A new 10-D hyperchaotic system with coexisting attractors and high fractal dimension: Its dynamical analysis, synchronization and circuit design

This work introduce a new high dimensional 10-D hyperchaotic system with high complexity and many of coexisting attractors. With the adjustment of its parameters and initial points, the novel system can generate periodic, quasi-periodic, chaotic, and hyperchaotic behaviours. For special values of parameters, we show that the proposed 10-D system has a very high Kaplan-Yorke fractal dimension, which can reach up to 9.067 indicating the very complexity of the 10-D system dynamics. In addition, the proposed system is shown to exhibit at least six varied attractors for the same values of parameters due to its multistability. Regions of multistability are identified by analysing the bifurcation diagrams of the proposed model versus its parameters and for six different values of initial points. Many of numerical plots are given to show the appearance of different dynamical behaviours and the existence of multiple coexisting attractors. The main problem with controlling chaos/hyperchaos systems is that they are not always fully synchronized. therefore, some powerful synchronization techniques should be considered. The synchronization between the high-dimensional 10-D system and a set of three low-dimensional chaotic and hyperchaotic systems is proposed. Ten control functions are designed using the active control method, ensuring synchronisation between the collection of systems and the 10-D hyperchaotic system. Finally, using Multisim 13.0 software to construct the new system’s electronic circuit, the feasibility of the new system with its extremely complicated dynamics is verified. Therefore, the novel 10-D hyperchaotic system can be applied to different chaotic-based application due to its large dimension, complex dynamics, and simple circuit architecture.


