Dynamical analysis,circuit implementation and deep belief network control of new six-dimensional hyperchaotic system

Abstract

In this paper, a new six-dimensional hyperchaotic system is proposed and some basic dynamical properties including bifurcation diagrams, Lyapunov exponents and phase portraits are investigated. Furthermore, the electronic circuit of this novel hyperchaotic system is simulated on the Multisim platform, and the simulation results are agreed well with the numerical simulation of the same hyperchaotic system on the Matlab platform. Finally, a control method based on Deep Belief Network is proposed to track and control the proposed hyperchaotic system. In this method, the function of the hyperchaotic system is studied by Deep Belief Network and a high precision fitting function is obtained. Then a controller which is composed of the fitting function and the tracking reference signal is designed to achieve the tracking control of hyperchaotic systems. Simulation results verify the effectiveness and feasibility of this method.

Introduction

Hyperchaos and hyperchaotic system were firstly proposed by Rössler. ¹ Compared with chaotic system, hyperchaotic system has at least two positive Lyapunov exponents, and it displays more complicated dynamical behaviors. The complex hyperchaotic signals can improve the securities of chaotic secure communication and chaotic information encryption. Therefore, researchers have conducted extensive and in-depth research in this field. In 2004, an improved Rössler hyperchaotic system was proposed by Nikolov and Clodong. ² In 2005, Li et al. designed a simple nonlinear state feedback controller to generate hyperchaos from a three-dimensional autonomous chaotic system. ³ In 2006, a new hyperchaotic system based on Lu system was proposed by Chen et al. ⁴ In the same year, Gao et al. constructed a new hyperchaotic system based on Chen's chaotic system by adding a controller. ⁵ In 2007, a four-dimensional hyperchaotic Lorenz system was proposed by Wang and Wang. ⁶ In 2009, a four-dimensional hyperchaotic system based on three-dimensional Bao's chaotic system was proposed by Bao et al. ⁷ In 2010, four-scroll hyperchaos phenomenon was found from a novel four-dimensional complex nonlinear autonomous system by Dadras and Momeni. ⁸ In 2012, Wei and Zhou constructed a new five-dimensional hyperchaotic system by adding a linear feedback controller to the Lorenz hyperchaotic system. ⁹ In 2014, a new four-dimensional hyperchaotic system and circuit implementation of the system were proposed by El-Sayed et al. ¹⁰ In 2015, Sun et al. constructed a multi-scroll hyperchaotic system by designing a continuous nonlinear function from a six-order hyperchaotic system. ¹¹ In 2016, a memristor-based Lorenz hyperchaotic system was proposed by Ruan et al. ¹² In 2017, by introducing a feedback control to the Liu–Su–Liu chaotic system, Vaidyanathan et al. ¹³ obtained a novel hyperchaotic system.

Hence, the chaotic system is characterized by well pseudorandom, long inscrutability, high sensibility to the initial value. Thus, it is necessary to control and eliminate the chaos. In recent years, the control of chaotic systems has made significant progress. In Wang et al., ¹⁴ a hybrid control method based on Particle Swarm Optimization and OGY method was proposed, in which Particle Swarm Optimization is used to guide the chaotic orbit firstly, and then the chaotic system was stabilized in the fixed point by OGY method. In T Ma et al., ¹⁵ an improved impulsive control method was proposed. The linear feedback of the error between the response system and the drive system is used as the pulse control signal to realize the global asymptotic synchronization control of two Chen systems. In Cao, ¹⁶ a new numerical method is presented to solve optimal control problem of a chaotic system based on Gauss pseudospectral method (GPM). This method can avoid solving HJB equation, and it can solve the optimal control problems of chaotic systems effectively and fastly. In Li et al., ¹⁷ a three-dimensional autonomous chaotic system was proposed, which used a single state controller to realize the synchronization of fractional-order chaotic system. In Ren and Liu, ¹⁸ a nonlinear feedback control method was proposed, which could reduce state values of the controlled system to any values according to the actual demand. In Yongjian et al., ¹⁹ a new chaotic system with two nonlinear terms was controlled by the negative feedback controller. In Effati et al., ²⁰ an optimal control technique and a feedback control approach were proposed, which can control the three-dimensional autonomous system and the four-dimensional hyperchaotic system. In Yau and Yan, ²¹ a sliding mode control was designed to stabilize the hyperchaos of Rössler system, which is robust against both the input nonlinearity and external disturbance. In Chen, ²² a simple adaptive feedback control method was proposed to control the chaos and hyperchaos. In Fu and Zhang, ²³ a controller was designed to stabilize the chaotic state of hyperchaotic Bao system to equilibrium by using the linear feedback control method and adaptive backstepping control method. Aiming at the complex chaotic behavior of permanent magnet synchronous motor (PMSM), ²⁴ they proposed a sliding mode control strategy based on second-order sliding mode control. In Singh and Roy, ²⁵ an adaptive proportional integral SMC was proposed for controlling of the many equilibria hyperchaotic system.

