Research on cascading high-dimensional isomorphic chaotic maps

Abstract

In order to overcome the security weakness of the discrete chaotic sequence caused by small Lyapunov exponent and keyspace, a general chaotic construction method by cascading multiple high-dimensional isomorphic maps is presented in this paper. Compared with the original map, the parameter space of the resulting chaotic map is enlarged many times. Moreover, the cascaded system has larger chaotic domain and bigger Lyapunov exponents with proper parameters. In order to evaluate the effectiveness of the presented method, the generalized 3-D Hénon map is utilized as an example to analyze the dynamical behaviors under various cascade modes. Diverse maps are obtained by cascading 3-D Hénon maps with different parameters or different permutations. It is worth noting that some new dynamical behaviors, such as coexisting attractors and hyperchaotic attractors are also discovered in cascaded systems. Finally, an application of image encryption is delivered to demonstrate the excellent performance of the obtained chaotic sequences.

Introduction

The biological neural network is a complex nonlinear dynamic system, and it is a hot topic to study the dynamics of neuronal networks in nonlinear science and neuroscience. Since the brain has functions such as self-learning, association, memory and reasoning, the cognitive system, situational memory, cooperative communication, autonomous decision-making and brain neurodynamics based on the biological neural network were studied (Zommara et al. 2018; Irak et al. 2019; Puanhvuan et al. 2017; Mora-Sánchez et al. 2019). Animal neural electrophysiological experiments have shown that chaos exists in biological neural systems. Aihara chaotic neural network reflects the chaotic characteristics of nervous systems and has been widely concerned (Aihara et al. 1990). Quite a few neuro-dynamical models are typically developed in the form of ordinary differential equations, which feature the dynamical behaviors of biological processes in the neuron (Dong et al. 2014; Xiao and Cao 2010). By substituting the resistive coupling weights with memristive coupling weights, several memristor-based neural networks were presented. Among these models, complex dynamical behaviors of limit cycle, chaotic attractor, and hyperchaotic attractor, as well as coexisting attractors are exhibited by numerical simulations (Bao et al. 2017a, b; Chen et al. 2019). There are also many other neural networks that can exhibit complex dynamical behaviors of chaotic oscillations (Zheng and Tonnelier 2009; Wang 2007; Jia et al. 2012; Liu et al. 2016). Additionally, the synchronization of neuronal network with different network connectivity patterns (Qu et al. 2012), coupling strength (Du et al. 2015), gap junction (Wang et al. 2013), and time-varying delay (Balasubramaniam et al. 2011) have been explored based on the stability and bifurcation theories.

Since chaos was discovered by Lorenz in the study of weather prediction in Lorenz (1963), the exploration of chaos dynamical behavior has aroused extensive attention of scholars around the world. There are a number of excellent researches on chaos disciplines, which have led to the full development of chaos. Recently, plenty of researches have devoted to the construction of new chaotic dynamic systems, such as multi-scroll and multiwing chaotic attractors (Muñoz-Pacheco et al. 2018; Zhang et al. 2018b), hyperchaotic attractors (Li et al. 2018), coexisting chaotic attractors (Vaidyanathan et al. 2019b), memristor-based attractors (Wang et al. 2018; Chen et al. 2018), nonautonomous chaotic attractor (Hong et al. 2018, 2019), hidden attractors (Leonov and Kuznetsov 2013; Dudkowski et al. 2016; Zhang et al. 2018a; Vaidyanathan et al. 2019a; Sambas et al. 2019a, b), and so on.

Considering that chaos is widely used in the field of secure communication, constructing chaotic systems with wonderful flexibility is of great value to information engineering applications. However, with the development of chaotic recognition technology, chaotic systems with poor performance are vulnerable to attack by different methods. For example, some simple chaotic sequence encryption can be deciphered by the spatial reconstruction method and exhaustive method (Daniel and Ion 2014; Li et al. 2016). Therefore, it is an urgent task to develop chaotic systems with more excellent chaotic performance.

There have been many kinds of research on the design of discrete chaotic maps and continuous-time chaotic systems. The mathematical model of continuous chaotic system is a multi-variable coupled system of differential equations, which has a lot of parameters and initial conditions. However, the algorithm of continuous chaotic system is complex, thus resulting in slow chaotic operation rate and low sequence bit rate. On the other hand, the discrete chaotic map has the advantages of fast operation speed and a high bit rate due to its simple algorithm. Therefore it is more suitable for various chaos-based applications.

Common discrete chaotic maps, such as Logistic map, Tent map, Sine map, and Gauss map, are usually designed with simple structures (Hua et al. 2017). In order to create chaotic map with better performance, different nonlinear operations have been introduced into seed maps, including cascade operation (Wang and Yuan 2013), parameter modulation (Hua and Zhou 2015), wheel-switching (Wu et al. 2014), scalar cascade and scalar modulation (Hua and Zhou 2017), sine-based control (Hua et al. 2017), etc. The utilization of these operations can improve the chaotic performance to some extent. However, almost all these operations are based on 1-D chaotic maps. Considering that there are limitations in 1-D chaotic maps, like few types, simple structure, and small parameter space, which make them tend to be deciphering through phase space reconstruction, reverse iteration or other methods. This paper attempts to cascade high-dimensional isomorphic chaotic maps and analyze the dynamic behavior of the generated chaos.

