Dynamic behavior and bifurcation analysis of a deterministic and stochastic coupled logistic map system

Abstract

In the present work, a system of two linear coupled logistic map is studied. Local stability analysis of the fixed points of the proposed system is investigated. The system occurs transcritical, flip, and Neimark-Sacker bifurcations, which are analyzed by both center manifold theory and bifurcation theory. For any non-linear system that represents a real-world model affected by noise, white noise is included in the system and its effect on fixed points is analyzed via the technique of stochastic sensitivity function. The phenomenon of noise-induced transitions between closed invariant curves is discussed. Finally, numerical simulations are performed with the aid of Matlab to assure the agreements with analytical results obtained.

1 Introduction

It is well known that the logistic map is a very simple one-dimensional discrete system that exhibits very complicated behavior through a period-doubling bifurcation cascade and eventual emergence of chaos [1,2,3,4,5,6]. The celebrated logistic map is given by

$$\begin{aligned} x_{n+1}=a x_n(1-x_n),\quad n=0,1,2,3, \ldots , \end{aligned}$$

(1)

where $0<x<1$, and $0<a<4$. Following the formulation of Eq. (1), many researchers have studied different aspects of chaos in two-dimensional logistic map and investigated their applications in many fields [7,8,9]. Recently, an experimental study of coupled oscillators shows an increase of complexity due to coupling process [10]. Actually, when a system is composed of many nonlinear units, it forms a new complex system with more complex behaviors which are not held by the individual units. Indeed, one of the standard models for nonlinear dynamical systems is to deal with a system of two symmetrically coupled maps admitting move towards chaos via period doubling bifurcations [11,12,13,14,15,16,17]. Most of the experimental results that studied systems of coupled objects agree with the complicated dynamical behaviors of coupled systems [18,19,20,21]. The two logistic mapping system was applied as a model for the chemical reaction dynamics [22] and population dynamics [23]. Mathematically, there are two ways to couple two logistic maps: linear and bilinear coupling. These types of coupled logistic maps have been studied numerically and analytically [24, 25] in which the authors found a quasi-periodic behavior with frequency locking as well as bifurcations. Indeed, discrete dynamical systems (mappings) have attracted the attentions of many researches in the last few decades as they are of enormous relevance in biological and physical processes [26,27,28,29,30,31,32,33,34]. In addition, discrete time models reflect much richer dynamics than those detected in their continuous temporal counterparts, as they represent many real phenomena in communications, economics and biological sciences.

The field of encryption research is an important field in computer science for the preservation of important information and confidential information, so that information security using hybrid chaotic dynamics has become an important subject that attracts many researchers. The method of encryption on the basis of separate chaotic systems is proposed in [35, 36]. In fact, chaotic maps have been shown to have several significant advantages in relation to the basic requirements for encryption algorithms [37]. It is evident from this that discreet logistic maps showing chaotic dynamics or hybrid with any integral transformations or elliptical curves can be very useful in terms of secure communication, encryption and information security.

It’s quite well known the uncontrolled random disturbances are an inevitable attribute of any kind of realistic system. The weak noise with a nonlinear system can also dramatically change its dynamics. Analyzing the effect of random disturbances is therefore a challenge for modern dynamics theory in various fields of science, such as biological, engineering and economics. Thus, noise becomes an essential component of the evolution of the dynamic system. It is noted that there is a fundamental shift in the dynamics of coupled systems due to random noise. The effect of noise on the non-linear dynamic behavior of many coupled maps with different goals has been discussed in [38,39,40,41].

In this paper, a symmetrically coupled logistic map is considered as follows

$$\begin{aligned} \left\{ \begin{array}{ll} x_{n+1}=&{}ax_n(1-x_n)+b(y_n-x_n),\\ y_{n+1}=&{}ay_n(1-y_n)+b(x_n-y_n), \end{array} \right. \end{aligned}$$

(2)

where $0<x_n,y_n<1$, $0<a<4$, and $-2\le b\le 2$ is called connection parameter. The system (2) is symmetrical with respect to the exchange of x and y and was represented in [25]. Other researchers have reconsidered the system (2) in [42, 43].

In this paper, a itemized bifurcation analysis for system (2) is carried out which is not addressed in [25, 42, 43]. The key contributions and outcomes of this work are defined as follows: It provides a first, thoroughly analytical study of the various types of codimension—one bifurcation that can occur in the linear coupled logistic map (2). Analyzing the effects of white noise on the dynamic behavioral of the system is discussed both analytically and numerically. The system (2) has a number of periodic cycles, such as transcritical, flip, and Neimark-Sacker bifurcations, which are analyzed by both center and bifurcation theories. These are interesting dynamic behaviors that have not been analytically analyzed in the literature for this system. In addition, the impact of each white noise parameter on the dynamic behavior was examined. Moreover, the extensive simulation results will be presented to detect the effect of the parameters on the change of the stability and bifurcation thresholds.

The paper is structured as follows. Section 2 discusses the existence and stability of fixed points of the deterministic system. In Sect. 3, a detailed bifurcation analysis is investigated. A white noise is added to the system and its influence is discussed in Sect. 4. In Sect. 5, some numerical simulations are performed using Matlab to verify the analytical results obtained in Sect. 3. Finally, the conclusion and discussion can be found in Sect. 6.

2 The existence of fixed points and their stability

At most, the system (2) has four fixed points:

• 1. The fixed point $E_1=(0,0)$ exists for all parameters values.

• 2. For $a\ne 1$, there exists $E_2=(\frac{a-1}{a},\frac{a-1}{a})$,

• 3. Furthermore, there are two fixed points

  $$\begin{aligned} E_3= & {} \left( \frac{1}{2a}((a-1-2b)+\sqrt{(1-a+2b)(1-a-2b)}),\right. \\&\left. \frac{1}{2a}((a-1-2b)-\sqrt{(1-a+2b)(1-a-2b)}\right) ,\\ E_4= & {} \left( \frac{1}{2a}((a-1-2b)-\sqrt{(1-a+2b)(1-a-2b)}),\right. \\&\left. \frac{1}{2a}((a-1-2b)+\sqrt{(1-a+2b)(1-a-2b)}\right) , \end{aligned}$$

  which are real if and only if $a\le 1-2|b|$ or $a\ge 1+2|b|$.

Lemma 1

[44] Let $F(\lambda ) = \lambda ^{2} + P\lambda +Q$. Suppose that $F(1) > 0$, $\lambda _{1}$ and $\lambda _{2}$ are two roots of $F(\lambda ) = 0$. Then

• 1. $|\lambda _{1}|<1$ and $|\lambda _{2}|<1$ if and only if $F(-1)<0$, $Q<1$;

• 2. $|\lambda _{1}|<1$ and $|\lambda _{2}|>1$ (or $|\lambda _{1}|>1$ and $|\lambda _{2}|<1$) if and only if $F(-1) < 0$;

• 3. $|\lambda _{1}|>1$ and $|\lambda _{2}|>1$ if and only if $F(-1) > 0$ and $Q > 1$;

• 4. $\lambda _{1}=-1$ and $|\lambda _{2}|\ne 1$ if and only if $F(-1) = 0$ and $P \ne 0, 2$;

• 5. $\lambda _{1}$ and $\lambda _{2}$ are complex and $|\lambda _{1}| = 1$and $|\lambda _{2}| = 1$ if and only if $P^{2}-4Q < 0$ and $Q = 1$.