In this paper, a novel six-dimensional hyperchaotic system is proposed, which adds an extra nonlinear control coefficient in the five-dimensional hyperchaotic system in the literature. ⁹ And a control method based on Deep Belief Network (DBN) is proposed to track and control the proposed hyperchaotic system. This method not only removes the limitation that mathematical model must be known clearly, but also can track any signal. Simulation results show the effectiveness and feasibility of this method.

The rest of the paper is organized as follows: A new six-dimensional hyperchaotic system is proposed in Section “The design of proposed new hyperchaotic system”. In Section “Dynamics of the proposed hyperchaotic system”, some related features are researched and analyzed, including symmetry, stability of equilibriums, dissipativity, bifurcations and Lyapunov exponent of hyperchaotic system. Circuit implementation of the hyperchaotic system is presented in Section “The circuit implementation of proposed six hyperchaotic system”. A chaos control method based on DBN is proposed in Section “The proposed control method based on Deep Belief Network”. Section “The simulations and discussions” carried on the simulation experiments to illustrate the effectiveness and feasibility of the proposed method. Finally, conclusion is given in Section “Conclusion”.

The design of proposed new hyperchaotic system

In Wei and Zhou, ⁹ a five-dimensional hyperchaotic system was proposed. Based on the system, nonlinear control parameter is added into the system, thus, the new six-dimensional hyperchaotic system is defined by

$$\{\begin{array}{l}\dot{x}=a(y-x)+w\hfill \\ \dot{y}=cx-y-xz-v\hfill \\ \dot{z}=-bz+xy\hfill \\ \dot{w}=dw-yz\hfill \\ \dot{v}=ry\hfill \\ \dot{u}=-eu+zw\hfill \end{array}$$

(1)

where a, b, c, d, r and e are system parameters, while x, y, z, w, u and v are six independent state variables. When a = 10, b = 8/3, c = 28, d = −1, e = 10 and r = 3, the Lyapunov exponents are L₁ = 0.362485, L₂ = 0.24709, L₃ = 0, L₄ = −0.225698, L₅ = −10.0017 and L₆ = −15.0708. Obviously, two parts of Lyapunov exponents are positive, while others are negative. And the sum of all Lyapunov exponents is less than zero. So when a = 10, b = 8/3, c = 28, d = −1, e = 10 and r = 3, system is in hyperchaotic condition.

Dynamics of the proposed hyperchaotic system

In the proposed six-dimensional hyperchaotic system, hyperchaos clearly appears when parameters are expressed as a = 10, b = 8/3, c = 28, d = −1, e = 10 and r = 3. Compared with chaotic systems, hyperchaotic system is more separated on the phase orbit, and its dynamic behavior is more complicated. The complex hyperchaotic signals can improve the security of chaotic secure communication and chaotic information encryption, therefore, it is particularly important to analyze the dynamic behavior of hyperchaotic system. In this section, the features of proposed six-dimensional hyperchaotic system are well studied, including symmetry, stability of equilibriums, dissipativity, bifurcations and Lyapunov exponent.

