A New Variable-Boostable 3D Chaotic System with Hidden and Coexisting Attractors: Dynamical Analysis, Periodic Orbit Coding, Circuit Simulation, and Synchronization

Abstract

The study of hidden attractors plays a very important role in the engineering applications of nonlinear dynamical systems. In this paper, a new three-dimensional (3D) chaotic system is proposed in which hidden attractors and self-excited attractors appear as the parameters change. Meanwhile, asymmetric coexisting attractors are also found as a result of the system symmetry. The complex dynamical behaviors of the proposed system were investigated using various tools, including time-series diagrams, Poincaré first return maps, bifurcation diagrams, and basins of attraction. Moreover, the unstable periodic orbits within a topological length of 3 in the hidden chaotic attractor were calculated systematically by the variational method, which required six letters to establish suitable symbolic dynamics. Furthermore, the practicality of the hidden attractor chaotic system was verified by circuit simulations. Finally, offset boosting control and adaptive synchronization were used to investigate the utility of the proposed chaotic system in engineering applications.

1. Introduction

Chaos theory has grown tremendously in recent decades and holds great promise for practical applications [1,2]. The study of chaotic systems began with the discovery of strange attractors by Lorenz in 1963 [3], when he constructed a three-dimensional (3D) quadratic chaotic system that exhibited the famous butterfly effect. Many other chaotic systems have since been presented, some of which satisfy the Šhil’nikov theorem [4], such as the Chen system [5], Qi system [6], Lü system [7], and Rössler system [8]. Recently, two types of attractors were classified by Leonov and Kuznetsov [9], namely self-excited attractors and hidden attractors, and the difference between them is reflected in whether the attractor intersects a neighborhood of any unstable fixed points. The Lorenz system and the systems mentioned above are all referred to as self-excited attractors, while others that do not satisfy the Šhil’nikov theorem are referred to as hidden attractors. Owing to the unique features of hidden attractors, they are difficult to locate and thus play a vital role in encryption and communication [10,11,12]. However, hidden attractors also bring disadvantages and present difficulties in the simulation of drilling systems and phase-locked loops [13].

Based on this, the study of hidden attractors has become an attractive research direction and has received considerable attention from researchers. There are three main types of hidden attractors, which are systems with no equilibria [14,15,16,17], only stable equilibria [18,19,20], and infinitely many equilibria [21,22]. The types of hidden attractors with infinite numbers of equilibria can be divided into various systems, including systems with line equilibria, ellipsoidal equilibria, and circular equilibria. Currently, for hidden attractors with infinite equilibria, researchers have primarily focused on systems with line equilibria [23,24,25]. By using a computer search, Jafari et al., discovered nine chaotic flows with line equilibria, all of which were hidden attractor chaotic systems [26]. Some new systems with hidden chaotic attractors were constructed by introducing perturbations or nonlinear terms into existing hidden attractor chaotic systems [27,28], while many new systems have also been obtained on the basis of modifying the Sprott system and Jafari system [29,30]. The existence of multi-stability can be found in hidden attractor chaotic systems, as reflected by the fact that coexisting attractors have been discovered, and thus, performance flexibility can be achieved [31,32]. In addition, researchers have also concentrated on studying hidden chaotic attractors in fractional-order systems [33,34,35,36], memristor systems [37,38,39,40] and jerk systems [41,42]. Hyperchaotic systems with planes or surfaces of equilibrium points that have hidden attractors are of particular interest, as they exhibit more complex dynamical behaviors than low-dimensional chaotic systems [43,44]. Multiscroll chaotic systems have exceptional benefits in the areas of digital image encryption and private communication [45]. The multi-stability in asymmetric systems, conditional symmetric systems, and self-reproducing systems also have attracted widespread attention [46,47,48]. In Ref. [49], the complex dynamic behaviors and hidden attractors in delayed impulsive systems were explored by means of various bifurcation analyses.

In this paper, we constructed a new hidden attractor chaotic system and explored its dynamical behavior using attraction basins, power spectra, bifurcation diagrams, and other nonlinear analysis tools. Our motivation was to develop an effective method to devise a novel chaotic system with hidden and coexisting attractors based on the existing ones, enabling us to further understand the properties of hidden attractors and multi-stability. The main difficulties in constructing such a system is that there is no general method to clarify which form of feedback controller can be added to produce a new variable-boostable system with both hidden attractors and coexisting attractors. The application of the proposed design lies in the new system being easy to control and synchronize, and the variable can be boosted to any level, so it can reduce the number of components required for signal conditioning. Moreover, offset boosting can be combined with amplitude control to achieve the full range of linear transformations of the signal. Furthermore, the existence of coexisting attractors can also make the system more flexible without adjusting parameters, and it can be used with suitable control strategies to cause switching between various coexisting states. Therefore, it has potential application prospects in the engineering field.

The main contributions and novelty of this work are summarized as follows. (1) We proposed a new 3D chaotic system and explored the adaptive synchronization of the new system. Compared to the above contributions in the literature, the prominent feature of the new system is that it belongs to the variable-boostable chaotic flow, which indicated that it is convenient for chaotic applications. (2) We found self-excited attractors and hidden attractors with two stable equilibrium points in this dissipative system when the parameters were varied. In addition, we investigated the existence of various coexisting asymmetric attractors. To the best of our knowledge, this combination of novel characteristics has rarely been reported. (3) We developed a topological classification method and built complicated symbolic dynamics with six letters instead of four, encoding the unstable periodic orbits embedded in the hidden chaotic attractor, which allowed one to perform a more comprehensive analysis of the periodic orbits.