Lemma 2

[44] Let $F(\lambda ) = \lambda ^{2} + P\lambda +Q$ is characteristic equation corresponding with the Jacobian matrix computed at a fixed point $(x^*, y^*)$, then $(x^*, y^*)$ is called

• 1. a sink if $|\lambda _{1}| < 1$ and $|\lambda _{2}| < 1$, so the sink is locally asymptotically stable;

• 2. a source if $|\lambda _{1}| > 1$ and $|\lambda _{2}| > 1$, so the source is locally unstable;

• 3. a saddle if $|\lambda _{1}| > 1$ and $|\lambda _{2}| < 1$ (or $|\lambda _{1}| < 1$ and $|\lambda _{2}| > 1)$;

• 4. non-hyperbolic if either $|\lambda _{1}| = 1$ or $|\lambda _{2}| = 1$.

In order to study stability and bifurcation, it is necessary to calculate the Jacobin matrix of the system (2) at any fixed point $(x ^ *, y ^ *)$ reads as

$$\begin{aligned} J(x^*,y^*)=\left( \begin{array}{ll} a(1-2x)-b &{} \quad b \\ b &{} \quad a(1-2y)-b \end{array} \right) . \end{aligned}$$

(3)

3 Analysis of local bifurcations

A more detailed description of the bifurcation in this section is being performed for the fixed points of system (2). Both center manifold theorem and bifurcation theory [45,46,47,48,49,50] are used to study bifurcation types in the system (2).

Proposition 1

The fixed point $E_1=(0,0)$ of system (2) is

• 1. A sink if $-1<a<1$, and $\frac{a-1}{2}<b<\frac{a+1}{2}$,

• 2. A source if (i) $a>1$ or $a<-1$ and (ii) $b<\frac{a-1}{2}$ or $b>\frac{a+1}{2}$,

• 3. A saddle if (i) $a>1$ or $a<-1$ and (ii) $\frac{a-1}{2}<b<\frac{a+1}{2}$,

• 4. A non-hyperbolic if (i) $a=\pm 1$ and (ii) $b=\frac{a-1}{2}$ or $b=\frac{a+1}{2}$.

Proposition 2

The fixed point $E_2=(\frac{a-1}{a},\frac{a-1}{a})$ of system (2) is

• 1. A sink if $1<a+2b<3$, and $1<a<3$,

• 2. A source if (i) $a+2b<1$ or $a+2b>1$ and (ii) $1<a<3$,

• 3. A saddle if $2ab-6(a+b)<-5$,

• 4. A non-hyperbolic if (i) $3(a+b)-ab=\frac{5}{2}$ and (ii) $a+b\notin \{-2,1\}$.

It is worth to mention here that system (2) admits no bifurcation at $E_1(0,0)$.

3.1 Bifurcation of the fixed point $E_2$

The Jacobian matrix (4) at $E_2$ reads as

$$\begin{aligned} J(E_2)=\left( \begin{array}{ll} -a-b+2 &{} \quad b \\ b &{} \quad -a-b+2 \end{array} \right) , \end{aligned}$$

it owns two eigenvalues $\lambda _1=-a-2b+2$ and $\lambda _2=-a+2$. If $a+2b=3$, thus we have $\lambda _1=-1$, $|\lambda _2|\ne 1$ provided that $a\ne 1,3$.

Theorem 1

If $b=\frac{3-a}{2}$, and $a\ne 1,3$, then system (2) exhibits a flip bifurcation at $E_2$. In addition, at this fixed point the stable period-doubling orbit bifurcates.

Proof

The system (2) can be used as follows

$$\begin{aligned} \left\{ \begin{array}{ll} x\rightarrow a x(1-x)+b(y-x), \\ y \rightarrow a y(1-y)+b(x-y). \end{array} \right. \end{aligned}$$

(4)

Let $b^*$ is a parameter bifurcation, consider the perturbation of (4) is given by

$$\begin{aligned} \left\{ \begin{array}{ll} x\rightarrow a x(1-x)+(b+b^*)(y-x), \\ y \rightarrow a y(1-y)+(b+b^*)(x-y), \end{array} \right. \end{aligned}$$

(5)

which $|b^*|\ll 1$ is a small perturbation.

Consider $u = x-x^*$, $v = y-y^*$, thus map (5) changed as follows

$$\begin{aligned} \left\{ \begin{array}{ll} u \rightarrow &{}(-a+2-b)u+bv-u b^*+v b^*-au^{2}\\ {} &{}+O((|u|+|v|+|b^*|)^3),\\ v \rightarrow &{}bu+(-a+2-b)v+u b^*-v b^*-av^2\\ {} &{}+O((|u|+|v|+|b^*|)^3). \end{array} \right. \end{aligned}$$

(6)

Constructing an invertible matrix as follows

$$\begin{aligned}T= & {} \left( \begin{array}{ll} b&{} \quad b\\ a+b-3&{} \quad b \end{array} \right) , \end{aligned}$$

We use the transformation as follows

$$\begin{aligned} \left( \begin{array}{cc} u\\ v \end{array} \right) =T\left( \begin{array}{cc} \tilde{x}\\ \tilde{y} \end{array} \right) , \end{aligned}$$

then the system (6) will be changed to

$$\begin{aligned} \left( \begin{array}{cc} \tilde{x}\\ \tilde{y} \end{array} \right) \rightarrow \left( \begin{array}{ll} -1&{} \quad 0\\ 0&{} \quad -a+2 \end{array} \right) \left( \begin{array}{cc} \tilde{x}\\ \tilde{y} \end{array} \right) +\left( \begin{array}{cc} \phi (\tilde{x},\tilde{y},b^*)\\ \psi (\tilde{x},\tilde{y},b^*) \end{array} \right) , \end{aligned}$$

(7)

where

$$\begin{aligned}&\phi (\tilde{x},\tilde{y},b^*)\\&\quad =\frac{1}{3-a}\left( -ub^*+vb^*-au^2+O((|u|+|v|+|b^*|)^3)\right) ,\\&\psi (\tilde{x},\tilde{y},b^*)\\&\quad =\frac{1}{b(3-a)}\left( (3-a-b)(-ub^*+vb^*-au^2+bub^*\right. \\&\left. \quad -bvb^*-abv^2+O((|u|+|v|+|b^*|)^3)\right) . \end{aligned}$$

and

$$\begin{aligned} u= & {} b(\tilde{x}+\tilde{y}),\\ v= & {} (a-b-3)\tilde{x}+b\tilde{y}. \end{aligned}$$