Symmetry

It is obvious that the new chaotic system (1) is invariant under the coordinate transforms

$$(x,y,z,w,v,u)\to (-x,-y,z,-w,-v,-u)$$

which shows that the system (1) is symmetrical about the coordinate axis Z. And this symmetry is well satisfied with all system parameters.

Stability and equilibria

The equilibria of the system (1) can be found by solving the following algebraic equations simultaneously

$$\{\begin{array}{l}a(y-x)+w=0\hfill \\ cx-y-xz-v=0\hfill \\ -bz+xy=0\hfill \\ dw-yz=0\hfill \\ ry=0\hfill \\ -eu+zw=0\hfill \end{array}$$

(2)

Obviously, for the only equilibrium $O(0,0,0,0,0,0)$ , the system (1) is linearized, the Jacobian matrix is defined as

$$J=\left[\begin{array}{llllll}-a\hfill & a\hfill & 0\hfill & 1\hfill & 0\hfill & 0\hfill \\ c\hfill & -1\hfill & 0\hfill & 0\hfill & 0\hfill & -1\hfill \\ 0\hfill & 0\hfill & -b\hfill & 0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill & d\hfill & 0\hfill & 0\hfill \\ 0\hfill & r\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill & 0\hfill & -e\hfill & 0\hfill \end{array}\right]$$

(3)

By letting $\left|J-\lambda I\right|=0$ , the characteristic equation is

$$\begin{array}{c}(\lambda -d)(\lambda +b)[{\lambda }^4+(a+1){\lambda }^3\\ +(a-ac){\lambda }^2+er\lambda -aer]=0\end{array}$$

(4)

When a = 10, b = 8/3, c = 28, d = −1, e = 10 and r = 3, six eigen values are ${\lambda }_1=\text{11}\text{.8065}$ , ${\lambda }_2=\text{0}\text{.0328+}\text{1.0528i},{\lambda }_3=\text{0}\text{.0328}-\text{1}\text{.0528i},{\lambda }_4=-\text{2}\text{.6667,\hspace{0.17em}}{\lambda }_5=-\text{10},{\lambda }_6=-\text{22}\text{.882}$ . Obviously, three eigenvalues are positive and others are negative. So, we can draw the conclusion that the $O$ is a saddle-point which is unstable.

Dissipativity

When a = 10, b = 8/3, c = 28, d = −1, e = 10 and r = 3, the divergence of flow of the system (1) is given by

$$\begin{array}{c}\nabla V=\frac{\partial \dot{x}}{x}+\frac{\partial \dot{y}}{y}+\frac{\partial \dot{z}}{z}+\frac{\partial \dot{w}}{w}+\frac{\partial \dot{u}}{u}+\frac{\partial \dot{v}}{v}\\ =-a-1-b+d-e=-\frac{74}{3}<0\end{array}$$

Obviously, system (1) is dissipative with an exponential contraction rate: $\frac{dV}{dt}=e^{-\frac{74}{3}t}$ . That is, a volume element $V(0)$ is contracted by the flow into a volume element $V(0)e^{-\frac{74}{3}t}$ in time t. When $t\to \infty$ , the every volume element which contains the system trajectory converges to 0 with exponent rate of $-\frac{74}{3}$ . Therefore, all system orbits are ultimately confined to some subset of zero volume, and the asymptotic motion settles on some attractors.

Nonlinear dynamic behaviors

When analyzing hyperchaos, bifurcation and Lyapunov exponents are two useful tools. The former is a kind of special nonlinear phenomenon related to hyperchaos, chaos and mutation, and the common way to obtain bifurcation is Maximum with parameter. With $a=10,b=8/3,c=28,d=-1,e=10$ and initial condition $(x_0,y_0,z_0,w_0,v_0,u_0)=(1,1,1,1,1,1)$ , bifurcation diagram and Lyapunov exponents spectrum of the system (1) with various linear control coefficient r are shown in Figures 1 and 2.