The rest of the paper is organized as follows. A new hidden attractor chaotic system is proposed, and its fundamental properties and dynamical characteristics under parameter variations are explored in Section 2. In Section 3, a numerical method for calculating periodic orbits, the variational method, is introduced. Section 4 uses the variational method to systematically calculate the periodic orbits of the new system. In Section 5, a corresponding circuit is designed to verify its practicality. Offset boosting control and adaptive synchronization of the novel system are investigated in Section 6. Finally, Section 7 presents the conclusions.

2. New Hidden Chaotic System with Two Stable Equilibria

In retrospect, Wei and Yang introduced the generalized Sprott C system with three real parameters [50]:

$$\begin{array}{ccc}\hfill \frac{dx}{dt}& =& a(y-x),\hfill \\ \hfill \frac{dy}{dt}& =& -cy-xz,\hfill \\ \hfill \frac{dz}{dt}& =& y^2-b.\hfill \end{array}$$

(1)

When $a=10$, $b=100$, and $c=0.4$, there is a hidden chaotic attractor in system (1), which is characterized by two stable fixed points.

With the use of the Bendixson theorem, a hidden attractor was found in a complex variable Lorenz chaotic system [51]. In this work, we discovered the hidden attractor by adding a disturbance term to the existing chaotic system, which could lead to the generation of a new system, but there is no universal method. We first found that the construction of hidden attractor chaotic systems cannot be realized by adding a constant or linear term to the generalized Sprott C system. Therefore, we attempted to add a nonlinear term to the system, and it was further confirmed that adding it to the third equation of Equation (1) could generate hidden chaotic attractors. Inspired by system (1), we propose a new system by adding the $kxy$ term to the third equation as follows:

$$\begin{array}{ccc}\hfill \frac{dx}{dt}& =& a(y-x),\hfill \\ \hfill \frac{dy}{dt}& =& -cy-xz,\hfill \\ \hfill \frac{dz}{dt}& =& y^2-b+kxy,\hfill \end{array}$$

(2)

where $x,y,z$ are the state variables, and a, b, c, k are positive constant parameters. When we select the parameter values of $a=12$, $b=100$, $c=10$, and $k=4.6$, the three Lyapunov exponents of system (2) can be estimated. To avoid transient chaos, we extended the time, and the Lyapunov exponents after 20,000 s were $L_1=0.9861$, $L_2=0$, and $L_3=-22.9857.$ The largest Lyapunov exponent was greater than 0, which confirmed the existence of chaos, as shown in Figure 1. Meanwhile, according to the Kaplan–Yorke formula,

$$D_{KY}=j+\frac{1}{\left|L_{j+1}\right|}\sum _{i=1}^j{L}_i=2+\frac{L_1+L_2}{\left|L_3\right|}=2.0429,$$

(3)

the fractal dimension also further verifies that the new system is chaotic.

2.1. Basic Properties of New Chaotic System

The basic properties of the new chaotic system are described as follows.

(1) Symmetry about the z-axis: When the coordinates are transformed, $(x,y,z)\to (-x,-y,z),$ the form of system (2) remains unchanged.

(2) Dissipativity:

$$\nabla ·V=\frac{\partial \stackrel{·}{x}}{\partial x}+\frac{\partial \stackrel{·}{y}}{\partial y}+\frac{\partial \stackrel{·}{z}}{\partial z}=-a-c.$$

(4)

Since a and c are positive constants, the new system (2) is dissipative. Based on the equation,

$$\frac{dV}{dt}=e^{-a-c},$$

(5)

the system converges to a set of measure zero exponentially as the volume of the phase space is contracted, $V=V_0{e}^{-a-c}$. Therefore, the system will end up fixed to an attractor.

(3) Equilibrium: The new system has two fixed points:

$$\begin{array}{ccc}\hfill E_1& =& (\sqrt{\frac{b}{k+1}},\sqrt{\frac{b}{k+1}},-c),\hfill \\ \hfill E_2& =& (-\sqrt{\frac{b}{k+1}},-\sqrt{\frac{b}{k+1}},-c).\hfill \end{array}$$

(6)

The Jacobi matrix can be obtained as follows:

$$J=\left(\begin{array}{ccc}-a& a& 0\\ -z& -c& -x\\ ky& 2y+kx& 0\end{array}\right).$$

(7)

The characteristic equation is

$$f\left(\lambda \right)={\lambda }^3+(a+c){\lambda }^2+(ac+2xy+k{x}^2+az)\lambda +2axy+ak{x}^2+akxy.$$

(8)

By substituting the coordinates of the two fixed points separately, we obtain the same characteristic equation:

$$f\left(\lambda \right)=a_3{\lambda }^3+a_2{\lambda }^2+a_1\lambda +a_0,$$

(9)

where

$$\begin{array}{ccc}\hfill a_3& =& 1,\hfill \\ \hfill a_2& =& a+c,\hfill \\ \hfill a_1& =& b+\frac{b}{k+1},\hfill \\ \hfill a_0& =& 2ab.\hfill \end{array}$$

(10)

From the Routh–Hurwitz criterion, the two fixed points are stable if the following conditions are satisfied: $a_i>0,(i=0,1,2,3),a_2{a}_1-a_3{a}_0>0$. The condition that needs to be satisfied for this system to have hidden attractors is $(c-a)k>-2c$. For the current parameters $(a,b,c,k)=(12,100,10,4.6)$, the Routh–Hurwitz stability criterion can be satisfied. As a result, the two fixed points $E_1$ and $E_2$ are both stable node-focus points. The new system is a chaotic system in which a strange attractor is hidden.