In the implementation of a multilayer cascade of high dimensional isomorphism chaotic maps, the input of the state variable in each layer is the output of the upper layer. To realize a closed-loop drive, the output of the last layer is employed as the input of the first layer for the next iteration. This framework is suitable for arbitrary high-dimensional and low-dimensional chaotic maps. Compared with the cascaded 1-D chaotic map, the system generated by cascading high-dimensional isomorphic chaotic maps has a more complex structure and much larger keyspace. Moreover, some rich dynamical behaviors, e.g. hyperchaotic attractors and coexisting chaotic attractors, could also emerge. Taking the generalized 3-D Hénon map as an example, the cascade of subsystems with different parameters is discussed firstly. Then the impact of different permutations is analyzed when multiple subsystems are cascaded. The simulation results show that the Lyapunov exponents of the cascaded chaotic system are closely related to the sum of the Lyapunov exponents of its subsystems. Besides, an image encryption application is further proposed to verify the excellent performance of cascaded chaotic systems. The main contributions of this work are summarised as follows:

1. The general framework of cascading isomorphic chaotic map is presented for both high-dimensional and low-dimensional chaotic maps.

2. The dynamical behaviors of new chaotic maps generated by cascading subsystems with different parameters and different permutations are analyzed separately.

3. Random properties of the constructed chaotic sequences are evaluated using the National Institute of Standards and Technology (NIST) SP800-22. And the performance analyses of image encryption are conducted to demonstrate the complexity of the sequence.

The rest of this paper is organized as follows. The general framework and the analysis of chaotic dynamical behavior are introduced in section “cascading of high-dimensional isomorphic chaotic maps”. The dynamical behaviors of cascaded 3-D Hénon map with different parameters and different permutations are analyzed in the following two sections. In the fifth section, the application of image encryption based on the generated chaotic sequence is introduced. Finally, conclusions are drawn in the last section.

Cascading of high-dimensional isomorphic chaotic maps

In order to improve the randomness and security of discrete chaotic sequence, which means, to improve the Lyapunov exponents and expand the keyspace of chaotic system, a cascade scheme for the discrete high-dimensional isomorphic chaotic map is presented in this section.

General framework

Consider the n-D chaotic map $\mathbf{x}(k+1)=F(\mathbf{x}(k))$ as follows

$$\left\{\begin{array}{c}{x}_1(k+1)=f_1(x_1(k),x_2(k),\dots ,x_n(k))\hfill \\ x_2(k+1)=f_2(x_1(k),x_2(k),\dots ,x_n(k))\hfill \\ ⋮\hfill \\ x_n(k+1)=f_n(x_1(k),x_2(k),\dots ,x_n(k)),\hfill \end{array}\right)$$ 1

where $\mathbf{x}(k)={[x_1(k),x_2(k),\dots ,x_n(k)]}^T$. When m isomorphic maps are cascaded, the generated system $\mathbf{y}(k+1)=\Gamma (\mathbf{y}(k))$ satisfies $\mathbf{y}(k+1)=F_m\circ F_{m-1}\circ F_{m-2}\circ \cdots \circ F_1(\mathbf{y}(k))$. The relationship between the cascaded chaotic map and its isomorphic subsystems is shown in Fig. 1. In order to achieve the cascade of m subsystems, the ith output of the upper-level ${\mathbf{x}}^i(j)$ needs to be the ith input of the next level ${\mathbf{x}}^i(j+1)$. The ith output of the last level ${\mathbf{x}}^i(m)$ is the input of the first level for the next round ${\mathbf{x}}^{i+1}(0)$. When $i=1,2,3,\dots ,N$, the output sequence of the iterative system is represented in matrix B as follows:

$$\begin{array}{c}\hfill B=\left[\begin{array}{ccccc}{\mathbf{x}}^1(1)& {\mathbf{x}}^2(1)& \dots & {\mathbf{x}}^{N-1}(1)& {\mathbf{x}}^N(1)\\ {\mathbf{x}}^1(2)& {\mathbf{x}}^2(2)& \dots & {\mathbf{x}}^{N-1}(2)& {\mathbf{x}}^N(2)\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ {\mathbf{x}}^1(m)& {\mathbf{x}}^2(m)& \dots & {\mathbf{x}}^{N-1}(m)& {\mathbf{x}}^N(m)\end{array}\right],\end{array}$$ 2

where ${\mathbf{x}}^i(j+1)=F_{j+1}({\mathbf{x}}^i(j))$, $j=0,1,2,\dots ,m-1$. The last row in matrix B is the output sequence of the cascaded chaotic map, that is $\mathbf{y}(i)={\mathbf{x}}^i(m)$.

Fig. 1.

The general framework of cascading high-dimensional isomorphic chaotic map

Suppose each subsystem has P parameters, then the parameter space of the chaotic map cascaded by m subsystems is mP, which is more in line with the requirement of large key space.

Chaotic dynamical behavior analysis

Lyapunov exponent represents the average exponential divergence rate of adjacent trajectories in phase space after a long iteration. The distance between two near initial points is $\delta {\mathbf{y}}_0=(\delta y_{10},\delta y_{20},\dots ,\delta y_{n0})$. After an iteration, its evolution is determined by the Jacobian matrix:

$$\begin{array}{c}\hfill \left[\begin{array}{c}\delta y_{11}\\ \delta y_{21}\\ ⋮\\ \delta y_{n1}\end{array}\right]=\left[\begin{array}{cccc}\frac{\partial {\Gamma }_1}{\partial y_1}& \frac{\partial {\Gamma }_1}{\partial y_2}& \dots & \frac{\partial {\Gamma }_1}{\partial y_n}\\ \frac{\partial {\Gamma }_2}{\partial y_1}& \frac{\partial {\Gamma }_2}{\partial y_2}& \dots & \frac{\partial {\Gamma }_2}{\partial y_n}\\ ⋮& ⋮& \ddots & ⋮\\ \frac{\partial {\Gamma }_n}{\partial y_1}& \frac{\partial {\Gamma }_n}{\partial y_2}& \dots & \frac{\partial {\Gamma }_n}{\partial y_n}\end{array}\right]\left[\begin{array}{c}\delta y_{10}\\ \delta y_{20}\\ ⋮\\ \delta y_{n0}\end{array}\right].\end{array}$$ 3