Figure 1.

Bifurcation diagram with r.

Figure 2.

Lyapunov exponent spectrum with r.

As we can see from Figures 1 and 2: When r values in the interval of (0,26), two parts of Lyapunov exponents are positive, so system (1) has a hyperchaotic attractor; when $26<r<27$ , one part of Lyapunov exponents are positive, so system (1) has a chaotic attractor; when $27<r<33$ , system (1) has a quasi-periodic orbit; when $r>33$ , system (1) has a periodic orbit.

Fixing $a=10,b=8/3,c=28,d=-1,e=10$ , the representative dynamical routes are summarized as follows:

when $r=3$ , the Lyapunov exponents of system (1) are $L_1=0.381654,$ $L_2=0.24709,$ $L_3=0,L_4=-0.225698,L_5=-10.0017,$ and $L_6=-15.0708$ . Obviously, system (1) is now in hyperchaotic status. Related phase portraits of the system are shown in Figure 3.

when $r=26.5$ , the Lyapunov exponents of system (1) are $L_1=0.0809075,$ $L_2=0,$ $L_3=\text{-}000567957,L_4=-0.357747,L_5=-10.0014,L_6=-\mathrm{14.3833.}$ Obviously, the system is now in chaotic status. Related phase portraits of the system are shown in Figure 4.

when $r=31$ , the Lyapunov exponents of system (1) are $L_1=0,$ $L_2=0,$ $L_3=-0.0141266$ , $L_4=-0.681787,L_5=-10.0016,$ $L_6=-\mathrm{13.9721.}$ Obviously, the system is now in quasi-periodic status. Related phase portraits of the system are shown in Figure 5.

when $r=80$ , the Lyapunov exponents of system (1) are $L_1=0,$ $L_2=-0.430864,$ $L_3=-0.436144,$ , $L_4=-1.07812,L_5=-10.0017,$ $L_6=-\mathrm{12.7198.}$ Obviously, the system is now in periodic status. Related phase portraits of the system are shown in Figure 6.

Figure 3.

Related phase portraits of system (1) with $r=3$ : (a) phase portrait of x versus y; (b) phase portrait of x versus z; (c) phase portrait of y versus z; (d) phase portrait of x versus w.

Figure 4.

Related phase portraits of system (1) with $r=26.5$ : (a) phase portrait of x versus y; (b) phase portrait of x versus z; (c) phase portrait of y versus z; (d) phase portrait of x versus w.

Figure 5.

Related phase portraits of system (1) with $r=31$ : (a) phase portrait of x versus y; (b) phase portrait of x versus z; (c) phase portrait of y versus z; (d) phase portrait of x versus w.

Figure 6.

Related phase portraits of system (1) with $r=80$ : (a) phase portrait of x versus y; (b) phase portrait of x versus z; (c) phase portrait of y versus z; (d) phase portrait of x versus w.

The circuit implementation of proposed six hyperchaotic system

Circuit verification is essential to guarantee correctness of proposed hyperchaotic system. Meanwhile, the key issue in circuit implementation is how to transform theoretical expression of six hyperchaotic system into realistic circuit expression by using electron devices, such as resistance and capacitance. In order to verify the effectiveness of the proposed six hyperchaotic system, the circuit implementation is designed and explained in detail in this section.