(4) Power spectrum: The power spectrum of the chaotic state is almost fully covered with background and broad peaks, as shown in Figure 2.

(5) Phase portraits: Using the fourth-order Runge–Kutta numerical integration method, the 2D phase diagrams of the chaotic system for a time-length of 200 s with $a=12$, $b=100$, $c=10$, and $k=4.6$ were obtained from the initial conditions $(x_0,y_0,z_0)=(1,1,1)$, as shown in Figure 3.

In addition, based on the definition of a hidden attractor, we checked the basins of attraction to determine whether a chaotic attractor of the new system could be found from the initial conditions near the equilibrium points. A cross-section at $z=-10$ was selected, and the basins of attraction were captured in three regions, as shown in Figure 4a, in which the blue dots represent the crossing trajectories of the chaotic attractor. The initial values in the red and yellow regions converged to the fixed points $E_1$ and $E_2$, respectively. The orange region represents chaos. In simple terms, the initial values in this region result in a chaotic state. Therefore, it can be clearly observed from the basins of attraction that system (2) contains a hidden chaotic attractor.

The exact correspondence is illustrated in Figure 4b, where the phase diagram trajectory finally converges to the fixed point $E_1$ for the initial value $I_1=(12,-5,-10)$ in the red attraction basin and to the fixed point $E_2$ for the initial value $I_2=(-12,5,-10)$ in the yellow attraction basin, while the initial value $(1,1,-10)$ in the orange region finally evolves to chaos. In Figure 4c, coexisting time series for different initial values are also shown, which indicates the multi-stability in system (2).

2.2. Observation of Chaotic and Complex Dynamics

The system parameters change to enrich the dynamical behaviors of the new system (2). In order to completely explore the diverse dynamical behaviors, we investigated the bifurcations under parameter variations and verified the complicated dynamical behaviors with the aid of bifurcation diagrams, largest Lyapunov exponent spectra, and division diagrams of two parameters.

2.2.1. Fix $b=100,c=10$, and $k=4.6$ and Vary a

We fixed the parameters as $b=100$, $c=10$, and $k=4.6$ while letting a vary in the region $[0,30]$. A summary of the results of the bifurcation diagram and the corresponding largest Lyapunov exponent spectrum are shown in Figure 5. When a increased, the new system converged to a fixed point and then transitioned to chaos, after which the solution became periodic through an inverse period-doubling bifurcation, then evolved to chaos, and finally degenerated to periodic solutions again.

Different types of coexisting attractors can also be found in system (2), which shows that the multi-stability of the new system is very rich. As the 2D phase portraits show in Figure 6a,b, system (2) included two coexisting chaotic attractors for the parameter $a=27.35$ when different initial values $(x_0,y_0,z_0)=(1,1,1)$ and $(x_0,y_0,z_0)=(-1,-1,1)$ were selected. Moreover, when the parameter was set at $a=29$, the system entered a periodic state, and due to the symmetry of the system, the coexistence of two periodic attractors appeared, as depicted in Figure 6c,d. A periodic attractor whose tip faced right or left was obtained depending on the initial values.

2.2.2. Fix $a=12$, $c=10$, and $k=4.6$ and Vary b

Keeping the parameters $a=12$, $c=10$, and $k=4.6$ constant, we let b vary from 40 to 140. Figure 7 shows the bifurcation diagram and the maximum Lyapunov exponent diagram versus b. The system went through a process of period-doubling bifurcations to chaos, transitioning from a periodic to a chaotic solution. The solution was chaotic over a large range, from 68 to 140, which was accompanied by periodic windows. In Figure 8, we present the exact details of the various periodic solutions that occur when b varies. It is worth noting that asymmetric periodic attractors coexist when $b=47$ with the initial values of $(x_0,y_0,z_0)=$$(1,1,1)$ and $(x_0,y_0,z_0)$$=(-1,-1,1)$.

2.2.3. Fix $a=12$, $b=100$, and $k=4.6$ and Vary c

We varied c from $-10$ to 20 while keeping $a=12$, $b=100$, and $k=4.6$. The bifurcation diagram and largest Lyapunov exponent spectrum are shown in Figure 9. The system exhibited intriguing dynamical behaviors in this case, undergoing a pitchfork bifurcation followed by a period-doubling bifurcation route to chaos, interspersed with several periodic windows, and finally, it converged to the equilibrium point. Rich dynamical behaviors can also be observed from the phase diagrams with different parameter values, as displayed in Figure 10. The complexity of the chaos varied when parameter c was changed, and there was a significant difference, which can also be reflected by the size of the largest Lyapunov exponent. Compared with the strange attractor when $c=8$ (see Figure 10c), we conclude that the chaotic behavior of the system was more complex when $c=-2$ (see Figure 10b).

2.2.4. Fix $a=12$, $b=100$, and $c=10$ and Vary k

We varied k from 0 to 20, again keeping the other parameters constant at $a=12$, $b=100$, and $c=10$. From the results in Figure 11, it is evident that the system transitioned from convergence to a fixed point into chaos, which was accompanied by periodic windows in between.