Introduction
Scientific communities have been interested in chaotic systems study over the past 60 years, especially since the work of Edward Lorenz, the famous American meteorologist in 1963 [1]. The essential trait of chaotic systems, he discovered, is their great sensitivity to initial conditions. A small change in the chaotic system's initial parameters results in significantly varied and unpredictable behaviour. This type of system's tremendous complexity makes it beneficial in a variety of fields, including secure communication [2][3][4][5].
The Kaplan-Yorke dimension and the Lyapunov exponents are the most important tools for describing chaotic behaviour in a dynamical system [6]. Kaplan-Yorke dimension, on the other hand, is an effective measure of the fractal dimension and chaotic complexity of the normal n-dimensional dynamical system, and it is calculated using Lyapunov exponent values. When calculating the Lyapunov exponent, the dynamical system's two adjacent starting values are taken into account. The paths produced via the initial guesses will exponentially diverge if this system exhibits chaotic behaviour, and the coefficients that characterises the divergence rate is a Lyapunov exponent. There is, absolutely, a Lyapunov exponent for every state-space dimension. At least one of the exponents must be positive for a dynamic system to display chaotic behavior. When there are many non-negative exponents, the related systems' dynamics expand in multiple directions, resulting in a more complex behaviour, which we name a hyperchaotic system in this situation.
Many papers have been published on hyperchaotic system. Vaidyanathan et al. [7] proposed of the new 4-D hyperchaotic system with no equilibrium and analysis of global hyperchaos synchronization results of the new hyperchaotic system using Integral Sliding Mode Control (ISMC). Singh et al. [8] proposed of the 5-D hyperchaotic system with stable equilibrium point and the proposed system exhibits multistability and transient chaotic behavior. Alattas et al. [9] proposed of the synchronization problem of hyperchaotic systems using integral-type sliding mode control for the 6-D hyperchaotic systems and presented of the analog electronic circuit using MultiSIM. Lagmiri et al. [10] constructed of the two new 7D hyperchaotic systems and to investigate the dynamics and synchronization of these new systems using the theory of observers. Kang et al. [11] proposed a color image encryption method combining with 2D-VMD and 8D hyperchaotic system. Zhu et al [12] presented a nine-dimensional eight-order chaotic system, and the corresponding circuit implementation. Mahmoud et al. [13] presented another complex nonlinear hyperchaotic model, spoke to by nine first-order nonlinear ordinary differential equations and proposed new nine-dimensional chaotic Lorenz System with quaternion variables [14]. Jianliang et al. [15] proposed a ten-dimensional nineorder chaotic system and the electronic circuit implementation. However, there is still a need for discovering systems with different 10D hyperchaotic system. Synchronization of chaotic systems has attracted much attention in recent years due to their applications in neuron model, robotic and cryprosystem. Yu et al. [16] presented a novel 5D hyperchaotic four-wing memristive system with multiline equilibrium and synchronization of the 5D hyperchaotic system with different structures by active control. Zambrano-Serrano and Anzo-Hernández [17] proposed a novel chaotic oscillator derived from the generic four-dimensional autonomous jerk systems and analyze the synchronization behavior of the chaotic oscillator via feedback control. Munoz-Pacheco et al. [18] analyzed the effect of a nonlocal fractional operator in an asymmetrical glucose-insulin regulatory system and proposed an active control scheme for forcing the chaotic regime (an illness) to follow a periodic oscillatory state, i.e., a disorder-free equilibrium. However, to the best of the authors' knowledge, neither the control nor the synchronisation of the new 10-D hyperchaotic system has been investigated yet with the active control method.
Secure transmissions utilising various methods and schemes is one of chaotic system's most essential applications. Chaotic systems generate complex signals with a random appearance, which are used to conceal the secret information to be communicated. As a result, many literature have studied the chaotic systems, so as to address the huge gap for the type of complicated system in the disciplines of chaotic encryption and secure communication [19,20]. Nazari et al. [21] proposed secure transmission of authenticated medical images using a novel chaotic IWT-LSB blind watermarking approach. design an embedded cryptosystem based on a pseudo-random number generator (PRNG). Trujillo-Toledo et al. [22] proposed design an embedded cryptosystem based on a pseudo-random number generator (PRNG)using enhanced sequences from the Logistic 1D map, and it reaches a throughput of up to 47.44 Mbit/s using a personal computer with a 2.9 GHz clock, and 10.53 Mbit/s using a Raspberry Pi 4. Hemdan [23] presented a medical image watermarking approach based on Wavelet Fusion (WF), Singular Value Decomposition (SVD), and Multi-Level Discrete Wavelet Transform (M-DWT) with scrambling techniques for securing the watermarks images. García-Guerrero et al. [24] introduces a process to improve the randomness of five chaotic maps that are implemented on a PIC-microcontroller. They have improved chaotic maps tested to encrypt digital images in a wireless communication scheme, particularly on a machine-to-machine (M2M) link, via ZigBee channels. Silva-Juárez et al. [25] proposed the use of first-order all-pass and low-pass filters to design the ratio of the polynomials that approximate the fractional-order. Also, the filters are implemented using amplifiers and synthesized on a field-programmable analog array (FPAA) device. Tlelo-Cuautle et al. [26] provides guidelines to implement fractional-order derivatives using commercially available devices and describes details on using FPGAs to approach fractional-order chaotic systems, programming in VHDL and reducing hardware resources.
In addition, as previously stated, several studies discovered that the hyperchaotic systems with high dimensional (n > 3) whose positive Lyapunov exponent is more than one and having a high Kaplan-Yorke dimension is capable of generating more random and complex signals with greater uncertainty, which improves the chaotic transmissions security. Based of these reasons, several types of these high dimensional systems have been developed having two positive Lyapunov exponents since after the emergence the first system by Rossler in 1979 [27]. Some nonlinear dynamical systems can develop many forms of complexity such as chaos, hyperchaos, bifurcation and multistability. A dynamical system that generate two or more synchronize different attractors for a given set of coefficients is defined to be multistable.
In the recent years, construction new high dimensional (n > 5) hyperchaotic systems with high fractal dimension [28] and extreme multistability become an interesting area of research in chaos theory because of the need of these kinds of hyper-complex systems in recent engineering applications especially in secure communications. In this work, we generate the first 10-D hyperchaotic system which exhibit up to six synchronize attractors having high Kaplan-Yorke fractal dimension. The new 10-D hyperchaos system's dynamic properties is discussed, its Regions of multistability identified, its active control synchronization and design its equivalent electronic circuit described.
The novelty and contributions of the paper are summarised as follows: 4. System has unstable and self-excited family.
5. This work studied the synchronization of the proposed 10D system with three diverse Hyperchaotic and chaotic systems via active controllers.
6. The equivalent electronic circuit for the new 10-D hyperchaotic system (1) is developed using Multisim 13.0 software.
The rest of this paper is organized as follows. Section 2 describes the dynamics of the new 10D Hyperchaotic system. Dynamical analyses of the new 10D Hyperchaotic system are shown in Section 3. multistability and coexisting attractors in the new 10D Hyperchaotic system is discussed in Section 4. In Section 5 we discuss the synchronization of the new 10D hyperchaotic systems using active control. Circuit implementation of the new 10-D hyperchaotic system are presented in Section 6. Finally, the conclusions of this paper are summarized in Section 7.