In Section “The design of proposed new hyperchaotic system”, when a = 10, b = 8/3, c = 28, d = −1, e = 10 and r = 3, system (1) is in hyperchaotic condition. As we can see, multiplication, addition and differential exist in system equations. Hence, the realistic expression of hyperchaotic system is obtained by summator, integrator and inverter. And the system parameters are set by resistance and capacitance. Based on above explanations, the related circuit implementation is shown in Figure 7, which is filled with operational amplifier LM741 with $\pm 12V$ , analogy multiplier AD633, resistance and capacitance. In order to conduct circuit experiments reliably, the output level of the system is adjusted to 1/20 of the original. So the hyperchaotic system (1) can be depicted by the equations as follows:

$$\{\begin{array}{l}\dot{x}=a(y-x)+w\hfill \\ \dot{y}=cx-y-20xz-v\hfill \\ \dot{z}=-bz+20xy\hfill \\ \dot{w}=dw-20yz\hfill \\ \dot{v}=ry\hfill \\ \dot{u}=-eu+20zw\hfill \end{array}$$

(5)

Figure 7.

Circuit diagram of the system (1).

According to the system circuit schematic and circuit theory, the system circuit implementation equation is given by

$$\{\begin{array}{l}\dot{x}=\frac{R_4{R}_7}{R_5{C}_1{R}_2{R}_6}y-\frac{R_4{R}_7}{R_5{C}_1{R}_1{R}_6}x+\frac{R_4{R}_7}{R_5{C}_1{R}_3{R}_6}w\hfill \\ \begin{array}{l}\dot{y}=\frac{R_{12}{R}_{15}}{R_{13}{C}_2{R}_8{R}_{14}}x-\frac{R_{12}{R}_{15}}{R_{13}{C}_2{R}_9{R}_{14}}y-\frac{R_{12}{R}_{15}}{R_{13}{C}_2{R}_{10}{R}_{14}}xz\\ -\frac{R_{12}{R}_{15}}{R_{13}{C}_2{R}_{11}{R}_{14}}v\end{array}\hfill \\ \dot{z}=-\frac{R_{18}{R}_{21}}{R_{19}{C}_3{R}_{16}{R}_{20}}z+\frac{R_{18}{R}_{21}}{R_{19}{C}_3{R}_{17}{R}_{20}}xy\hfill \\ \dot{w}=-\frac{R_{24}{R}_{27}}{R_{25}{C}_4{R}_{22}{R}_{26}}w-\frac{R_{24}{R}_{27}}{R_{25}{C}_4{R}_{23}{R}_{26}}yz\hfill \\ \dot{v}=\frac{R_{35}{R}_{38}}{R_{36}{C}_6{R}_{34}{R}_{37}}y\hfill \\ \dot{u}=-\frac{R_{30}{R}_{33}}{R_{31}{C}_5{R}_{28}{R}_{32}}u+-\frac{R_{30}{R}_{33}}{R_{31}{C}_5{R}_{29}{R}_{32}}zw\hfill \end{array}$$

(6)

So, resistances and capacitances in equation (6) are set as follows:

$$\begin{array}{l}{R}_1=1\text{k}\Omega ,R_2=1\text{k}\Omega ,R_3=10\text{k}\Omega ,R_4=10\text{k}\Omega ,\\ R_5=100\text{k}\Omega ,R_6=10\text{k}\Omega ,R_7=10\text{k}\Omega ,R_8=1\text{k}\Omega ,\\ R_9=28\text{k}\Omega ,R_{10}=1.4\text{k}\Omega ,R_{11}=28\text{k}\Omega ,\\ R_{12}=28\text{k}\Omega ,R_{13}=100\text{k}\Omega ,R_{14}=10\text{k}\Omega ,R_{15}=10\text{k}\Omega ,\\ R_{16}=3\text{k}\Omega ,R_{17}=0.4\text{k}\Omega ,R_{18}=8\text{k}\Omega ,R_{19}=100\text{k}\Omega ,\\ R_{20}=10\text{k}\Omega ,R_{21}=10\text{k}\Omega ,R_{22}=10\text{k}\Omega ,R_{23}=0.5\text{k}\Omega ,\\ R_{24}=10\text{k}\Omega ,R_{25}=100\text{k}\Omega ,R_{26}=10\text{k}\Omega ,R_{27}=10\text{k}\Omega ,\\ R_{28}=1\text{k}\Omega ,R_{29}=0.5\text{k}\Omega ,R_{30}=10\text{k}\Omega ,R_{31}=100\text{k}\Omega ,\\ R_{32}=10\text{k}\Omega ,R_{33}=10\text{k}\Omega ,R_{34}=1\text{k}\Omega ,R_{35}=3\text{k}\Omega ,\\ R_{36}=100\text{k}\Omega ,R_{37}=10\text{k}\Omega ,R_{38}=10\text{k}\Omega ,\\ C_1=C_2=C_3=C_4=C_5=C_6=10\text{uF}\end{array}$$