Table 1. Lyapunov exponents and Kaplan–Yorke dimension of system (2): $(a,b,c)=(12,100,10)$ and $(x_0,y_0,z_0)=(1,1,1)$.

k	$L_1$	$L_2$	$L_3$	$D_{\mathit{KY}}$	Dynamics
1	−0.794149	−0.795286	−20.4107	0	Equilibrium
3	0.961912	0	−22.9613	2.0419	Chaos
4.225	0	−0.0340439	−21.9686	1.0	Period
5	0.920123	0	−22.9146	2.0399	Chaos
5.54	0	−0.0956672	−21.9091	1.0	Period
13	0.742037	0	−22.7375	2.0324	Chaos

lists the Lyapunov exponents and Kaplan–Yorke dimensions for the different parameter values, demonstrating the diverse dynamical behaviors with the change of the k value.

2.2.5. Division of Different Parameters

The division diagram for the parameters c and k is shown to investigate the characteristics of the dynamical behaviors of the new system when the other parameters remained constant at $a=12$ and $b=100.$ The parameter c was set to vary between $-10$ and 30, whereas the parameter k was altered between 0 and 25. Figure 12a shows the division of this region by a pseudo-colored map, which was obtained by computing the largest Lyapunov exponents. There are a variety of colors in the division diagram, corresponding to rich variations. The different colors represent different dynamical behaviors. Red, orange, yellow, and green represent chaotic states, cyan corresponds to a periodic state, and blue indicates an equilibrium state. The complexity of the chaos increased as the color became redder. The division diagram of the parameters c and k coincides well with the different dynamical behaviors for the individual parameters shown in Figure 9 and Figure 11. Similarly, we fixed $c=10$ and $k=4.6$, and the division diagram for parameters a and b was obtained, as shown in Figure 12b, in which most regions are periodic. The division diagrams in Figure 12 indicate that the dynamical behaviors of the new system were very rich.

3. Variational Method

Chaotic motion consists of multiple unstable periodic orbits embedded in the strange attractor [52]. The study of periodic orbits gives us a better understanding of the chaotic properties of dynamical systems. If the system is high-dimensional or strongly chaotic, many existing methods for finding unstable periodic orbits will become inefficient or even fail. Here, we use the new method proposed by Lan and Cvitanović, namely the variational method [53]. This method is robust and converges at a fast rate. The variational method employs the logical limit of the multi-point shooting method. First, we have to make an initial loop guess for the overall topology of the unstable periodic orbit and then drive it toward the evolution of the real periodic orbit. The following partial differential equation dominates the loop evolution toward the cycle:

$$\frac{{\partial }^2\stackrel{\sim }{x}}{\partial s\partial \tau }-\lambda A\frac{\partial \stackrel{\sim }{x}}{\partial \tau }-v\frac{\partial \lambda }{\partial \tau }=\lambda v-\stackrel{\sim }{v}.$$

(11)

In Equation (11), $\lambda$ is used to control the period, the deformation of the loop is described by the fictitious time $\tau$, the intrinsic coordinate used to parameterize the loop is $s\in [0,2\pi ]$, $A_{ij}=\frac{\partial v_i}{\partial x_j}$ denotes the gradient matrix of the velocity field, v is the dynamic flow vector field, defined by the derivative of $x,$ and $\stackrel{\sim }{v}$ represents the tangential velocity of the loop.

The stability of the numerical method can be achieved using the Newton descent method. At this point, the cost function obtained by the evolution of the loop toward the cycle is monotonically decreasing:

$$F^2\left[\left(\stackrel{\sim }{x}\right)\right]=\frac{1}{2\pi }{\oint }_{L\left(\tau \right)}d\stackrel{\sim }{x}{[\stackrel{\sim }{v}\left(\stackrel{\sim }{x}\right)-\lambda v\left(\stackrel{\sim }{x}\right)]}^2.$$

(12)

Through iteration, the tangential velocity direction of the loop is continuously brought closer to the velocity direction of the dynamical flow. When $\tau \to \infty$, the two directions become consistent, and thus, the loop converges to the true periodic orbit defined by the dynamical system flow. Consequently, the period of the periodic orbit can be calculated from the following equation:

$$T_p={\int }_0^{2\pi }\lambda \left(\stackrel{\sim }{x}(s,\infty )\right)ds.$$

(13)

Discretization of the loop derivatives is required to ensure numerical stability:

$${\stackrel{\sim }{v}}_n\equiv \frac{\partial \stackrel{\sim }{x}}{\partial s}{|}_{\stackrel{\sim }{x}=\stackrel{\sim }{x}\left(s_n\right)}\approx {\left(\stackrel{\wedge }{D}\stackrel{\sim }{x}\right)}_n.$$

(14)

A five-point approximation is used for the numerical calculations, and the matrix is

$$\stackrel{\wedge }{D}=\frac{N}{24\pi }\left(\begin{array}{ccccccccccc}0& 8& -1& & & & & & & 1& -8\\ -8& 0& 8& -1& & & & & & & 1\\ 1& -8& 0& 8& -1& & & & & & \\ & & & & & ···& & & & & \\ & & & & & & 1& -8& 0& 8& -1\\ -1& & & & & & & 1& -8& 0& 8\\ 8& -1& & & & & & & 1& -8& 0\end{array}\right).$$

(15)