By the center manifold theorem [50, 51], there exists a center manifold $W_c(0,0,0)$ of (7) at the fixed point (0, 0) in a small neighborhood of $b^*$ which may take the form

$$\begin{aligned} W_c(0,0,0)= & {} \{(\tilde{x},\tilde{y},b^*)\in R^3,\tilde{y}\\= & {} h(\tilde{x},b^*), h(0,0)=0, Dh(0,0)=0\}, \end{aligned}$$

for $\tilde{x}$ and $\delta ^*$ sufficiently small. We suppose that the center manifold of the form

$$\begin{aligned} h(\tilde{x},b^*)=\mu _0\tilde{x}^2+\mu _1\tilde{x}b^*+\mu _2{b^*}^2+O((|\tilde{x}|+|b^*|)^3). \end{aligned}$$

(8)

The center manifold should achieve the equation

$$\begin{aligned}&h(-\tilde{x}+\phi (\tilde{x},h(\tilde{x},b^*),b^*),b^*)=(-a+2)h(\tilde{x},b^*)\nonumber \\&\qquad \qquad +\psi (\tilde{x},h(\tilde{x},b^*),b^*). \end{aligned}$$

(9)

By replacing (8) for (9) and matching similar power coefficients for (9), we obtain

$$\begin{aligned} \mu _0= & {} \frac{ab}{(a-1)(a-3)},\\ \mu _1= & {} \frac{a+b-3}{b(a-3)},\\ \mu _2= & {} 0. \end{aligned}$$

Hence, we realize the system (7) which is restrictive to the center manifold:

$$\begin{aligned}&F:\tilde{x}\rightarrow -\tilde{x}+A\tilde{x}^2+B\tilde{x}b^*+C\tilde{x}^2b^*+D\tilde{x}{b^*}^2+E\tilde{x}^3\nonumber \\&+ F0\tilde{x}^3b^*+G\tilde{x}^2{b^*}^2+H\tilde{x}^4+ O((|\tilde{x}|+|b^*|)^4), \end{aligned}$$

(10)

where

$$\begin{aligned} A= & {} \frac{-ab^2}{3-a},\quad B=\frac{-1}{3-a},\\ C= & {} \frac{ab}{(3-a)(a-1)(a-3)}+\frac{ab^2}{(a-1)(a-3)}\\&+\frac{(-2ab^2)(a+b-3)}{b(a-3)(3-a)},\\ D= & {} \frac{a+b-3}{b(3-a)(a-3)},\quad E=\frac{-2a^2b^3}{(a-1)(a-3)(3-a)},\\ F0= & {} \frac{(-2a^2b^3)(a+b-3)}{b(a-3)(3-a)},\quad G=\frac{(-ab^2)(a+b-3)^2}{b^2(a-3)^2(3-a)},\\&\quad H=\frac{-a^2b^3}{(a-1)^2(a-3)^2(3-a)} \end{aligned}$$

To allow the map (10) to occur a flip bifurcation, we order that two preferential quantities $\alpha _1$ and $\alpha _2$ are not zero [51]:

$$\begin{aligned} \alpha _1= & {} \Big (2\frac{\partial ^2F}{\partial b^* \partial \tilde{x}}+\frac{\partial F}{\partial b^*}\frac{\partial F}{\partial \tilde{x}}\Big )_{(0,0)}=-2\ne 0,\\ \alpha _2= & {} \Big (\frac{1}{2}\Big (\frac{\partial ^2F}{\partial \tilde{x}^2}\Big )^2+\frac{1}{3}\Big (\frac{\partial ^3F}{\partial \tilde{x}^3}\Big )\Big )_{(0,0)}\\= & {} \frac{-2a^2b^4}{3-a}(1+\frac{2ab}{(a-1)(a-3)})^2\\- & {} \frac{4a^2b^3}{3(3-a)(a-1)(a-3)}(1+\frac{2ab}{(a-1)(a-3)})\ne 0. \end{aligned}$$

$\square$

Now we discuss the transcritical bifurcation of $E_2$.

Theorem 2

If $b=\frac{1-a}{2}$, and $a\ne 1,3$, then system (2) shows a transcritical bifurcation at $E_2$.

Proof

Use the $b^*$ as a bifurcation parameter, and realize the disturbance of (4) as in the system (5). Taking $u = x-x^*$, $v = y-y^*$, then the map (5) has form

$$\begin{aligned} \left\{ \begin{array}{ll} u \rightarrow &{}(1+b)u+bv-u b^*+v b^*-(1-2b)u^{2}\\ {} &{}+O((|u|+|v|+|b^*|)^3),\\ v \rightarrow &{}bu+(1+b)v+u b^*-v b^*-(1-2b)v^2\\ {} &{}+O((|u|+|v|+|b^*|)^3). \end{array} \right. \end{aligned}$$

(11)

Design the inverse matrix as follows

$$\begin{aligned} T= & {} \left( \begin{array}{ll} b&{} \quad b\\ -b&{} \quad b \end{array} \right) , \end{aligned}$$

and to use transformation

$$\begin{aligned} \left( \begin{array}{cc} u\\ v \end{array} \right) =T\left( \begin{array}{cc} \tilde{x}\\ \tilde{y} \end{array} \right) , \end{aligned}$$

thus (11) turn into

$$\begin{aligned} \left( \begin{array}{cc} \tilde{x}\\ \tilde{y} \end{array} \right) \rightarrow \left( \begin{array}{ll} 1&{} \quad 0\\ 0&{} \quad \lambda _2 \end{array} \right) \left( \begin{array}{cc} \tilde{x}\\ \tilde{y} \end{array} \right) +\left( \begin{array}{cc} \theta (\tilde{x},\tilde{y},b^*)\\ \vartheta (\tilde{x},\tilde{y},b^*) \end{array} \right) , \end{aligned}$$

(12)

where

$$\begin{aligned} \theta (\tilde{x},\tilde{y},b^*)= & {} \frac{1}{2b}(-2ub^*+2vb^*-(1-2b)(u^2+v^2))\\&+O((|u|+|v|+|b^*|)^3)),\\ \vartheta (\tilde{x},\tilde{y},b^*)= & {} \frac{-(1-2b)}{2b}(u^2+v^2)+O((|u|+|v|+|b^*|)^3)), \end{aligned}$$

with

$$\begin{aligned} u= & {} b(\tilde{x}+\tilde{y}),\\ v= & {} b(-\tilde{x}+\tilde{y}). \end{aligned}$$

Assuming there is a center manifold $W_c(0,0,0)$ of (12) at the fixed point (0, 0) in a small neighborhood of $b^*$ which may take the form

$$\begin{aligned} W_c(0,0,0)= & {} \{(\tilde{x},\tilde{y},b^*)\in R^3,\\ \tilde{y}= & {} l(\tilde{x},b^*), l(0,0)=0, Dl(0,0)=0\}, \end{aligned}$$

for $\tilde{x}$ and $b^*$ sufficiently small. Consider a center manifold as follows

$$\begin{aligned} l(\tilde{x},b^*)=m_0\tilde{x}^2+m_1\tilde{x}b^*+m_2{b^*}^2+O((|\tilde{x}|+|b^*|)^3). \end{aligned}$$