For simplicity, Eq. (3) is noted as $\delta {\mathbf{y}}_1={\mathbf{J}}_{\Gamma }^1({\mathbf{y}}_0)\delta {\mathbf{y}}_0$. According to the differential principle of the composite function:

$$\begin{array}{c}\hfill {\mathbf{J}}_{\Gamma }^1({\mathbf{y}}_0)=\frac{d{F}_m({\mathbf{x}}_{m-1}^1)}{d{\mathbf{x}}_{m-1}^1}\frac{d{F}_{m-1}({\mathbf{x}}_{m-2}^1)}{d{\mathbf{x}}_{m-2}^1}\dots \frac{d{F}_2({\mathbf{x}}_1^1)}{d{\mathbf{x}}_1^1}\frac{d{F}_1({\mathbf{x}}_0^1)}{d{\mathbf{x}}_0^1},\end{array}$$ 4

that is

$$\begin{array}{cc}\hfill {\mathbf{J}}_{\Gamma }^1({\mathbf{y}}_0)& ={\mathbf{J}}_{F_m}^1({\mathbf{x}}_{m-1}^1){\mathbf{J}}_{F_{m-1}}^1({\mathbf{x}}_{m-2}^1)\dots {\mathbf{J}}_{F_2}^1({\mathbf{x}}_1^1){\mathbf{J}}_{F_1}^1({\mathbf{x}}_0^1)\hfill \\ \hfill & =\prod _{j=0}^{m-1}{\mathbf{J}}_{F_{j+1}}^1({\mathbf{x}}_j^1),\hfill \end{array}$$ 5

where ${\mathbf{x}}_j^i={\mathbf{x}}^i(j)$.

After N iterations, the difference between two sequences is

$$\begin{array}{c}\hfill \delta {\mathbf{y}}_N={\mathbf{J}}_{\Gamma }^N({\mathbf{y}}_{N-1}){\mathbf{J}}_{\Gamma }^{N-1}({\mathbf{y}}_{N-2})\dots {\mathbf{J}}_{\Gamma }^2({\mathbf{y}}_1){\mathbf{J}}_{\Gamma }^1({\mathbf{y}}_0)\delta {\mathbf{y}}_0,\end{array}$$ 6

and

$$\begin{array}{c}\hfill {\mathbf{J}}_{\Gamma }^i({\mathbf{y}}_{i-1})=\frac{d{F}_m({\mathbf{x}}_{m-1}^i)}{d{\mathbf{x}}_{m-1}^i}\frac{d{F}_{m-1}({\mathbf{x}}_{m-2}^i)}{d{\mathbf{x}}_{m-2}^i}\dots \frac{d{F}_1({\mathbf{x}}_0^i)}{d{\mathbf{x}}_0^i}.\end{array}$$ 7

Then the Lyapunov exponent of the cascaded system is calculated as

$$\begin{array}{c}\hfill L{E}_{\Gamma }=\frac{1}{N}\underset{N\to \infty }{lim}\ln \frac{∥\delta ,{\mathbf{y}}_N∥}{∥\delta ,{\mathbf{y}}_0∥}=\frac{1}{N}\underset{N\to \infty }{lim}\ln \left(\prod _{i=1}^N,\prod _{j=0}^{m-1},{\mathbf{J}}_{F_{j+1}}^i,({\mathbf{x}}_j^i)\right).\end{array}$$ 8

Note that the Lyapunov exponent of $F_{j+1}$ is $L{E}_{F_{j+1}}$, the relationship between $L{E}_{\Gamma }$ and $L{E}_{F_{j+1}}$ is discussed hereinafter.

Case 1 When the cascaded subsystems have the same parameters, which means $F_1=F_2=\cdots =F_m$, then

$$\begin{array}{cc}\hfill L{E}_{F_{j+1}}& =\underset{N\to \infty }{lim}\frac{1}{\mathit{mN}}\ln \frac{∥\delta ,{\mathbf{x}}_m^N∥}{∥\delta ,{\mathbf{x}}_0^1∥}\hfill \\ \hfill & =\frac{1}{\mathit{mN}}\underset{N\to \infty }{lim}\ln \left(\prod _{i=1}^N,\prod _{j=0}^{m-1},{\mathbf{J}}_{F_{j+1}}^i,({\mathbf{x}}_j^i)\right)\hfill \\ \hfill & =\frac{1}{m}L{E}_{\Gamma }.\hfill \end{array}$$ 9

So it is obtained that when m identical chaotic maps are cascaded, the Lyapunov exponent of the generated chaotic map satisfies $L{E}_{\Gamma }=mL{E}_{F_{j+1}}$. In this way, we can enhance the initial sensitivity of the chaotic map without changing its state.