Thus, Equation (11) can be changed to the following form with a fictitious time Euler step $\delta \tau$:

$$\left(\begin{array}{cc}\stackrel{\wedge }{A}& -\stackrel{\wedge }{v}\\ \stackrel{\wedge }{a}& 0\end{array}\right)\left(\genfrac{}{}{0pt}{}{\delta \stackrel{\sim }{x}}{\delta \lambda }\right)=\delta \tau \left(\genfrac{}{}{0pt}{}{\lambda \stackrel{\wedge }{v}-\stackrel{\wedge }{\stackrel{\sim }{v}}}{0}\right),$$

(16)

where $\stackrel{\wedge }{A}=\stackrel{\wedge }{D}-\lambda$ diag $[A_1,A_2,...,A_N],$$\stackrel{\wedge }{v}={(v_1,v_2,...,v_N)}^t$, $\stackrel{\wedge }{\stackrel{\sim }{v}}={(\stackrel{\sim }{v_1},\stackrel{\sim }{v_2},...\stackrel{\sim }{,v_N})}^t$, and $\stackrel{\wedge }{a}$ is an $Nd$-dimensional row vector, which restricts the change of the coordinates. By inverting the matrix on the left of Equation (16), we can solve for $\delta \stackrel{\sim }{x}$ and $\delta \lambda$ to acquire the deformation of the loop coordinates and period. The banded lower–upper decomposition method can be used to accelerate the computation, and the Woodbury formula can be employed to deal with periodic and boundary terms [54]. The variational method can be effectively used to calculate the unstable periodic orbits of various chaotic systems [55,56,57]. In the next section, we utilize the variational method to locate the unstable periodic orbits in the hidden chaotic attractor of system (2).

4. Symbolic Encoding of Unstable Periodic Orbits in the Hidden Chaotic Attractor with Six Letters

Periodic orbit theory can be used to calculate many physical quantities, such as the fractal dimension and topological entropy [58,59]. In most cases, the theory is acquired by performing calculations for the required unstable short-period orbits. Here, we explore the unstable periodic orbits in the hidden chaotic attractor of system (2) for the parameter values of ($a,b,c,k)=(12,100,10,4.6)$. When the Poincaré section $z=-10$ is chosen, the first return map can be plotted, which contains a large number of dense points with a certain hierarchical structure, as shown in Figure 13. There were five branches, and thus, it was necessary to build complex symbolic dynamics to encode the periodic orbits of the new system (2) for the current parameters [60]. Taking this into account, we used the variational method to locate the cycles of the new system, and six cycles with simple topological structures were found, as shown in Figure 14. The symbolic encoding rules of these periodic orbits are as follows:

(1) For a cycle with a smooth ellipse shape around a fixed point, the symbol 0 is used to denote the cycle around the left and the symbol 1 is used to denote the cycle around the right.

(2) For an irregular cycle around a fixed point with a smaller extension on the z-axis around 100, which has a blunt fold, in the shape of a raised wing, the symbol 2 is used to denote the cycle around the left and the symbol 3 is used to denote the cycle around the right.

(3) For an irregular cycle around a fixed point with a larger extension on the z-axis around 140, which has a very sharp fold, forming the shape of a ginkgo leaf, the symbol 4 is used to denote the cycle around the left and the symbol 5 is used to denote the cycle around the right.

The six cycles presented above are the building blocks that make up the orbits of the new system, and the other complex long-period orbits are composed of them. Thus, we can calculate the cycles systematically by utilizing the six-letter symbolic dynamics. It can be clearly seen that cycles 0 and 1 were symmetric to each other, as were cycles 2 and 3 and cycles 4 and 5. They were cycles with a topological length of 1, while cycles had a topological length of 2 when they rotated once around each of the left and right fixed points or twice around a fixed point. Since there was z-axis symmetry in the system, we could likewise find symmetric and asymmetric cycles with a topological length of 2, as shown in Figure 15.

The cycles could be classified into two types: self-conjugated and mutually conjugated. The periods of the cycles that were conjugated to themselves were not equal to those of the other cycles, such as cycles 01, 23, and 45. Two mutually conjugated cycles had equal periods and symmetry, e.g., cycles 03 and 12. We discovered that several cycles with topological length 2 were pruned, which means that they did not exist, such as cycles 02, 13, and 04. We also found that building blocks with symbols 0 or 1 could not be combined with the building blocks with symbols 4 or 5 to form a periodic orbit. Therefore, for example, for the cycles of topological length 3, there would not be cycles 045 and 124. After some attempts, we found that cycles 012, 123, 002, and 022 were also pruned. To obtain a clear picture of the periodic orbits of the new system, we show the 2D phase diagrams of nine cycles with topological length 3 in Figure 16. All the cycles and their periods $T_p$ within a topological length 3 are tabulated in

Table 2. Unstable cycles in the new system within a topological length of 3.

Length	Itineraries	Periods	Length	Itineraries	Periods	Length	Itineraries	Periods
1	0	0.645509	3	223	3.552630	3	001	2.324411
	1	0.645509		233	3.552630		011	2.324411
	2	1.186404		033	2.981070		123	—
	3	1.186404		122	2.981070		032	—
2	01	1.559290		021	2.653639		003	2.401931
	23	2.366105		013	2.653639		112	2.401931
	12	1.792160		031	—		113	—
	03	1.792160		012	—		002	—
	02	—		132	2.962501		022	—
	13	—		023	2.962501		133	—
1	4	1.515729	3	445	4.815813	3	354	4.221816
	5	1.515729		455	4.815813		234	3.878345
2	24	2.695403		344	4.221765		325	3.878345
	35	2.695403		255	4.221765		225	3.892262
	25	2.706161		335	3.881997		334	3.892262
	34	2.706161		224	3.881997		254	4.211233
	45	3.031718		244	4.211320		345	4.211233
3	235	3.889210		355	4.211320
	324	3.889210		245	4.221816

.