New 10-D hyperchaotic system
There are four positive parameters in the new 10D hyperchaotic system, as well as twentythree terms with two quadratic and one quartic nonlinearity. The new system is describe using the algebraic equations (1): where the state variables are given as x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 , x 9 and x 10 while a, b, c and d parameters denote the positive constant. When the initial guess are selected as: ð1; 0; 0; 0; 0; 0; 0; 0; 0; 0Þ: and the coefficient values are selected as: System (1) exhibit a complex hyperchaotic behavior with high fractal dimension and its phase portraits are described in The Lyapunov exponents (LE) for the new 10D hyperchaotic system (1) whose initial conditions is given in Eq (2) and the parameters values as in Eq (3)

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The obtained ten LE of the new 10D hyperchaotic system (1) are: As shown in Fig 2, system (1) has LE 1,2 > 0, LE 3,4,5,6 = 0, LE 7,8,9,10 < 0, which means that it exhibits a hyperchaotic behavior with two positive, four zero and four negative LE. The sum of the LE is also negative, indicating that our suggested 10-D system (1) is dispersed.
According to chaos theory, Kaplan-Yorke dimensions high value directly corresponds to system dynamics' high complexity. For the proposed system (1), the analogous Kaplan-Yorke

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dimension is calculated as follows: with j representing the index such that: So, for (1), we discover that: We can observe from (7) that fractal dimension of Kaplan-Yorke is very large in comparison to other systems. Thus, the proposed 10-D system (1) displays a very complex hyperchaotic behaviour. In 2011, J. C. Sprott [29] proposed three criteria for the publication of a new hyperchaotic system. It is said in [29], that a new system must satisfy at least one criterion. Among the three criteria, one criterion is that the system should exhibit some behavior previously unobserved. The new behavior of the new 10D hyperchaotic is compared in Table 1.
It can be seen from Table 1 that the 10-D system (1) has a more advanced fractal dimension than some famous high dimensional chaotic systems reported in literature, which indicate and prove the high complexity of system (1).

Dynamical analysis of the new 10-D hyperchaotic system
In this part, the effect of initial conditions and coefficient on the complexity and properties of system (1) would be studied. Stability of equilibrium points, Lyapunov exponents, fractal dimension and coexisting attractors will be the main properties of investigation.

Bifurcation, Lyapunov exponents and fractal dimension
The LE spectrum and bifurcation diagram are two most significant tools for analyzing a system's dynamical behavior. The Kaplan-Yorke fractal dimension is also a useful indicator of system complexity. The dynamical behavior and complexity of the novel 10-D system (1) are examined using numerical simulations in this section of the study, with variable positive coefficient a, b, c, and d.
Parameter a varying. To investigate the sensitivity of (1) to the value of parameter a, we let b = 0.1, c = 0.5, d = 0.01 and vary a between 0 and 0.2. The bifurcation diagram (BD) of (1) with corresponding Lyapunov exponents spectrum when a belongs to the following set of values [0;0.2] and for initial conditions (3) are depicted in Fig 3, we can observe that BD and Lyapunov exponents spectrum are in good agreement.  (1) can exhibits periodic behaviour with Kaplan-yorke fractal dimension equal to zero indicating no complexity of the dynamics. Also, the 10-D system can involves into a chaotic attractor with one positive LE and a higher fractional Kaplan-Yorke dimension which indicates complexity of the dynamics. In addition, more complexity is observed when the new system generate a hyperchaotic behaviour with more than one positive LE and higher values of Kaplan-Yorke fractal dimension, which indicates a very complicated dynamic behavior generated by the new 10-D system (1).
When, the new 10-D system (1) Table 4 shows the LE and fractional Kaplan-Yorke dimension for various values of c.   Table 5 shows the LE, the Kaplan-Yorke fractal dimension and the dynamics for different values of d. To the best of the authors knowledge, this study on the new 10-D hyperchaotic system with a Kaplan-Yorke fractal dimension higher than 9 has never been studied by any researcher.