(13)

The center manifold must be satisfied

$$\begin{aligned} l(\tilde{x}+\theta (\tilde{x},l(\tilde{x},b^*),b^*),b^*)=\lambda _2l(\tilde{x},b^*)+\vartheta (\tilde{x},l(\tilde{x},b^*),b^*). \end{aligned}$$

(14)

By replacing (13) for (14) and matching similar power coefficients for in (14), we obtain

$$\begin{aligned} m_0=\frac{-2(1-2b)}{1-\lambda _2},m_1=m_2=0. \end{aligned}$$

Hence, we realize the system (12) which is restricted to the center manifold:

$$\begin{aligned}&F_1:\tilde{x}\rightarrow \tilde{x}-b(1-2b)\tilde{x}^2-2\tilde{x}b^*-b(1-2b)m_0^2\tilde{x}^4\nonumber \\&\qquad \qquad + O((|\tilde{x}|+|b^*|)^5), \end{aligned}$$

(15)

one can check that conditions of transcritical bifurcation are satisfies as

$$\begin{aligned} F_1(0,0)= & {} 0, \Big (\frac{\partial F_1}{\partial \tilde{x}}\Big )_{(0,0)}=1, \Big (\frac{\partial ^2 F_1}{\partial \tilde{x}^2}\Big )_{(0,0)}\\= & {} -2b(1-b)\ne 0, \Big (\frac{\partial ^2 F_1}{\partial b^* \partial \tilde{x}}\Big )_{(0,0)}= -2\ne 0. \end{aligned}$$

$\square$

3.2 Bifurcation for the fixed point $E_3$

The characteristic equation at the positive fixed point $E_3(x^*,y^*)=(\frac{1}{2a}((a-1-2b)+\sqrt{(1-a+2b)(1-a-2b)}),\frac{1}{2a}((a-1-2b)-\sqrt{(1-a+2b)(1-a-2b)})$ has the following form:

$$\begin{aligned} \lambda ^2+P(x^*,y^*)\lambda +Q(x^*,y^*)=0, \end{aligned}$$

where

$$\begin{aligned} P(x^*,y^*)= & {} -2a+2a(x^*+y^*)+2b,\\ Q(x^*,y^*)= & {} a^2(1-2x^*)(1-2y^*)-b(2a-2a(x^*+y^*)), \end{aligned}$$

let

$$\begin{aligned} F(\lambda )=\lambda ^2+P(x^*,y^*)\lambda +Q(x^*,y^*), \end{aligned}$$

If $a>1+2b$, then

$$\begin{aligned} F(1)=4b^2-(1-a)^2>0,\qquad F(-1)=4+4b+4b^2-(1-a)^2 \end{aligned}$$

It is very important here to pay attention that we cannot use Lemma 1 to classify topological properties of the fixed points $E_3$ and $E_4$. To make this clear, we need for Lemma 1 that $F(1)>0$ which ends up with $4b^2-(1-a)^2 >0$. The last inequality contradicts with the condition $4b^2-(1-a)^2<0$ which is necessary for the fixed points $E_3$ and $E_4$ to be real.

Now, by solving the characteristic equation

$$\begin{aligned} \lambda ^2-2(1+b)\lambda +(1+2b+4b^2-(1-a)^2)=0, \end{aligned}$$

has two eigenvalues:

$$\begin{aligned} \lambda _{1,2}=(1+b)\pm \sqrt{(1-a^2)-3b^2}. \end{aligned}$$

(16)

Proposition 3

The fixed point $E_3$ of the system (2) is

• a sink if $|(1+b)+ \sqrt{(1-a^2)-3b^2}|<1$, and $|(1+b)-\sqrt{(1-a^2)-3b^2}|<1$,

• a source if $|(1+b)+ \sqrt{(1-a^2)-3b^2}|>1$, and $|(1+b)- \sqrt{(1-a^2)-3b^2}|>1$,

• a saddle if either: $|(1+b)+ \sqrt{(1-a^2)-3b^2}|<1$, and $|(1+b)- \sqrt{(1-a^2)-3b^2}|>1$, or $|(1+b)+ \sqrt{(1-a^2)-3b^2}|>1$, and $|(1+b)- \sqrt{(1-a^2)-3b^2}|<1$,

• a non-hyperbolic if either $a=1\pm 2|b|$, or $a=1\pm 2\sqrt{b^2+b+1}$, $b\ne -1,-2$.

Let

$$\begin{aligned} FB_1= \left\{ (a,b): a=1-2\sqrt{(1+b)^2-b}, b\ne -1,-2\right\} , \end{aligned}$$

or

$$\begin{aligned} FB_2= \left\{ (a,b): a=1+2\sqrt{(1+b)^2-b}, b\ne -1,-2\right\} . \end{aligned}$$

Theorem 3

The system (2) can admit a flip bifurcation at the fixed point $E_3$ when parameters vary in a small neighborhood of $FB_1$ or $FB_2$.

Proof

Since $(a,b)\in FB_1$, choosing b represents the bifurcation parameter. Assuming the perturbation of (4):

$$\begin{aligned} \left\{ \begin{array}{ll} x\rightarrow a x(1-x)+(b_1+b^*)(y-x), \\ y \rightarrow a y(1-y)+(b_1+b^*)(x-y), \end{array} \right. \end{aligned}$$

(17)

such that $|b^*|\ll 1$ is the perturbation parameter.

Put $u = x-x^*$, $v = y-y^*$, thus the map (17) transformed as follows:

$$\begin{aligned} \left\{ \begin{array}{ll} u \rightarrow &{}a_1u+a_2v+a_{13}u b^*+a_{23}v b^*+a_{11}u^{2}\\ {} &{}+O((|u|+|v|+|b^*|)^3),\\ v \rightarrow &{}b_1u+b_2v+b_{13}u b^*+b_{23}v b^*+b_{22}v^2\\ {} &{}+O((|u|+|v|+|b^*|)^3), \end{array} \right. \end{aligned}$$

(18)

where

$$\begin{aligned} \left\{ \begin{array}{lllll} &{}a_{1}=a-2ax^*-b,\;a_{2}=b,\;a_{13}\\ {} &{}=-1,\;a_{23}=1,\; a_{11}=-a &{} \\ &{}b_{1}=b,\;b_{2}=a-2ay^*-b,\;b_{13}\\ {} &{}=1,\;b_{23}=-1,\; b_{23}=-1 ,\;b_{22}=-a. &{} \end{array} \right. \end{aligned}$$

(19)