Figure 13 shows that the multi-branch structure of the first return map created some difficulties for the analysis of the unstable periodic orbits. The establishment of the symbolic encoding approach based on the topological structure of the trajectory and its circuiting property with respect to different equilibrium points showed the effectiveness of the analysis of the cycles in the hidden chaotic attractors. It is hoped that this method can also be used to encode periodic orbits embedded in hidden hyperchaotic attractors.

5. Circuitry of Proposed System

Multi-scroll chaotic systems often use current–feedback operational amplifiers (CFOAs) to implement the circuit, which helps to enhance the frequency bandwidth [61]. FPGA implementation has strong universality and is less limited by hardware resources [62], while circuit simulation has the characteristics of simple debugging and a low cost. To verify the correctness and feasibility of the new proposed system, circuit simulations were performed in this study, and we selected the NI Multisim 14 software (accessed on 1 May 2015 and website address https://www.ni.com/zh-cn/suppot/downloads) to simulate the circuit. The state variables $x,$ $y,$ and z were reduced by a factor of 10 to avoid the state variables being out of the dynamic range of the device. Therefore, system (2) is rewritten as

$$\begin{array}{ccc}\hfill \stackrel{·}{X}& =& a(Y-X),\hfill \\ \hfill \stackrel{·}{Y}& =& -cY-10XZ,\hfill \\ \hfill \stackrel{·}{Z}& =& 10Y^2-0.1b+10kXY,\hfill \end{array}$$

(17)

where $X=0.1x,$$Y=0.1y,$ and $Z=0.1z$.

By performing a time-scale transformation of Equation (17), in which the time-scale factor is set to ${\tau }_0=\frac{1}{R_0{C}_0}=1000$ and $t={\tau }_0\tau$, we obtain

$$\begin{array}{ccc}\hfill \stackrel{·}{X}& =& 1000a(Y-X),\hfill \\ \hfill \stackrel{·}{Y}& =& -1000cY-10,000XZ,\hfill \\ \hfill \stackrel{·}{Z}& =& 10,000Y^2-100b+10,000kXY.\hfill \end{array}$$

(18)

Based on Kirchhoff’s law, the following equation can be obtained from the circuit diagram in Figure 17:

$$\begin{array}{ccc}\hfill \stackrel{·}{X}& =& \frac{R_3}{R_2{R}_4{C}_1}Y-\frac{R_3}{R_1{R}_4{C}_1}X,\hfill \\ \hfill \stackrel{·}{Y}& =& -\frac{R_9}{R_7{R}_{10}{C}_2}Y-\frac{R_9}{R_8{R}_{10}{C}_2}0.1XZ,\hfill \\ \hfill \stackrel{·}{Z}& =& \frac{R_{16}}{R_{13}{R}_{17}{C}_3}0.1Y^2+\frac{R_{16}}{R_{15}{R}_{17}{C}_3}{V}_1+\frac{R_{16}}{R_{14}{R}_{17}{C}_3}0.1XY.\hfill \end{array}$$

(19)

The circuit consisted of three functional modules: addition, integration, and inversion, and the three channels corresponded to the three variables of the system. As shown in Figure 17, the circuit included nineteen resistors, three capacitors, nine TL082CP operational amplifiers, and three analog multipliers (the output gain was 0.1). The power supply voltage was ±17 V. The coefficients of system (2) were $a=12,$ $b=100,$ $c=10,$ and $k=4.6,$ and the values of the circuit components were $C_1=C_2=C_3=100$ nF, $R_1=R_2=8.333$ k $\mathsf{\Omega }$, $R_3=R_9=R_{16}=100$ k $\mathsf{\Omega }$, $R_4=R_5=R_6=R_7=R_{10}=R_{11}=R_{12}=R_{15}=R_{17}=R_{18}=R_{19}=10$ k $\mathsf{\Omega }$, $R_8=R_{13}=1$ k $\mathsf{\Omega }$, $R_{14}=0.217$ k $\mathsf{\Omega }$, and $V_1=-1$ V.

The designed circuit was successfully implemented in Multisim, and the results are reported in Figure 18. The results of the circuit implementation agreed with the numerical simulation results, which validated the realizability of the proposed new system (2).

6. Offset Boosting Control and Adaptive Synchronization of New System

Engineering applications for variable-boostable systems show considerable promise, and they are simple to control once offsets are added [63,64]. As the offset changes, bipolar or unipolar signals may be produced. We select z as the state variable, since it only occurs once in system (2). The control parameter w has the ability to boost the state variable z. As a result, the offset-boosted system can be written as

$$\begin{array}{ccc}\hfill \frac{dx}{dt}& =& a(y-x),\hfill \\ \hfill \frac{dy}{dt}& =& -cy-x(z+w),\hfill \\ \hfill \frac{dz}{dt}& =& y^2-b+kxy,\hfill \end{array}$$

(20)

where the control parameter w is a constant.

We select parameter values of $a=12,b=100,c=10$, and $k=4.6$, and the initial values of the variables were all set to 1. As shown in Figure 19, it is evident from the attractors with various offsets into the y-z phase space and the time sequence diagram that a chaotic signal could change from being a bipolar signal to a unipolar signal. As the value of the control parameter w changed, the attractor moved up and down along the z-axis. For example, when w was taken as 0, a bipolar signal existed. When the value of w was taken as −75, a positive unipolar signal appeared, while a negative unipolar signal appeared when w was taken as 40.