Multistability and coexisting attractors in the new 10D hyperchaotic system
To study the effect of initial criteria on the behaviour of (1), the bifurcation diagrams of (1) versus its three parameters (a, b and c) for six different initial conditions are calculated and plotted. The obtained bifurcation diagrams allow us to examine the phenomena of multistability; this strange occurrence demonstrates system (1)'s extraordinary sensitivity to initial conditions, which is attributable to its extremely complicated dynamics [34]. Let ξ 1 , ξ 2 , ξ 3 , ξ 4 , ξ 5 and ξ 6 be six different initial conditions for the new 10-D hyperchaotic system (1), where:

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Multistability when parameter a varying. Here the BD of (1) with respect to coefficient a is calculated and plotted starting from six different initial points ξ 1 , ξ 2 , ξ 3 , ξ 4 , ξ 5 and ξ 6 . Fix b = 0.1, c = 1.8 and d = 0.01, from the bifurcation diagram, it can be observed that the new 10D system (1) has six different dynamical evolutions when a 2 [0;0.2] as depicted in Fig 11. When a 2 [0;0.04], we can see that system (1) has coexistence of one chaotic attractor starting from and five quasi-periodic attractors as shown in Fig 12(a). Coexistence of four quasiperiodic attractors starting from ξ 2 , ξ 3 , ξ 4 and ξ 3 . and two chaotic attractors starting from ξ 1 and ξ 4 are determined when a 2 [0.05;0.02] as depicted in Fig 12(b). Dynamics, Kaplan-Yorke fractal dimension and Lyapunov exponents for all coexisting attractors when a 2 [0;0.2] are listed in Table 6.
Multistability when parameter b varying. Here system (1) bifurcation diagram with respect to b is calculated and plotted starting from the six different initial points ξ 1 , ξ 2 , ξ 3 , ξ 4 , ξ 5 and ξ 6 . Fix a = 0.1, c = 1.8 and d = 0.01, from the bifurcation diagram, it can be observed that the new 10D hyperhaotic system (1) exhibit six different dynamical evolutions when b 2 [0.1;3] as depicted in Fig 13. When b 2 [0;1.2], we can see that system (1) has coexistence of

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two chaotic attractors starting from ξ 1 and ξ 4 and four periodic attractors starting from the remaining initial points as shown in Fig 14(a). Coexistence of one chaotic attractor starting from ξ 4 and five periodic attractor starting from the other initial conditions are observed when a 2 [0.13;0.19], (see Fig 14(b)). When b 2 [2;3], the new 10-D hyperchaotic system has coexistence of three chaotic attractors starting from ξ 2 , ξ 3 and ξ 4 and three periodic attractors starting from ξ 1 , ξ 5 and ξ 6 as shown in Fig 14(c). Dynamics, Kaplan-Yorke fractal dimension and Lyapunov exponents for all coexisting attractors when b 2 [0.1;3] are listed in Table 7.    Table 8.  https://doi.org/10.1371/journal.pone.0266053.g016 Table 8. Lyapunov exponents, Kaplan-Yorke dimensin and dynamics of system (1) coexisisting attractors with parameter c varying. To the best of the authors knowledge, no research has been done on the new 10-D hyperchaotic chaotic system that exhibiting different coexisting attractors with the variation of its parameters.

Synchronization of the new 10D hyperchaotic system with a set of chaotic systems
This section study the synchronization of the proposed new 10-D hyperchaotic with three diverse Hyperchaotic and chaotic systems via active controllers. One considers a set of three systems as master system. The slave system will be the new 10D Hyperchaotic system. The idea is to synchronize the first three coordinates of the new 10-D hyperchaotic system with coordinates of the 3D system (17), the second three state coordinates of the new 10-D hyperchaotic system will be synchronized with the state coordinates of the 3D system (18). Finally, we will synchronize the last four coordinates of the new 10-D hyperchaotic system with the coordinates of the 4D hyeprchaotic system (19).