Case 2 When the cascaded subsystems have different parameters, Eq. (8) can be expressed as

$$\begin{array}{c}\hfill L{E}_{\Gamma }=\sum _{j=0}^{m-1}\frac{1}{N}\underset{N\to \infty }{lim}\ln \left(\prod _{i=1}^N,{\mathbf{J}}_{F_{j+1}}^i,({\mathbf{x}}_j^i)\right).\end{array}$$ 10

The Lyapunov exponent of subsystem $F_{j+1}$ is defined as

$$\begin{array}{c}\hfill L{E}_{F_{j+1}}=\frac{1}{N}\underset{N\to \infty }{lim}\ln \left(\prod _{k=1}^N,{\mathbf{J}}_{F_{j+1}},({\mathbf{x}}_k)\right),\end{array}$$ 11

the difference between the product terms of (10) and (11) is that the state variables brought into ${\mathbf{J}}_{F_{j+1}}$ in (11) is a continuous iterative sequence while the sequence ${\mathbf{x}}_j^i$ carried in (10) is the column vector of B, which is not continuous iteration of the $F_{j+1}$ map.

It can be concluded that, in general, the Lyapunov exponent of the new system derived by cascading different chaotic maps is not equal to the sum of Lyapunov exponent of each subsystem (except that the characteristic roots of the characteristic equations of all subsystems are independent to state variables). If we consider the difference between ${\mathbf{x}}_j^i$ in (10) and ${\mathbf{x}}_k$ in (11) as a perturbation, then each product term of (10) can be seen as the Lyapunov exponent with input perturbation, denoted as $L{E}_{F_{j+1}}^{\ast }$:

$$\begin{array}{c}\hfill L{E}_{\Gamma }=\sum _{j=0}^{m-1}\frac{1}{N}\underset{N\to \infty }{lim}\ln \left(\prod _{i=1}^N,{\mathbf{J}}_{F_{j+1}}^i,({\mathbf{x}}_j^i)\right)=\sum _{j=0}^{m-1}L{E}_{F_{j+1}}^{\ast }.\end{array}$$ 12

Therefore, when the isomorphic systems with different parameters are cascaded, the Lyapunov exponent of the generated new system is equal to the sum of the perturbed Lyapunov exponent of each subsystem. For a specific chaotic map, the influence of this disturbance is different.

In order to intuitively analyze the dynamical behaviors of chaotic maps constructed based on the proposed method, various cascaded systems based on the generalized 3-D Hénon map are analyzed in the following two sections.

Cascading of 3-D Hénon maps with different parameters

A n-D invertible map which generalizes the 2-D Hénon map is presented in Baier and Klein (1990). For $n=3$, the system equation is described as

$$\left\{\begin{array}{c}x(i+1)=a-y{(i)}^2-bz(i)\hfill \\ y(i+1)=x(i)\hfill \\ z(i+1)=y(i),\hfill \end{array}\right)$$ 13

where a and b are system parameters. Fixed $b=0.1$, the Lyapunov exponent spectrum and bifurcation diagram varying with a are shown in Fig. 2a, b respectively. It can be seen that (13) exhibits rich dynamical behaviors for different parameter a. When $a\in (0.81,1.09)\cup (1.37,1.70]$, the generalized Hénon map is a chaotic attractor for there exist at least one Lyapunov exponent greater than zero. When $a\in (1.51,1.70]$, since there are two Lyapunov exponents greater than zero, it appears as a hyperchaotic attractor. Set $a=1.68$, the phase portrait with initial condition $(x_0,y_0,z_0)=(1,0.1,0)$ is shown in Fig. 3.

Fig. 2.

Dynamical behaviors of (13). a The Lyapunov exponent spectrum with a varying from 0.5 to 1.7, b the corresponding bifurcation diagram

Fig. 3.

Phase portrait of (13) with $a=1.68$ and $b=0.1$. a Projection on y–z plane, b 3D view

Note $a_p$ and $b_p$ ($p=1$, 2, $\dots$, m) are parameters of the pth subsystem when m generalized 3-D Hénon maps are cascaded in the order of $\mathbf{u}(k+1)=F_m\circ F_{m-1}\circ F_{m-2}\circ \cdots \circ F_1(\mathbf{u}(k))$ with $\mathbf{u}(k)={(x(k),y(k),z(k))}^T$. It is easy to understand that the parameters of the subsystem will affect the dynamical behaviors of the cascaded system. Here, the cascade of two generalized Hénon maps $\mathbf{u}(k+1)=F_2\circ F_1(\mathbf{u}(k))$ is analyzed as an example. The equation of the constructed system can be expressed by

$$\begin{array}{c}\hfill \left\{\begin{array}{c}x(i+1)=a_2-x{(i)}^2-b_{2}y(i)\hfill \\ y(i+1)=a_1-y{(i)}^2-b_{1}z(i)\hfill \\ z(i+1)=x(i),\hfill \end{array}\right)\end{array}$$ 14

where ($a_1$, $b_1$) and ($a_2$, $b_2$) correspond to the parameters of $F_1$ and $F_2$ maps. For the sake of simplicity, $b_1$ and $b_2$ are fixed to 0.1. When $a_1$ takes different values, the dynamical behaviors with $a_2\in [0.5,1.7]$ are analysed by the Lyapunov exponent spectrum and bifurcation diagram.

As can be seen from Fig. 2a, when $a=1.68$, the Lyapunov exponents of the generalized Hénon map are $(0.1884,0.1496,-2.6407)$, which correspond to the hyperchaotic attractor. When $a=1.05$, system (13) behaves as an iteration of period-2 with the Lyapunov exponents $(-2.3658,$ $-0.4393,$ $-1.8630)$. These two parameters are representative and therefore selected as paradigms in the cascaded system. The dynamical behaviors of (14) with $a_2$ varying from 0.5 to 1.7 when $a_1=a_2$, $a_1=1.68$, and $a_1=1.05$ are shown in Fig. 4a–c respectively. The dotted lines in Fig. 4a–c depict the largest Lyapunov exponent of (13) with $a=a_2$.