Construct an invertible matrix

$$\begin{aligned} T=\left( \begin{array}{ll} a_{2} &{} \quad a_{2} \\ -1-a_{1} &{} \quad \lambda _{2}-a_{1} \end{array} \right) , \end{aligned}$$

and applying transformation:

$$\begin{aligned} \left( \begin{array}{c} u \\ v \end{array} \right) =T\left( \begin{array}{c} \tilde{x}\\ \tilde{y} \end{array} \right) , \end{aligned}$$

then the system (18) will be changed to

$$\begin{aligned} \left( \begin{array}{c} \tilde{x}\\ \tilde{y} \end{array} \right) \rightarrow \left( \begin{array}{ll} -1 &{} \quad 0 \\ 0 &{} \quad \lambda _{2} \end{array} \right) \left( \begin{array}{c} \tilde{x} \\ \tilde{y} \end{array} \right) +\left( \begin{array}{c} f(\tilde{x},\tilde{y},b ^{*}) \\ g(\tilde{x},\tilde{y},b ^{*}) \end{array} \right) , \end{aligned}$$

where

$$\begin{aligned} f(\tilde{x},\tilde{y},b ^{*})= & {} \left( \frac{a_{13}(\lambda _2-a_1)}{a_2(1+\lambda _2)}-a_2b_{13}\right) ub^*\\&+\left( \frac{a_{23}(\lambda _2-a_1)}{a_2(1+\lambda _2)}-a_2b_{23}\right) vb^*\\&+\frac{a_{4}(\lambda _2-a_1)}{a_2(1+\lambda _2)}u^2-\frac{a_2b_{22}}{a_2(1+\lambda _2)}v^2\\&+O((|u|+|v|+|b| ^{*}|)^{3}),\\ g(\tilde{x},\tilde{y},b ^{*})= & {} \left( \frac{(1+a_1)a_{13}}{a_2(1+\lambda _2)}+a_2b_{13}\right) ub^* \\&+\left( \frac{(1+a_1)a_{23}}{a_2(1+\lambda _2)}+a_2b_{13}\right) vb^*\\&+ \frac{(a_1+1)a_{11}}{a_2(1+\lambda _2}u^2+\frac{a_2b_{22}}{a_2(1+\lambda _2}v^2\\&+O((|u|+|v|+|b|^{*}|)^{3}). \end{aligned}$$

and

$$\begin{aligned} u= & {} a_{2}(\tilde{x}+\tilde{y}),\;v=-(1+a_{1})\tilde{x}+(\lambda _{2}-a_{1}) \tilde{y}, \\ u^{2}= & {} a_{2}^{2}(\tilde{x}^{2}+\tilde{x}\tilde{y}+\tilde{y}^{2}), \\ v^2= & {} (1+a_2)^2\tilde{x}^{2} + (\lambda _2-a_1)^2\tilde{y}^{2}-2(1+a_1)(\lambda _2-a_1)\tilde{x}\tilde{y}. \end{aligned}$$

Based on the center manifold theory, there exists the following center manifold:

$$\begin{aligned} W_{c}(0,0,0)= & {} \{(\tilde{x},\tilde{y},b ^{*})\in R^{3},\\ \tilde{y}= & {} h(\tilde{x},b ^{*}),h(0,0)=0,Dh(0,0)=0\}, \end{aligned}$$

for $\tilde{x}$ and $b ^{*}$ sufficiently small. To compute the center manifold, we assume that

$$\begin{aligned} h(\tilde{x},b ^{*})=n_{0}\tilde{x}^{2}+n_{1}\tilde{x}b ^{*}+n_{2}{b ^{*}}^{2}+O((|\tilde{x}|+|b ^{*}|)^{3}). \end{aligned}$$

(20)

The center manifold has to satisfy

$$\begin{aligned}&h(-\tilde{x}+f(\tilde{x},h(\tilde{x},b ^{*}),b ^{*}),b ^{*})\nonumber \\&\qquad =\lambda _{2}h(\tilde{x},b ^{*})+g(\tilde{x} ,h(\tilde{x},b ^{*}),b ^{*}). \end{aligned}$$

(21)

Replacing (20) in (21) and matching similar power coefficient values of (21), we have

$$\begin{aligned} n_{0}= & {} \frac{a_{11}a_2(1+a_1)+b_{22}(1+a_1)^2}{(1-\lambda _2^2)}, \\ n_{1}= & {} \frac{-(1+a_1)a_{13}-a_2b_{13}}{(1+\lambda _2)^2}, \\ n_{2}= & {} 0. \end{aligned}$$

The system (18) constrained by the center manifold is given as follows:

$$\begin{aligned}&\!\!\!\!F_2:\tilde{x}\rightarrow -\tilde{x}+A_1\tilde{x}^{2}+A_2\tilde{x}b ^{*}+A_3\tilde{x}^{2}b ^{*}+A_4\tilde{x}{b ^{*} }^{2}\nonumber \\&\!\!\!\!\quad +A_5\tilde{x}^{3}+A_6\tilde{x}^{3}b^*+A_7\tilde{x}^{2}{b ^{*} }^{2}+A_8\tilde{x}^{4}+ O((|\tilde{x}|+|b ^{*}|)^{5}).\nonumber \\ \!\!\! \end{aligned}$$

(22)

where

$$\begin{aligned} A_1= & {} a_{11}a_2^2-a_2b_{22}(1+a_1),\\ A_2= & {} \frac{\lambda _2-a_1}{a_2(1+\lambda _2)}(a_{13}a_2-a_{23}(1+a_1))\\&-a_2^2b_{13}+b_{23}a_2(1+a_1),\\ A_3= & {} \frac{\lambda _2-a_1}{a_2(1+\lambda _2)}(a_{13}a_2n_0+a_{23}(\lambda _2-a_1)n_0+2a_{11}a_2^2n_1)\\&-a_2^2b_{13}n_0-a_2b_{23}(\lambda _2-a_1)n_0\\&+a_2b_{22}(1+a_1)(\lambda _2-a_1)n_1,\\ A_4= & {} \frac{\lambda _2-a_1}{a_2(1+\lambda _2)}(a_{13}a_2n_1+a_{23}(\lambda _2-a_1)n_1)\\&-a_2^2b_{13}n_1-a_2b_{23}(\lambda _2-a_1)n_1,\\ A_5= & {} 2a_{11}a_2^2n_0\frac{\lambda _2-a_1}{a_2(1+\lambda _2)}+a_2b_{22}(1+a_1)(\lambda _2-a_1)n_0,\\ A_6= & {} 2a_{11}a_2^2n_0n_1\frac{\lambda _2-a_1}{a_2(1+\lambda _2)}-2a_2b_{22}(\lambda _2-a_1)^2n_0n_1,\\ A_7= & {} a_{11}a_2^2n_1^2\frac{\lambda _2-a_1}{a_2(1+\lambda _2)}-a_2b_{22}(\lambda _2-a_1)^2n_1^2,\\ A_8= & {} a_{11}a_2^2n_0^2\frac{\lambda _2-a_1}{a_2(1+\lambda _2)}-a_2b_{22}(\lambda _2-a_1)^2n_0^2. \end{aligned}$$

Thus, map (22) undergoes a flip bifurcation because the following conditions are satisfied

$$\begin{aligned} \beta _{1}= & {} \Big (2\frac{\partial ^{2}F_2}{\partial b ^{*}\partial \tilde{x}}+\frac{\partial F_2}{\partial b ^{*}}\frac{\partial F_2}{ \partial \tilde{x}}\Big )_{(0,0)}=2A_2\ne 0, \\ \beta _{2}= & {} \Big (\frac{1}{2}\Big (\frac{\partial ^{2}F_2}{\partial \tilde{x} ^{2}}\Big )^{2}+\frac{1}{3}\Big (\frac{\partial ^{3}F_2}{\partial \tilde{x}^{3}} \Big )\Big )_{(0,0)}=2(A_1^2+A_5)\ne 0. \end{aligned}$$

$\square$

The same procedure can be applied to the points in the neighborhood of $FB_{2}$.