The experimental results of the circuit are shown in Figure 8, which shows that the simulation results are agreed well with the numerical simulation of the same hyperchaotic system on the Matlab platform. It verifies the correctness and validity of the proposed six-dimensional hyperchaotic system.

Figure 8.

Related phase portraits of system (1) in circuit: (a) phase portrait of x versus y; (b) phase portrait of x versus z; (c) phase portrait of y versus z; (d) phase portrait of x versus w.

The proposed control method based on Deep Belief Network

Chaos is widely existing, but it is usually a bad phenomenon. Therefore, chaos control has become the key to the application of chaos. A control method based on DBN is proposed to track and control the system (1). In this method, the function of the hyperchaotic system is realized by DBN and a high precision fitting function is obtained. Finally, a controller which is composed of the fitting function and the tracking reference signal is designed to achieve the tracking control of hyperchaotic systems.

Deep Belief Network

Deep learning, which is a machine learning process of multiple levels of deep network structure based on the sample data, has been reported by Hinton firstly. ²⁶ As an important model in deep learning, DBN is widely used in the field of image modeling, feature extraction and recognition.^27,28

A DBN consists of a stack of restricted Boltzmann machines (RBMs), which can not only be regarded as probabilistic generative models, but also be used as a discriminant model. An RBM consists of two layers of neurons: one is called visible layer which is used for input training data, and another one is called hidden layer as a feature detector. The visible and hidden layers are fully inter-connected via connections with symmetric undirected weights, but there are no intra-layer connections within either the visible or the hidden layer. The classic network structure of DBN, which is composed of several layers of RBMs and one layer of BP, is shown in Figure 9.

Figure 9.

Structure of DBN.

As shown in Figure 9, $V$ is the visible layer, $h^i$ is the ith hidden layer. The incentive value of hidden unit is defined by

$$h=Wx$$

(7)

where W represents the symmetric interaction weight and x is the input data

Proposed control strategy based on DBN

The system given in equation (1) can be described by

$${(x_{t+1},y_{t+1},z_{t+1},w_{t+1},v_{t+1},u_{t+1})}^T=f{({(x_t,y_t,z_t,w_t,v_t,u_t)}^T,k)}^T$$

(8)

where, f is the system function, $f\hspace{0.17em}\text{=(}{f}_1,f_2,f_3,f_4,f_5,f_6)$ , $f_1,f_2,f_3,f_4,f_5,f_6$ are the system equations in six directions, $x_{t+1},y_{t+1},z_{t+1},w_{t+1},v_{t+1},u_{t+1}$ represent the state variables in each direction of the system at time t, k is system parameter. In order to facilitate writing, the following will use $O_t^T$ instead of ${(x_t,y_t,z_t,w_t,v_t,u_t)}^T$ .

By adding a controller to the system equation, the system equation becomes

$$O_{t+1}{T}^{=}f{(O_t^T,k)}^T+u$$

(9)

where controller is defined as

$$u=-F{(O_t^T,k)}^T+{O^{\prime }}_{t+1}{T}^{+}\alpha (O_t^T-O_t{\prime }^T)$$

(10)

where F is the fitting function, $F\text{=},F_1,F_2,F_3,F_4,F_5,F_6,$ $O_t^{\prime }$ is the reference signal at time t, and $\alpha$ is the convergence parameters of system.