Fig. 4.

Chaotic dynamical behaviors of (14) when $a_2\in [0.5,1.7]$. a–c The Lyapunov exponent spectrums with $a_1=a_2$, $a_1=1.68$, and $a_1=1.05$ respectively, d–f the corresponding bifurcation diagrams

It is necessary to make an intuitive comparison of chaotic dynamical behavior for the above three cases. As can be seen from Fig. 4a, the largest Lyapunov exponent shown in red solid line is twice as that of the dotted line, which verifies the conclusion that the cascade of two identical subsystems expands the Lyapunov exponent and maintains the dynamic states. According to Fig. 4b, e, the cascaded system is a chaotic attractor in the whole parameter domain. When $a_2\in (1.39,1.70]$, the cascaded system is a hyperchaotic attractor. Therefore, the domains of chaotic and hyperchaotic are effectively expanded by cascading subsystem with $a_1=1.68$. In addition, the value of the largest Lyapunov exponent also increases compared with the generalized Hénon map. By contrast, the hyperchaotic state in Fig. 4c disappeared, and the chaotic domain reduced to $a_2\in (0.97,1.07)\cup (1.57,1.70]$. So, the cascade of the subsystem with $a_1=1.05$ enlarges the periodic domain while reduces the chaotic domain. Table 1 summarizes the chaotic and hyperchaotic domains of generalized 3-D Hénon map and three cascaded chaotic maps.

Table 1.

Comparison of chaotic and hyperchaotic domains of four chaotic maps

Chaotic maps	Chaotic domain	Hyperchaotic domain
3-D Hénon map	$(0.81,1.09)\cup (1.37,1.70]$	(1.51, 1.70]
$F_2\circ F_1(a_1=a_2)$	$(0.81,1.09)\cup (1.37,1.70]$	(1.51, 1.70]
$F_2\circ F_1(a_1=1.68)$	[0.50, 1.70]	(1.39, 1.70]
$F_2\circ F_1(a_1=1.05)$	$(0.97,1.07)\cup (1.57,1.70]$	$\mathrm{\varnothing }$

Three specific cascaded chaotic maps are given with ${\Gamma }_1$($a_1$ $=1.68$, $a_2=1.68$), ${\Gamma }_2$($a_1=1.42$, $a_2=1.68$) and ${\Gamma }_3$($a_1=1.05$, $a_2=1.68$). Their phase portraits are shown in Fig. 5a–c respectively. In order to verify the initial value sensitivity of these systems, sequences $X_1$ and $X_2$ are generated by setting the initial values with (1, 0.1, 0) and (1.000001, 0.1, 0), and the first 50 iteration values are plotted in Fig. 5d–f respectively. Their differences with $X_2-X_1$ are shown in Fig. 5g–i respectively. It can be seen that after about 30 iterations, the output sequence with tiny change in initial values are significant different.

Fig. 5.

Phase portraits and initial value sensitivity analysis of cascaded systems ${\Gamma }_1$, ${\Gamma }_2$, and ${\Gamma }_3$, a–c phase portraits with initial values (1, 0.1, 0) of ${\Gamma }_1$, ${\Gamma }_2$, and ${\Gamma }_3$ respectively. d–f The first 50 iteration values of first variable x(k) with initial values (1, 0.1, 0) and (1.000001, 0.1, 0). g–i Difference of two trajectories $X_1$ and $X_2$

In summary, the parameters of the subsystems have a great influence on the dynamical behaviors of the cascaded chaotic map. The chaotic domain of the obtained system could expand, shrink or remain unchanged by cascading operation. At the same time, the state of the system may be transformed from a chaotic state to a hyperchaotic state or periodic state.

Cascading of 3-D Hénon maps with different permutations

It is necessary to explore the influence of different permutations on the dynamical behaviors of the generated chaotic map with m subsystems cascaded. Considering that the parameters of $m-1$ subsystems are fixed and the dynamical behaviors of the resulting system with another parameter will be analyzed.

When three generalised Hénon maps are cascaded, set $b_1=b_2=b_3=0.1$, $m_2=1.4$, $m_3=1.5$, and let $m_1$ be a variable, there exist 6 different permutations, that is $(a_1,a_2,a_3)$={$(m_1,m_2,m_3)$, $(m_2,m_3,m_1)$, $(m_3,m_1,m_2)$, $(m_1,m_3,m_2)$, $(m_3,m_2,m_1)$, $(m_2,m_1,m_3)\}$. The cascaded systems ${\Gamma }_i$ ($i=1,2,\dots ,6$) are induced in Table 2. The bifurcation diagrams and Lyapunov exponent spectrums are plotted in Fig. 6, where Fig. 6a–c, g–i represent the bifurcation diagrams of ${\Gamma }_1$, ${\Gamma }_2$, ${\Gamma }_3$, ${\Gamma }_4$, ${\Gamma }_5$, and ${\Gamma }_6$. Figure 6d–f, j–l are corresponding to the Lyapunov exponent spectrums respectively.

Table 2.