We pay attention here that a Neimark-Sacker bifurcation can not occur neither at the fixed point $E_{3}$ nor at $E_{4}$. This is because the eigenvalues in (16) are complex only if $3b^2>(1-a)^2$ which contradicts the fact that the fixed points $E_3$ and $E_4$ are real only if $4b^2<(1-a)^2$. On the other hand, we may discuss the possibility of occurence of Neimark-Sacker bifurcation at $E_3$ if it is not real. If the (a, b) parameters vary in a small neighborhood of $NS_{1,2}$ which is expressed by

$$\begin{aligned} NS_1= \left\{ (a,b):b=\frac{-1+\sqrt{1+4(1-a)^2}}{2}, a\ne -1,3\right\} , \end{aligned}$$

$$\begin{aligned} NS_2= \left\{ (a,b):b=\frac{-1-\sqrt{1+4(1-a)^2}}{2}, a\ne -1,3 \right\} , \end{aligned}$$

Considering parameters $(c,s,b_2)$ arbitrarily from $NS_1$, Take into account the system (5) with $(c,s,b_2)$, that is described in

$$\begin{aligned} \left\{ \begin{array}{ll} x\rightarrow a x(1-x)+b_2(y-x), \\ y \rightarrow a y(1-y)+b_2(x-y), \end{array} \right. \end{aligned}$$

(23)

The system (23) has a positive fixed point $E_3(x^*,y^*)$. Since parameters $(c,s,b_2)\;\in NS_1$, then $b_2=\frac{-1+ M}{4}$, where $M=\sqrt{1+4(1-a)^2}$. Choosing $b^*$ as the bifurcation parameter, we consider a perturbation of the system (23) as follows:

$$\begin{aligned} \left\{ \begin{array}{ll} x\rightarrow a x(1-x)+(b_2+\bar{b^*})(y-x), \\ y \rightarrow a y(1-y)+(b_2+\bar{b^*})(x-y), \end{array} \right. \end{aligned}$$

(24)

such that $\bar{b^*}\ll 1$ is a perturbation parameter.

Let $u = x-x^*$, $v = y-y^*$, thus the map (24) transformed to

$$\begin{aligned} \left\{ \begin{array}{ll} u \rightarrow &{} \quad a_1u+a_2v+a_{11}u^{2}+O((|u|+|v|+|b^*|)^3),\\ v \rightarrow &{} \quad b_1u+b_2v+b_{22}v^2+O((|u|+|v|+|b^*|)^3), \end{array} \right. \end{aligned}$$

(25)

where $a_1, a_2, a_{11}$, and $b_1, b_2, b_{22}$ are chosen to give (20) replacing $b_1$ by $b_2+\bar{b}^*$.

Now, the characteristic equation of system (25) can be written as

$$\begin{aligned} \lambda ^2+P(\bar{b}^*)\lambda +Q(\bar{b}^*)=0, \end{aligned}$$

where

$$\begin{aligned} P(\bar{b}^*)= & {} -2(1+\bar{b}^*+b_2),\\ Q(\bar{b}^*)= & {} 1+2(\bar{b}^*+b_2)+4(\bar{\delta }^*+\delta _2)^2-(1-a)^2. \end{aligned}$$

Now, we can write the pair of complex eigenvalues in the form

$$\begin{aligned} \lambda ,\bar{\lambda }=-\frac{P(\bar{b}^*)}{2}\pm \frac{i}{2}\sqrt{4Q(\bar{b}^*)-{P^2(\bar{b}^*)}}, \end{aligned}$$

and so

$$\begin{aligned} |\lambda |_{\bar{b}^*=0}= & {} \sqrt{Q(0)}=1,\\ \frac{d|\lambda |}{d\bar{b}^*}|_{\bar{b}^*=0}= & {} -\frac{M}{\sqrt{\frac{1}{4}(3+M^2)-(1-a)^2}}\ne 0. \end{aligned}$$

Moreover, we required that when $\bar{b}^*=0$, $\lambda ^m,{\bar{\lambda }}^m\ne 1, (m=1,2,3,4)$, which is equivalent to $P(0)\ne -2, 0, 1, 2$. Since we choose $(c,s,b_2)\in NS_1$. So, $P(0)\ne -2, 2$. We require only that $P(0)\ne 0,1$, which ends up with

$$\begin{aligned} M\ne -3,3. \end{aligned}$$

(26)

Therefore, the eigenvalues $\lambda ,\bar{\lambda }$ at origin of the system (25) do not lie in the intersection of the unit circle with the coordinate axes when $\bar{b}^*=0$ and the condition (26) holds.

Next, we analyze the normal system form (25) at $\bar{b}^*=0$. Let $\bar{b}^*=0$, $\mu =1+b_2$, $\omega =\sqrt{3b^2-(1-a)^2}$. Construct an invertible matrix

$$\begin{aligned} T= & {} \left( \begin{array}{ll} a_2&{} \quad 0\\ \mu -a_1&{} \quad -\omega \end{array} \right) , \end{aligned}$$

using transformation

$$\begin{aligned} \left( \begin{array}{cc} u\\ v \end{array} \right) =T\left( \begin{array}{cc} \tilde{x}\\ \tilde{y} \end{array} \right) , \end{aligned}$$

then the system (25) has form

$$\begin{aligned} \left( \begin{array}{cc} \tilde{x}\\ \tilde{y} \end{array} \right) \rightarrow \left( \begin{array}{ll} \mu &{} \quad -\omega \\ \omega &{} \quad \mu \end{array} \right) \left( \begin{array}{c} u\\ v \end{array} \right) +\left( \begin{array}{cc} f_1(\tilde{x},\tilde{y})\\ f_2(\tilde{x},\tilde{y}) \end{array} \right) , \end{aligned}$$

(27)

where

$$\begin{aligned} f_1(\tilde{x},\tilde{y})= & {} \frac{a_{11}}{a_2}u^2+O((|\tilde{x}|+|\tilde{y}|)^3),\\ f_2(\tilde{x},\tilde{y})= & {} \frac{(\mu -a_1)a_{11}}{a_2\omega }u^2-\frac{b_{22}}{\omega }v^2+O((|\tilde{x}|+|\tilde{y}|)^3). \end{aligned}$$

with

$$\begin{aligned} u= & {} a_2 \tilde{x},\; v=(\mu -a_1)\tilde{x}-\omega \tilde{y},\\ u^2= & {} {a_2}^2{\tilde{x}}^2,\;v^2=(\mu -a_1)^2\tilde{x}^2+\omega ^2\tilde{y}^2-2\omega (\mu -a_1)\tilde{x}\tilde{y}. \end{aligned}$$