It can be seen from the above discussion that our new proposed system has an easy-to-control nature, and the adjustment of the overall signal can be achieved simply by changing a single parameter, i.e., adding an offset w to the variable z to achieve a shift in the z-direction, which has potential in engineering applications.

Chaotic synchronization is the key to achieving chaotic and confidential communication. There are various schemes for synchronization [65], and in this section, we take an adaptive synchronization approach to realize the chaotic synchronization of two identical systems with unknown parameters due to its robustness and simple implementation.

The following master system is the new hidden attractor chaotic system we introduced:

$$\begin{array}{ccc}\hfill \stackrel{·}{x_m}& =& a(y_m-x_m),\hfill \\ \hfill \stackrel{·}{y_m}& =& -c{y}_m-x_m{z}_m,\hfill \\ \hfill \stackrel{·}{z_m}& =& y_m^2-b+k{x}_m{y}_m,\hfill \end{array}$$

(21)

and the slave system adopts the following form by adding adaptive controls $u_x,u_y$, and $u_z$ for each of the three directions:

$$\begin{array}{ccc}\hfill \stackrel{·}{x_s}& =& a(y_s-x_s)+u_x,\hfill \\ \hfill \stackrel{·}{y_s}& =& -c{y}_s-x_s{z}_s+u_y,\hfill \\ \hfill \stackrel{·}{z_s}& =& y_s^2-b+k{x}_s{y}_s+u_z.\hfill \end{array}$$

(22)

The synchronization error is set to

$$\begin{array}{ccc}\hfill e_x& =& x_s-x_m,\hfill \\ \hfill e_y& =& y_s-y_m,\hfill \\ \hfill e_z& =& z_s-z_m.\hfill \end{array}$$

(23)

Then, the error dynamics of the slave system (22) and master system (21) can be written as

$$\begin{array}{ccc}\hfill \stackrel{·}{e_x}& =& a(e_y-e_x)+u_x,\hfill \\ \hfill \stackrel{·}{e_y}& =& -c{e}_y-x_s{z}_s+x_m{z}_m+u_y,\hfill \\ \hfill \stackrel{·}{e_z}& =& y_s^2-y_m^2+k{x}_s{y}_s-k{x}_m{y}_m+u_z.\hfill \end{array}$$

(24)

The examination of the stability of the error system is based on the transformation of the synchronization issue between the master and slave systems. The adaptive controller selected in this scheme is

$$\begin{array}{ccc}\hfill u_x& =& -\widehat{a}\left(t\right)(e_y-e_x)-k_1{e}_x,\hfill \\ \hfill u_y& =& \widehat{c}\left(t\right)e_y+x_s{z}_s-x_m{z}_m-k_2{e}_y,\hfill \\ \hfill u_z& =& -y_s^2+y_m^2-\widehat{k}\left(t\right)x_s{y}_s+\widehat{k}\left(t\right)x_m{y}_m-k_3{e}_z,\hfill \end{array}$$

(25)

in which $k_1,k_2$, and $k_3$ are positive gain constants and $\widehat{a}\left(t\right),\widehat{b}\left(t\right),\widehat{c}\left(t\right)$, and $\widehat{k}\left(t\right)$ are parameter estimates. Therefore, substituting Equation (25) into Equation (24) and simplifying it yields

$$\begin{array}{ccc}\hfill \stackrel{·}{e_x}& =& (a-\widehat{a}\left(t\right))(e_y-e_x)-k_1{e}_x,\hfill \\ \hfill \stackrel{·}{e_y}& =& -(c-\widehat{c}\left(t\right))e_y-k_2{e}_y,\hfill \\ \hfill \stackrel{·}{e_z}& =& (k-\widehat{k}\left(t\right))x_s{y}_s-(k-\widehat{k}\left(t\right))x_m{y}_m-k_3{e}_z.\hfill \end{array}$$

(26)

The parameter estimation error is set to

$$\begin{array}{ccc}\hfill e_a\left(t\right)& =& a-\widehat{a}\left(t\right),\hfill \\ \hfill e_b\left(t\right)& =& b-\widehat{b}\left(t\right),\hfill \\ \hfill e_c\left(t\right)& =& c-\widehat{c}\left(t\right),\hfill \\ \hfill e_k\left(t\right)& =& k-\widehat{k}\left(t\right).\hfill \end{array}$$

(27)

Then, we obtain

$$\begin{array}{ccc}\hfill \stackrel{·}{e_a}& =& -\stackrel{·}{\widehat{a}},\hfill \\ \hfill \stackrel{·}{e_b}& =& -\stackrel{·}{\widehat{b}},\hfill \\ \hfill \stackrel{·}{e_c}& =& -\stackrel{·}{\widehat{c}},\hfill \\ \hfill \stackrel{·}{e_k}& =& -\stackrel{·}{\widehat{k}}.\hfill \end{array}$$

(28)

The error dynamics can be rewritten as follows:

$$\begin{array}{ccc}\hfill \stackrel{·}{e_x}& =& e_a(e_y-e_x)-k_1{e}_x,\hfill \\ \hfill \stackrel{·}{e_y}& =& -e_c{e}_y-k_2{e}_y,\hfill \\ \hfill \stackrel{·}{e_z}& =& e_k{x}_s{y}_s-e_k{x}_m{y}_m-k_3{e}_z.\hfill \end{array}$$