So the residual is

$$\begin{array}{c}e(t+1)=O_{t+1}^T-{O^{\prime }}_{t+1}^T=f(O_t^T,k{)}^T-F(O_t^T,k{)}^T\\ +{O^{\prime }}_{t+1}^T+\alpha (O_t^T-{O^{\prime }}_t^T)-{O^{\prime }}_{t+1}^T\end{array}$$

(11)

Assuming that fitting function can match nonlinear system with high accuracy, and $\left|\alpha \right|<1$ . After many iterations, the system error becomes zero, that is, the system is stable and can effectively track the reference signal.

The structure of control strategy is shown in Figure 10, some related parameters are: m is the data of system (1) used to train and test DBN, $O^{\prime }$ is the reference signal, t is the current number of chaotic data, n is the number of neurons layers in the DBN, N is the number of neurons in each layer. M is the number of each batch, P is the number of iterations for training DBN and $p^{\prime }$ is the number of iterations to train RBM. W is the symmetric weight parameters between visible layer and hidden layer, b is the bias of visible node and c is the bias of hidden node.

Figure 10.

Structure of the control strategy.

From the descriptions above, the processes to control system (1) in this study as follows:

Step 1: Initializing the reference signal $O^{\prime }$ and the number of sample data m in system (1).

Step 2: Obtaining the training data and testing data $x_i,i=1,2,3\dots m$ .

Step 3: Initializing parameters in DBN: the number of neurons layers n, the number of neurons N in each layer, the number of each batch M, the maximum iterations P, and the maximum iterations $p^{\prime }$ in RBM, the current number of iterations T, the current number of iterations $T^{\prime }$ in RBM.

Step 4: Training the first RBM in the DBN: the train data is given to visible layer $v^{(0)}$ , the probability of hidden neurons being turned on is calculated.

$$P(h_j^{(0)}=1|v^{(0)})=\sigma (W_j{v}^{(0)})$$

(12)

where the superscript is used to distinguish between different vectors, subscript j represents the dimension.

• Step 5: Extracting a sample from the calculated probability distribution.

  $$h^{(0)}\sim P(h_j^{(0)}|v^{(0)})$$

  (13)

• Step 6: Reconstructing the apparent layer with $h^{(0)}$ and extracting a sample of the visible layer.

• Step 7: Calculating the probability of hidden neurons being turned on again with the reconstructed neurons.

  $$P(h_j^{(1)}=1|v^{(1)})=\sigma (W_j{v}^{(1)})$$

  (14)

• Step 8: Updating the weight W, b, c.

  $$\begin{array}{l}{W}_{i,j}=W_{i,j}+P(h_i^{(0)}=1|v^{(0)})v_j{{\hspace{0.17em}}^{(0)}}^T\\ -P(h_i^{(1)}=1|v^{(1)})v_j{{\hspace{0.17em}}^{(1)}}^T\\ b_j=b_j+v^{(0)}-v^{(1)}\\ c_i=c_i+P(h_i^{(0)}=1|v^{(0)})-P(h_i^{(1)}=1|v^{(1)})\end{array}$$

  (15)

• Step 9: Judging whether $T^{\prime }$ is equal to $P^{\prime }$ . If not, $T^{\prime }=T^{\prime }+1$ and back to Step 4.

• Step 10: Fixing the weight and biases of the first RBM, and using the state of its recessive neurons as the input vector to the second RBM.

• Step 11: Repeating steps 4–9 to train the second RBM.

• Step 12: Calculating output of the second RBM.

• Step 13: Calculating mean squared error (MSE)

  $$MSE=\sqrt{\frac{1}{m}{\displaystyle \sum _{i=1}^m{(y_i-Y_i)}^2}},i=(1,2,\mathrm{\dots }m)$$

  (16)

• Step 14: Using the gradient descent method to optimize the error function and the weights are fine-tuned via back-propagation algorithm.

• Step 15: Judging whether T is equal to P. If not, T = T + 1 and back to Step 12.

• Step 16: Repeating steps 1 to 15 five times to get the output fitting function in six directions.