Cascading of three 3-D Hénon maps with different permutations

Cascaded system	${\Gamma }_1$	${\Gamma }_2$	${\Gamma }_3$	${\Gamma }_4$	${\Gamma }_5$	${\Gamma }_6$
$F_1(a_1)$	$m_1$	$m_2$	$m_3$	$m_1$	$m_3$	$m_2$
$F_2(a_2)$	$m_2$	$m_3$	$m_1$	$m_3$	$m_2$	$m_1$
$F_3(a_3)$	$m_3$	$m_1$	$m_2$	$m_2$	$m_1$	$m_3$

Fig. 6.

The dynamical behaviors of the cascaded three 3-D Hénon maps with different permutations. a–c The bifurcation diagrams of ${\Gamma }_1$, ${\Gamma }_2$, and ${\Gamma }_3$ varying with $m_1$, d–f the corresponding Lyapunov exponent spectrums of ${\Gamma }_1$, ${\Gamma }_2$, and ${\Gamma }_3$, g–i the bifurcation diagrams of ${\Gamma }_4$, ${\Gamma }_5$, and ${\Gamma }_6$ varying with $m_1$, j–l the corresponding Lyapunov exponent spectrums of ${\Gamma }_4$, ${\Gamma }_5$, and ${\Gamma }_6$

It is interesting to find that the Lyapunov exponent spectrums of ${\Gamma }_1$, ${\Gamma }_2$, and ${\Gamma }_3$ are the same, so as to the Lyapunov exponent spectrums of ${\Gamma }_4$, ${\Gamma }_5$, and ${\Gamma }_6$. In order to explain this phenomenon, the definition of the shift-equivalent sequence is introduced.

Lemma 1

(Wu et al. 2014) Two controlling sequences $Q^{a\ast }$ and $Q^{b\ast }$ are shift-equivalent if and only if there exist integers k and l, such that

$$\begin{array}{c}\hfill q_i^{a\ast }=q_{mod(i+k,l)+1}^{b\ast }\end{array}$$ 15

for all controlling elements in $Q^{a\ast }$ or $Q^{b\ast }$.

The controlling sequences of ${\Gamma }_1$, ${\Gamma }_2$, and ${\Gamma }_3$ are $M^{1\ast }=[m_1,m_2,m_3]$, $M^{2\ast }=[m_2,m_3,m_1]$ and $M^{3\ast }=[m_3,m_1,m_2]$. $M^{1\ast }$ and $M^{2\ast }$ is shift-equivalent because there exist $k=1$ and $l=3$ to make (15) hold. Also, when $k=2$ and $l=3$, Eq. (15) holds for $M^{1\ast }$ and $M^{3\ast }$. Thus the controlling sequences $M^{1\ast }$, $M^{2\ast }$, and $M^{3\ast }$ are shift-equivalent to each other. The same results can be seen in ${\Gamma }_4$, ${\Gamma }_5$, and ${\Gamma }_6$. According to (8), the controlling sequences will be repeated during N iterations. Since N tends to be infinite, the effect of the first one or two subsystems’ iterations on the solution of the Lyapunov exponents of cascaded chaotic maps is negligible. So the dynamical behaviors of these cascaded systems are quite similar to each other.

Seen Fig. 6 vertically, the controlling sequences of the first three cascaded systems and the last three cascaded systems are shift-inequivalent. According to (12), the difference of the Lyapunov exponents can be explained by different disturbance mechanisms caused by shift-inequivalent controlling sequences.

The bifurcation diagrams coincide well with the corresponding Lyapunov exponent spectrums. In addition, it can be seen that although shift-equivalent cascaded chaotic systems have the same chaotic and non-chaotic domains, their phase trajectories are different. Figure 7 shows the phase portraits of the six cascaded systems when two different vales of $m_1$ are set. The blue chaotic attractors are obtained with $m_1=1.2$, and the red periodic points corresponding to $m_1=0.8$. Therefore, for a set of subsystems with given parameters, diverse attractors can be obtained by cascading them with different permutations.

Fig. 7.

The phase portraits of ${\Gamma }_1$, ${\Gamma }_2$, ${\Gamma }_3$, ${\Gamma }_4$, ${\Gamma }_5$, and ${\Gamma }_6$ with two specific values of $m_1$. a–f The blue chaotic attractors and the red periodic dots are corresponding to the simulation results with $m_1=1.2$ and $m_1=0.8$ respectively. (Color figure online)

As for subsystems with the same parameters, the control sequence has no effect on the cascaded system equation. The largest Lyapunov exponents when cascaded m ($m=1$, 2, 3, 4) generalized Hénon maps are shown in Fig. 8a. The simulation result verifies the conclusion that when m identical chaotic maps are cascaded, the Lyapunov exponent of the generated chaotic map satisfies $L{E}_{\Gamma }=mL{E}_{F_{j+1}}$. Novel coexisting attractors can be discovered when cascaded four generalized Hénon maps with the same parameters. When the parameter a varies from 1.1 to 1.4, the bifurcation diagrams of state variable $x_i$ plotted in Fig. 8b. The trajectories colored in red start from the initial conditions (0, 0, 0), and those colored in green, violet , and blue correspond to initial conditions (0, 1, 0), (1,0,0), and (1,1,0) respectively.

Fig. 8.

Chaotic dynamical behaviors of cascading multiple identical maps. a Maximum Lyapunov exponents with $m=1,2,3,4$ varying with parameter a, b bifurcation diagrams corresponding to $m=4$ when set different initial conditions. (Color figure online)

The basin of attraction of the different attracting sets provides more information about the coexisting attractors, which are defined as the set of initial conditions whose trajectories converge to the respective attractor. For $a=1.4$, $b=0.1$, and $z_0=0$, the basin of attraction varying with initial values $x_0$ and $y_0$ is given in Fig. 9a. For four different initial conditions located in the above four regions, the phase portraits are demonstrated in Fig. 9b, respectively. Therefore, it can be concluded that although the introduction of the cascade operator does not change the dynamic state of the system, it affects the attractor morphology and produces new dynamical behavior.