So, at the origin (0, 0), we have

$$\begin{aligned} f_{1\tilde{x}\tilde{x}}= & {} 2a_{11}a_2,\\ f_{1\tilde{x}\tilde{x}\tilde{x}}= & {} f_{1\tilde{x}\tilde{y}}=f_{1\tilde{y}\tilde{y}}=f_{1\tilde{x}\tilde{x}\tilde{y}}=f_{1\tilde{x}\tilde{y}\tilde{y}} =f_{1\tilde{y}\tilde{y}\tilde{y}}=0,\\ f_{2\tilde{x}\tilde{x}}= & {} \frac{2(\mu -a_1)}{\omega }(a_2-b_{22}(\mu -a_1)),\;f_{2\tilde{x}\tilde{y}}=b_{22}(\mu -a_1),\\ f_{2\tilde{y}\tilde{y}}= & {} -2b_{22}\omega ,\\ f_{2\tilde{x}\tilde{x}\tilde{x}}= & {} f_{2\tilde{x}\tilde{x}\tilde{y}}=f_{2\tilde{x}\tilde{y}\tilde{y}}=f_{2\tilde{y}\tilde{y}\tilde{y}}=0. \end{aligned}$$

The system (25) will encounter the Neimark-Sacker bifurcation when the following quantity is not equal to zero:

$$\begin{aligned} \theta= & {} \Big [-Re\Big (\frac{(1-2\lambda ){\bar{\lambda }}^2}{1-\lambda }L_{11}L_{12}\Big )\\&\qquad \qquad -\frac{1}{2}|L_{11}|^2 -|L_{21}|^2+Re(\bar{\lambda }L_{22})\Big ]_{\bar{b}^*=0}, \end{aligned}$$

where

$$\begin{aligned} L_{11}= & {} \frac{1}{4}((f_{1\tilde{x}\tilde{x}}+f_{1\tilde{y}\tilde{y}})+i(f_{2\tilde{x}\tilde{x}}+f_{2\tilde{y}\tilde{y}})),\\ L_{12}= & {} \frac{1}{8}((f_{1\tilde{x}\tilde{x}}-f_{1\tilde{y}\tilde{y}}+2f_{2\tilde{x}\tilde{y}})+i(f_{2\tilde{x}\tilde{x}}-f_{2\tilde{y}\tilde{y}}-2f_{1\tilde{x}\tilde{y}})),\\ L_{21}= & {} \frac{1}{8}((f_{1\tilde{x}\tilde{x}}-f_{1\tilde{y}\tilde{y}}-2f_{2\tilde{x}\tilde{y}})+i(f_{2\tilde{x}\tilde{x}}-f_{2\tilde{y}\tilde{y}}+2f_{1\tilde{x}\tilde{y}})),\\ L_{22}= & {} \frac{1}{16}((f_{1\tilde{x}\tilde{x}\tilde{x}}+f_{1\tilde{x}\tilde{y}\tilde{y}}+f_{2\tilde{x}\tilde{x}\tilde{y}}+f_{2\tilde{y}\tilde{y}\tilde{y}})\\&+i(f_{2\tilde{x}\tilde{x}\tilde{x}}+f_{2\tilde{x}\tilde{y}\tilde{y}}-f_{1\tilde{x}\tilde{x}\tilde{y}}-f_{1\tilde{y}\tilde{y}\tilde{y}})), \end{aligned}$$

The same arguments can be applied to $NS_2$.

4 Coupled logistic maps with white noise

In any real system, noise is present and this makes the interaction between nonlinearity and stochasticity very important in modeling dynamic behaviors of many systems such as epidemics, climate, optics and so on [52,53,54,55]. In [55], the authors have studied the effect of noise on the attractors of two coupled logistic maps. They have concluded that a very small noise can lead to attractor destruction. The aim of this part of the paper is to discuss the response of the fixed points of the deterministic system (2) to random disturbance.

Consider the following stochastically forced system

$$\begin{aligned} \left\{ \begin{array}{l} x_{n+1}= ax_n(1-x_n)+b(y_n-x_n)+\varepsilon _1\eta _n,\\ y_{n+1}= ay_n(1-y_n)+b(x_n-y_n)+\varepsilon _2\zeta _n, \end{array} \right. \end{aligned}$$

(28)

where $\varepsilon _1$ and $\varepsilon _2$ are noise intensities, and $\eta _n$ and $\zeta _n$ are independent Gaussian random values with parameters $E\eta _n=E\zeta _n=0$, $E\eta ^2=E\zeta ^2=1$. According to [56], the stochastic trajectories leave the deterministic attractor under the random noise and form a probabilistic distribution nearby. In our analysis, we assume that $\varepsilon _1 = \varepsilon _2= \varepsilon$.

4.1 Analysis of randomly forced fixed points

As the noise is present, the regular structure of the fixed points is smoothed. A dispersion of random states near the bifurcation points grows. Consider the influence of noise on fixed point $E_1$ of model (2). The following analysis is based on the stochastic sensitively function technique and confidence ellipses method represented in [56,57,58,59]. Let us consider the impact of the noise on $E_1$. According to this method, we need to construct a matrix $W=\left( \begin{array}{ll} w_{11} &{} \quad w_{12} \\ w_{21} &{} \quad w_{22} \end{array} \right)$, which is the stochastic sensitivity matrix for the fixed point $E_1(0,0)$. In fact, W is the unique solution to the matrix equation [56] $W=J W J^T+ Q, \,\,J=\frac{\partial f}{\partial x}(E_1), \,\, Q=\sigma (E_1)\sigma ^T(E_1)$, where $f=\left( \begin{array}{c} ax(1-x)+b(y-x) \\ ay(1-y)+b(x-y) \end{array} \right)$, and $\sigma (E_1)$ characterizes the dependence of random disturbance on state. Consequently, we have

$$\begin{aligned} w_{11}= & {} \frac{-a^2+2ab-2b^2+1}{a^4-4a^3b+4a^2b^2-2a^2+4ab-4b^2+1},\\ w_{12}= & {} w_{21}=\frac{2b(a-b)}{a^4-4a^3b+4a^2b^2-2a^2+4ab-4b^2+1},\\ w_{22}= & {} \frac{-a^2+2ab-2b^2-1}{a^4-4a^3b+4a^2b^2-2a^2+4ab-4b^2+1}. \end{aligned}$$

The eigenvalues associated to W are $\lambda _1=\frac{-1}{a^2-1}$, and $\lambda _2=\frac{1}{-a^2+4ab-4b^2+1}$ which at the fixed point $E_1$ describes the stochastic sensitivity of noise.

