the references cited in, which is a very interesting problem that will be the topic of our future research.

A framework for the detailed analysis of Neimark–Sacker bifurcation in delayed discrete-time dynamical systems with diffusion coupling is established, which might give some new ideas about the bifurcation theory of functional differential Equations [43,44,45]. The main techniques and method employed in the present are bifurcation theory [30,33] and Lyapunov exponents [26,35,36,37,38,46], but not those classical fixed point theorems [47,48]. The results in this paper may inspire future research in the coming, for instance, one may give a good further study of those discussed in [2,4,6,14,15,16,20,31,42,49] etc.