(29)

The quadratic Lyapunov function can be constructed as follows:

$$V=\frac{1}{2}(e_x^2+e_y^2+e_z^2+e_a^2+e_b^2+e_c^2+e_k^2).$$

(30)

Differentiating V along the trajectories of the system yields

$$\stackrel{·}{V}=-k_1{e}_x^2-k_2{e}_y^2-k_3{e}_z^2-e_a(\stackrel{·}{\widehat{a}}-e_x(e_y-e_x))-e_b\stackrel{·}{\widehat{b}}-e_c(\stackrel{·}{\widehat{c}}+e_y^2)-e_k(\stackrel{·}{\widehat{k}}+e_z{x}_m{y}_m-e_z{x}_s{y}_s).$$

(31)

Thus, the parameter estimates can be set as

$$\begin{array}{ccc}\hfill \stackrel{·}{\widehat{a}}& =& e_x(e_y-e_x)+k_4{e}_a,\hfill \\ \hfill \stackrel{·}{\widehat{b}}& =& k_5{e}_b,\hfill \\ \hfill \stackrel{·}{\widehat{c}}& =& -e_y^2+k_6{e}_c,\hfill \\ \hfill \stackrel{·}{\widehat{k}}& =& e_z{x}_s{y}_s-e_z{x}_m{y}_m+k_7{e}_k,\hfill \end{array}$$

(32)

where $k_4,k_5,k_6$, and $k_7$ are positive gain constants.

We obtained a negative definite Lyapunov function:

$$\stackrel{·}{V}=-k_1{e}_x^2-k_2{e}_y^2-k_3{e}_z^2-k_4{e}_a^2-k_5{e}_b^2-k_6{e}_c^2-k_7{e}_k^2.$$

(33)

According to Lyapunov stability theory, under the adaptive controller, all the synchronization errors $e_x$, $e_y$, and $e_z$ and parameter estimation errors $e_a,e_b,e_c$, and $e_k$ globally and exponentially converge to 0 when the initial values are chosen at random. Therefore, through the above theoretical analysis, it is known that the master system and the slave system can be fully synchronized.

The effectiveness of the proposed approach was verified by numerical simulations, which are described as follows. The parameters were set as $(a,b,c,k)=(12,100,10,4.6)$, which resulted in a hidden chaotic attractor. The gain constants were selected as $k_i=3(i=1,2,3,4,5,6,7)$. The initial values of the master system, slave system, and parameter estimates were taken as

$$(x_m\left(0\right),y_m\left(0\right),z_m\left(0\right))=(-1,2.5,-4),(x_s\left(0\right),y_s\left(0\right),z_s\left(0\right))=(-0.5,-0.5,-5),(\widehat{a}\left(0\right),\widehat{b}\left(0\right),\widehat{c}\left(0\right),\widehat{k}\left(0\right))=(2,113,15,10)$$

(34)

Figure 20 displays the full synchronization of the respective states of the master and slave systems. It can be seen that after a short time, the state trajectories of the master system $x_m,y_m$ and $z_m$ gradually overlapped with the slave system $x_s,y_s$ and $z_s$. The time-histories of the synchronization errors and parameter estimation errors are also shown in Figure 21, which indicate that all the errors asymptotically converged to zero with time. In summary, the simulation results of this new hidden chaotic system demonstrated the operability of the chaotic circuit and adaptive synchronization control.

7. Conclusions

On the basis of the generalized Sprott C system, we added a nonlinear cross-term to the third equation to construct a new hidden attractor chaotic system that has two stable equilibria. To quantitatively examine its chaotic properties, several tools, including the Lyapunov exponent spectrum, power spectrum, and the Poincaré first return map, were applied. The influences of four parameters on the dynamical behaviors of the system were explored by means of bifurcation diagrams, maximum Lyapunov exponent spectra, and division diagrams of two parameters, and the rich and complex dynamical behaviors of the system were also presented in combination with the phase portraits. Meanwhile, the existence of various coexisting attractors was discovered, which indicated a multi-stability phenomenon. Furthermore, we calculated the unstable periodic orbits embedded in the hidden chaotic attractor with the help of the variational method, and we encoded and classified the cycles using six-letter symbolic dynamics. Finally, a circuit simulation, offset boosting control, and adaptive synchronization linked this hidden attractor chaotic system to physical experiments and verified the practicality of the system.

We believe that the periodic orbit coding method used in this paper can provide a reference for analyzing periodic orbits in other hidden attractor chaotic systems. More results from studies on the applications of the proposed hidden attractor chaotic system will be revealed in future research. Moreover, the hidden attractors in fractional-order systems have also attracted extensive attention in recent years. When a single parameter changes, self-excited, hidden, or nonhyperbolic chaotic attractors will appear in a new fractional-order chaotic system with different families of hidden and self-excited attractors [66]. In a new fractional-order chaotic system without any equilibrium points based on a fracmemristor, the hidden chaotic attractors are propagated infinitely using a trigonometric function [67]. We can investigate the fractional-order system corresponding to this hidden attractor chaotic system to gain a better grasp of the complexity of chaotic systems and their variety of practical applications. To present the multi-stability of coexisting attractors, memristor chaotic systems can also be introduced. Attention should also be paid to the FPGA implementation of chaotic systems, which will be the focus of our subsequent work.