Fig. 1

Eigenvalues of the matrix W of system (28) at $E_1$

The two eigenvalues have different attitude as it is depicted in Fig. 1. $\lambda _1(b)$ is constant while $\lambda _2(b)$ is monotonically increasing form 1.35 to 1.95 for $a=0.5$. These eigenvalues and the corresponding eigenvectors form confidence ellipses as spatial arrangement of random states around the fixed point $E_1(0,0)$. In fact, the eigenvalues determine the sizes of the semi-axes of the ellipses, and the eigenvectors demonstrate the directions of these axes. Figure 2 shows random states and confidence ellipse for $a=0.5$, $\varepsilon =0.001$ and $b=0.6$ of system (28) at $E_1$ with a trust probability of $P=0.95$.

Fig. 2

Random states and confidence ellipse for $a=0.5$ and $\varepsilon =0.001$ of the system (28) at $E_1$

4.2 Noise-induced transitions between attractors

The deterministic system (2) has variety of dynamic behavior such as regular attractors deformed in closed invariant curves. Consider the transition induced by noise between stochastic system attractors (28) for $a=3.2$, and $b=0.15$. For these values, the deterministic system (2) admits coexisting two closed invariant curves and a 6-discrete cycle as shown in Fig. 3.

Fig. 3

Attractors of the deterministic system (2) for $a=3.2$ and $b=0.15$

First of all, let the noise intensity be weak, that is $\varepsilon =0.002$. As depicted in Fig. 4, random trajectories which start near one of the closed invariant curves are well localized near it. As the intensity of the noise increases, that is $\varepsilon =0.02$, a dispersion of random states increases too.

Fig. 4

Random states of system (28) with $a=3.2$, $b=0.15$, and a)$\varepsilon =0.002$, b)$\varepsilon =0.02$, c)$\varepsilon =0.05$, d)$\varepsilon =0.1$, closed invariant curves of system (2) are plotted in red

5 Numerical simulations

Numerical simulations for the verification of analytical results obtained in Sects. 3 and 4 are shown in this section.

• 1. First of all, let us consider the deterministic system (2). Fix the parameter a and let b be free. In Figs. 5 and 6, we present the bifurcation diagram and corresponding maximal lyapunov exponent for the influence of the parameter b. Figure 7 represents the bifurcation diagram when $a=3$ with initial values $(x_0,y_0)=(0.1,0.2)$. The fixed point$E_2=(\frac{a-1}{a},\frac{a-1}{a})$ is given by $E_2=(0.6875,0.6875)$. At $b=-0.1$, $E_2$ loses its stability via a period-2 orbit that agrees with the theorem 1. The associated maximal lyapunov exponent is shown in Fig. 8. Next, let $a=3.4$, that is, $E_2=(0.7059,0.7059)$. At $b=-0.2$, $E_2$ loses its stability via a period-2 orbit which again agrees with theorem 1 as can be seen in Fig. 9. The corresponding maximal lyapunov exponent is shown in Fig. 10. The same results can be said to Figs. 11 and 12. The transcritical bifurcation at $E_2=(0.6875,0.6875)$ occurs at $b=-1.1$ as $a=3.2$ as can be seen in Fig. 7 which agrees with theorem 2. Again the system (2) admits a transcritical bifurcation at $E_2=(0.7059,0.7059)$ if $a=3.4$ and $b=-1.2$ and this agrees with theorem 2. Figure 11 illustrates the bifurcation diagram for $a=3.45$ and different b, while Fig. 12 illustrates the corresponding maximal lyapunov exponent. Finally, different phase portraits are plotted in Fig. 13 for different a and b. Figure 13a shows four closed invariant curves with $a=3.4$ and $b=-0.45$ which appear as a result of a Neimark-Sacker bifurcation, while Fig. 13b–e show chaotic attractors with $a=3.5,3.5,3,3.2,3.45,3.45$ and $b=-0.4,-0.2,0.4,0.3,-0.3,-0.25$ respectively. In [43], it was concluded that coupled logistic maps have new transitions to chaos such as quasiperiodicity and torus destruction as can be seen in Figs. 9 and 11.

• 2. Second of all, let us consider the stochastic system (28). Figure 14 shows the noise induced transformations of bifurcation diagrams for $a=3.4$ and for $\varepsilon =0.001,0.005,0.02,0.01$ in Fig. 14a–d, respectively. From above figures, the fine structures of the bifurcation diagrams become blemished especially near the bifurcation points. Fig. 15a–d show the influence of the white noise on the regular attractors (four closed invariant curves here) of the deterministic system (2) when $a=3.4$, $b=-0.25$ and different noise intensity $\varepsilon =0.001,0.005,0.02,0.01$.

Fig. 5

Bifurcation of (2) in (b, x) plane for $a=3$

Fig. 6

Maximal Lyapunov exponent for (2)

Fig. 7

Bifurcation for the system (2) in (b, x) plane for $a=3.2$

Fig. 8

Maximal Lyapunov exponent for (2)

Fig. 9

Bifurcation for the system (2) in (b, x) plane for $a=3.4$

Fig. 10

Maximal Lyapunov exponent for (2)

Fig. 11

Bifurcation for the system (2) in (b, x) plane for $a=1.7$

6 Conclusion

Linearly coupled logistic maps as a deterministic and a stochastic system is considered in this work. The form of the coupled system were already proposed in [25] and investigated further by other researchers but the detailed bifurcation analysis were not reported in any of them. The local stability conditions for the fixed points of the deterministic system are obtained. Dynamic behavior such as bifurcation and chaos is described in the proposed system. According to the center manifold theorem and the bifurcation theory, explicit conditions assure that the system admits transcritical, flip, and Neimark-Sacker bifurcations are given. The detailed bifurcation analysis introduced here supports the numerical observations given in [25]. Since we believe that noise is present in any nonlinear real system, we add a white noise to the deterministic system and study its influence on its fixed points using the stochastic sensitivity function technique. Finally, The phenomenon of noise-induced shifts between closed invariant curves is explored.

Fig. 12

Maximal Lyapunov exponent for (2)

Fig. 13

Phase portraits for the system (2) with different a and b

Fig. 14

The bifurcation diagrams for stochastic system (28) with $a=3.4$ and a $\varepsilon =0.001$, b $\varepsilon =0.005$, c $\varepsilon =0.02$, and d $\varepsilon =0.01$

Fig. 15

Phase portraits for the stochastic system (28) with $a=3.4$, $b=-0.25$, and a $\varepsilon =0.001$, b $\varepsilon =0.005$, c $\varepsilon =0.02$, and d $\varepsilon =0.01$

Now, the new results in this work enhance the understanding of the complexities of deterministic and stochastic logistic mapping system. The rich dynamics of the system have also been analyzed, including interesting chaotic sets. In addition, the studied system (2) can be used in various engineering applications such as secure communications, encryption and security of information which will be investigated in a forthcoming work. In addition, this paper provides an effective analytical technique for the thorough implementation of various discrete time systems.