• Step 17: Outputting the total fitting function to the system (1) for control.

The simulations and discussions

In order to verify the effectiveness of the control strategy, numerical simulation is conducted by using the MATLAB software. The step of sampling time is 0.01, system parameters are a = 10, b = 8/3, c = 28, d = −1, e = 10 and r = 3, initial condition $(x_0,y_0,z_0,w_0,u_0,v_0)=(1,1,1,1,1,1)$ . The total data m = 1200, where 1000 are used for training while the other 200 are for testing. The number of neuron layers n = 2, the number of neurons $N=[30,20]$ , the batch M = 5, the maximum iteration P = 100 and $p^{\prime }=10$ . The two reference signals are defined as follows:

$$\begin{array}{c}{O}^{\prime }(1)=\left[\begin{array}{c}5*\text{sin}(t*\pi \text{*}5\text{+}\pi )\text{+}4\\ 5*\text{sin}(t*\pi \text{*}5\text{+}\pi )\text{+}10\\ 5*\text{sin}(t*\pi \text{*}5\text{+}\pi )\text{+}16\\ 20*\text{sin}(t*\pi \text{*}5\text{+}\pi )\text{+}58\\ 100*\text{sin}(t*\pi \text{*}5\text{+}\pi )-98\\ 5*\text{sin}(t*\pi \text{*}5\text{+}\pi )-7\end{array}\right],\\ O^{\prime }(2)=\left[\begin{array}{c}5\hspace{0.17em}\text{*}\hspace{0.17em}\text{sawtooth}(t*\pi \text{*}10)\\ 5\hspace{0.17em}\text{*}\hspace{0.17em}\text{sawtooth}(t*\pi \text{*}10)\text{+16}\\ 5\hspace{0.17em}\text{*}\hspace{0.17em}\text{sawtooth}(t*\pi \text{*}10)\text{+21}\\ 15\hspace{0.17em}\text{*}\hspace{0.17em}\text{sawtooth}(t*\pi \text{*}10)\text{+74}\\ 2\hspace{0.17em}\text{*}\hspace{0.17em}\text{sawtooth}(t*\pi \text{*}10)-86\\ 2.5\hspace{0.17em}\text{*}\hspace{0.17em}\text{sawtooth}(t*\pi \text{*}10)-4.5\end{array}\right]\end{array}$$

The simulation results are shown in Figures 11 and 12, where the reference signal in Figure 11 is a sinusoidal signal, and the reference signal in Figure 12 is a sawtooth signal.

Figure 11.

The simulation results with sinusoidal signal: (a) state values x; (b) state values y; (c) state values z; (d) state values w; (e) state values u; (f) state values v.

Figure 12.

The simulation results with Sawtooth signal: (a) state values x; (b) state values y; (c) state values z; (d) state values w; (e) state values u; (f) state values v.

The controller is added in system when t = 1 s.

The results analyzed from Figures 11 and 12 show that the state variables in every dimension of the system are irregular before being controlled. During the tracking control process, the initial state variables fluctuate a little. Over time, however, the state variables of each dimension of the system can gradually track the reference signal accurately. The simulation results show that the proposed control method can effectively realize the tracking control of six-dimensional hyperchaotic system for any reference signal.

Conclusion

By adding an extra nonlinear control coefficient in the existing five-dimensional Lorenz hyperchaotic system, a novel six-dimensional hyperchaotic system is proposed in this paper. Some related characters are analyzed comprehensively, including symmetry, stability of equilibriums, dissipativity, bifurcations and Lyapunov exponent. An electronic circuit is designed to realize the hyperchaotic system, which verify the correctness and validity of the proposed six-dimensional hyperchaotic system. Furthermore, a novel control method based on DBN is proposed to track and control the hyperchaotic system. Simulation results show the effectiveness and feasibility of this method. This method can not only remove the limitation that mathematical model must be known clearly, but also can track any signal. And it can apply to other chaotic systems, with very broad application prospects.