Fig. 9.

The basin of attraction and phase portraits of the cascaded chaotic map when $m=4$. a The basin of attraction varying with initial values $x_0$ and $y_0$, b phase portraits for four different initial conditions

This section discusses the influence of different permutations on the cascaded system. The constructed systems with shift-equivalent controlling sequences have the same Lyapunov exponent spectrums but different bifurcation diagrams. While the Lyapunov exponent spectrums of constructed systems with shift-inequivalent controlling sequences are usually different from each other, so as to the bifurcation diagrams. Novel coexisting attractors are discovered when cascaded four generalized Hénon maps with the same parameters.

Application

In order to verify that the generated chaotic sequence has good performance of secure communication, the representative image encryption scheme is reviewed to test the performance of image encryption. Two cascaded systems ${\Gamma }_1^{\prime }$ and ${\Gamma }_2^{\prime }$, which are constructed by cascading three generated Hénon maps with same parameters $a_1=a_2=a_3=1.42$ and two generated Hénon maps with different parameters $a_1=1.42$ and $a_2=1.68$, are employed as examples to verify their excellent performance on image encryption. The initial value sensitivity, Randomness test of pseudo-random number sequence, and other performance analyses are conducted in this section.

Description of encryption process

A gray-scale image I is encrypted with the cascaded chaotic sequence. The process of the encryption and decryption can be described as follows.

1. The pixels of the plain-image are preprocessed by

   $$\begin{array}{c}\hfill g(i,j)=\mathrm{sort}(I),\end{array}$$ 16

   $$\begin{array}{c}\hfill s(i,j)=\mathrm{mod}(g(i,j)+g(j,i),256),\end{array}$$ 17

   where g(i, j) is the gray value of the pixel, $i\in [1,2,\dots ,M]$ and $j\in [1,2,\dots ,N]$ (M and N represent the row and column of the plain-image).
2. Preprocessing of Chaotic Sequences. Remove the first 5000 results of iteration, retain the next $M\ast N$ results, and perform the operation

   $$\begin{array}{c}\hfill n(i,j)=\mathrm{mod}(1000\ast (x(i)-\mathrm{floor}(\mathrm{abs}(x(i)))),256),\end{array}$$ 18

   where $i=[1,2,\dots ,M\ast N]$.
3. The plain-image is encrypted with XOR operation bit-by-bit

   $$\begin{array}{c}\hfill m(i,j)=s(i,j)\oplus n(i,j).\end{array}$$ 19

4. Decryption is the reverse version of the encryption process.

Simulation results and performance analyses

1. Histogram analysis The histogram clearly shows the number of pixels of each intensity values and the distribution of the overall pixel intensity values of the whole image. A secure encryption system should be able to resist attacks based on statistical analysis. So it requires the histogram of the encrypted image as uniform as possible. Take the chaotic sequence of ${\Gamma }_1^{\prime }$ as an example, Fig. 10a–c display the original image, encrypted image, and decrypted image, respectively. Meanwhile, their histograms are correspondingly shown in Fig. 10d–f, respectively. It can be seen that the histogram of the encrypted image is much more uniform than the original image, which means that the encryption scheme based on the cascaded chaotic system ${\Gamma }_1^{\prime }$ has good performance.

2. Information entropy The Shannon entropy of image I is defined by

   $$\begin{array}{c}\hfill H(I)=-\sum _{i=1}^{L}Pr(I_i){\log }_{2}Pr(I_i),\end{array}$$ 20

   where $I_i$ denotes the ith possible value of I. $Pr(I_i)$ represents the probability value of $I_i$, and L is the number of possible different intensities. If an encryption scheme can generate encrypted images owning the maximum information entropy close to eight, it means that it has excellent randomness property. The calculated entropy values of the encrypted images obtained from ${\Gamma }_1^{\prime }$ and ${\Gamma }_2^{\prime }$ are 7.9856 and 7.9903, respectively. It is found that there is a slight improvement in performance of cryptosystem comparing with the generalized Hénon map.
3. Correlation analysis The correlation of each pair of pixels can be calculated by

   $$\begin{array}{c}\hfill E(x)=\frac{1}{N}\sum _{i=1}^N{x}_i,E(y)=\frac{1}{N}\sum _{i=1}^N{y}_i;\end{array}$$ 21

   $$\begin{array}{c}\hfill cov(x,y)=\frac{{\sum }_{i=1}^N(x_i-E(x))(y_i-E(y))}{\sqrt{\sum _{i=1}^N{(x_i-E(x))}^2}\sqrt{\sum _{i=1}^N{(y_i-E(y))}^2}},\end{array}$$ 22

   where x and y are the intensity values of two adjacent pixels, and N is the total number of pixels. The correlation of adjacent pixels in diagonal direction about different encrypted images are given in Table 3, which indicates that the cascaded systems reveal better encryption performance.
4. Fixed point analysis If the gray value of pixel (i, j) in image I does not change after encryption, the pixel is a fixed point. The percentage of fixed points in image I to all pixels is called the fixed point ratio of the picture, which is defined as

   $$\begin{array}{c}\hfill BD(I,C)=\frac{{\sum }_{i=1}^M{\sum }_{i=1}^{N}f(i,j)}{\mathit{MN}}\times 100\%,\end{array}$$ 23

   where

   $$f(i,j)=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}{I}_{i,j}=C_{i,j}\hfill \\ 0.\hfill & \mathrm{others}\hfill \end{array}\right)$$ 24