The first 3-D chaotic system
This subsection review the 3D chaotic system [35], which has six terms with two nonlinearities and it was given by: 4 11 À gx 13 : Suppose the parameters are represented by e = 3, f = 1 and g = 1 and for the initial conditions (0.1; 0.1; 0.1), then, system (17) exhibits a chaotic behaviour with the following values of Lyapunov exponents: LE 1 = 1.386, LE 2 = 0, LE 3 = −5.386 The phase portraits of the 3D chaotic system (17) are depicted in Fig 17.

The second 3-D chaotic system
This subsection review the 3D chaotic system [36], which has three quadratic nonlinear terms. It was described as follows:

The 4-D hyperchaotic system
This subsection review the 4D hyperchaotic system [37], which has four nonlinear terms and line equilibrium. It was described as follows:

Design of active controllers for synchronization
Design of active controllers is considered in this subsection, in order to synchronize the new 10-D hyperchaotic system (1) and a set of three multidimensional systems (20). One considers

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the following set of chaotic systems (The first 3D chaotic system (17), the second 3D chaotic system (18) and the 4D hyperchaotic system (19) as master system: Then, the new 10-D system is studied as a master system and described as follows: where the functions of the active control to be found is u 1 , u 2 , u 3 , u 4 , u 5 , u 6 , u 7 , u 8 , u 9 and u 10 .
The state errors are defined as e − 1 = x 1 − x 11 , e 2 = x 2 − x 12 , e 3 = x 3 − x 13 , e 4 = x 4 − x 21 , e 5 = x 5 − x 2 , e 6 = x 6 − x 23 , e 7 = x 7 − x 31 , e 8 = x 8 − x 32 , e 9 = x 9 − x 33 , e 10 = x 10 − x 34 From the slave system (21), we subtract master system (20) including the control functions, thus, we obtain error system as follows: Our aim is to design the active control functions, which control the error system to be asymptotically stable; in order to ascertain synchronization between the new 10-D hyperchaotic system (21) and set of systems (20). By choosing the active control functions as the follows: u 10 ¼ px 32 þ x 9 À x 7 À e 10 : The dynamical equations of error system becomes: _ e 9 ¼ À e 9 ; _ e 10 ¼ À e 10 : It can be noted from (24)   hyperchaotic systems. To the best of the authors knowledge, no study has been done to investigates the synchronization of the new 10D hyperchaotic system with the active control strategy. Also, synchronization of the proposed 10D system (1) with a class of low dimensional systems making it very desirable to use in secure communications schemes that need high complexity.

Circuit implementation of the new 10-D system
In order to test system (1) physical feasibility, an equivalent electronic circuit for the new 10-D hyperchaotic system (1) is developed using Multisim 13.0 software as depicted in Fig 21. Using

Conclusion
In this work, a new ten-dimensional hyperchaotic system is first presented; the new system contains four positive parameters and twenty-three terms with two quadratic and a quartic nonlinearities. The new system has many specifics properties, it has three unstable equilibrium points, it can exhibits four different dynamical behaviours (periodic, quasi-periodic, chaos and hyperchaos) for special values of parameters. In addition, the new system may generate many

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coexisting attractors with high fractal dimension when fixing the parameters and changing the initial conditions. Dynamical properties of the new system is investigated using Lyapunov exponents, Kaplan-Yorke dimension, bifurcation diagrams, phase portraits, equilibrium points stability and dissipativity. The idea of synchronizing the new 10-D high dimensional system with a set of three low dimensional system is applied by using active controllers; which guarantee the convergence of the synchronization errors to zero asymptotically. Finally, in order to prove the real feasibility of the new system and the physical existence of the coexisting attractor, an equivalent electronic circuit was designed using Multisim. The obtained results show a good agreement with Matlab results, which confirm the feasibility of both the 10-D system and its dynamical behaviours. We strongly believe that the new 10-D Hyperchaotic system with its high dimension, very complex dynamic and easy to implement circuit schematic can be applied in various chaotic-based applications. The hardware implementations of the new systems along with their applications are considered as the future direction of the work.  (0,0,0,0,0,0,0,0,0,0.5), (0,0,0,0,0,0,0,0,0,±1); (d) periodic attractor with initial point (1,0,0,0,0,0,0,0,0,0). https://doi.org/10.1371/journal.pone.0266053.g023