   The fewer the number of fixed points, the better the encryption performance. The results of fixed point ratio based on three chaotic sequences are calculated in Table 3. Hence one can see that chaotic sequences generated by ${\Gamma }_1^{\prime }$ and ${\Gamma }_2^{\prime }$ can improve the secrecy performance of encrypted images.
5. Grayscale average variety estimate Both I and C are images of size $M\ast N$ and gray level L, the average value of grayscale variation of two images is

   $$\begin{array}{c}\hfill GAVE(I,C)=\frac{{\sum }_{i=1}^M{\sum }_{i=1}^N|I_{i,j}-C_{i,j}|}{\mathit{MN}}\times 100\%.\end{array}$$ 25

   The best case should be that the image gradation average value is equal to L/2. The results of GAVE between the original image and three encrypted images are given in Table 3, it can be seen that the calculated results based on ${\Gamma }_1^{\prime }$ and ${\Gamma }_2^{\prime }$ are closer to 128 than the generalized Hénon map.

Fig. 10.

Simulation results of the proposed encryption and decryption scheme: a original image, b encrypted image, c decrypted image, d histogram of the original image, e histogram of the encrypted image, f histogram of the decrypted image

Table 3.

Performances analyses of image encryption

Performances analyses	3D Hénon map	${\Gamma }_1^{\prime }$	${\Gamma }_2^{\prime }$
Information entropy	7.9853	7.9856	7.9903
Correlation analysis(diagonal)	0.0245	0.0022	0.0016
Fixed point analysis	0.6109	0.6096	0.5956
GAVE	95.5351	95.6550	103.1058

Randomness test of pseudo-random number sequence

To verify the performance of the pseudo-random number generators constructed by chaotic maps ${\Gamma }_1^{\prime }$ and ${\Gamma }_2^{\prime }$, the NIST SP 800-22 statistical test suite is used with the default significance level. Results of 16 tests items for 2 Mbit number series are given in Table 4, in which all P-Values are larger than 0.01. Moreover, there are three test results larger than 0.7 for the generalized Hénon map, while six terms larger than 0.7 for ${\Gamma }_1^{\prime }$ and ${\Gamma }_2^{\prime }$ chaotic attractors. Thus, the designed pseudo-random generators have excellent randomness and are suited for encryption application.

Table 4.

The percentage of sequences passing each item of NIST test suite

Statistical tests	P value
	3D Hénon map	${\Gamma }_1^{\prime }$	${\Gamma }_2^{\prime }$
Frequency	0.554420	0.350485	0.129620
BlockFrequency	0.202268	0.739918	0.678686
CumulativeSums-Forward	0.699313	0.911413	0.719747
CumulativeSums-Reverse	0.595549	0.867692	0.616305
Runs	0.534146	0.334538	0.779188
LongestRuns	0.759756	0.171867	0.137282
Rank	0.719747	0.924076	0.759756
FFT	0.401199	0.013569	0.162606
NonOverlappingTemplate	0.153763	0.366918	0.383827
OverlappingTemplate	0.739918	0.401199	0.350485
Universal	0.401199	0.983453	0.019188
ApproximateEntropy	0.171867	0.739918	0.115387
RandomExcursions	0.327854	0.010341	0.875539
RandomExcursionsVariant	0.229900	0.386280	0.988549
Serial	0.032923	0.366918	0.816537
LinearComplexity	0.595549	0.213309	0.55442
Success Counts	16/16	16/16	16/16

The value marked in bold indicate that the result of the corresponding test item is greater than 0.7

All the above analysis demonstrate that the performances of the image encryption scheme based on cascaded chaotic maps ${\Gamma }_1^{\prime }$ and ${\Gamma }_2^{\prime }$ are better than the original generalized Hénon map. So the security of cascaded maps is improved, and they can be used for protecting image data in cyberspace.

Conclusions

This paper presented a general framework for cascading high-dimensional isomorphic maps to enhance the chaos complexity. It is analyzed that the Lyapunov exponent of the cascaded system is equal to the sum of the perturbed Lyapunov exponent of each subsystem. The effectiveness of the proposed method is verified by cascading 3-D generalized Hénon map. It can be seen that the chaotic domain expanded by cascading a robust chaotic map and narrowed down when cascading a periodic oscillation. Also, the dynamical behaviors of cascading subsystems with the same parameters in different sequences are discussed. The result shows that the constructed systems with shift-equivalent controlling sequences have the same Lyapunov exponent spectrums but different bifurcation diagrams. While the constructed systems with shift-inequivalent controlling sequences are different from each other. Moreover, some diverse attractors, e.g. hyperchaotic attractors and coexisting chaotic attractors, could be obtained. Finally, the application of image encryption proves that the proposed cascaded high-dimensional chaotic map has excellent secure communication performance.

In the implementation of cascade chaotic system, it is extremely important that the output of a cascaded subsystem should match the chaotic domain of the next one. With this consideration, this paper discusses the cascade of simple isomorphic system. It is known that sine chaotification can be employed to normalize the output and chaotic domain of the system, which can naturally improve the chaos complexity, and thus, greatly enhance the robustness of chaotic system. Therefore, we will integrate sine function into the outputs of all variables to define their outputs and chaotic ranges, aiming to realize the cascaded and high-dimensional heterogeneous chaotic systems in future work.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.