A Simple Chaotic Flow with Hyperbolic Sinusoidal Function and Its Application to Voice Encryption

: In this article, a new chaotic system with hyperbolic sinusoidal function is introduced. This chaotic system provides a new category of chaotic ﬂows which gives better perception of chaotic attractors. In the proposed chaotic ﬂow with hyperbolic sinusoidal function, according to the changes of parameters of the system, the self-excited attractor and two forms of hidden attractors are occurred. Dynamic behavior of the o ﬀ ered chaotic ﬂow is studied through eigenvalues, bifurcation diagrams, phase portraits

Recently, Wei (2011) announced a chaotic system with no equilibrium point [19]. Jafari et al. (2013) discovered a set of 17 elementary quadratic chaos systems with no equilibrium points [20]. A chaos system possessing a stable equilibrium point was recently found in [21,22]. It is observed that Shilnikov method [23,24] is not applicable to check chaos behavior in special dynamical systems with no equilibrium point or with stable equilibrium points. Such dynamical systems can be viewed as systems with hidden chaotic attractors in scientific computing [24][25][26]. Chaotic systems with hidden attractors can result in unexpected disastrous behavior in mechanical systems and electronic circuits.
It is stimulating that chaotic flows containing infinite number of equilibrium points have achieved much consideration in the past decade. Especially, structures with uncountable equilibrium points

Chaotic Flow with a Hyperbolic Sinusoidal Function
In the search for chaos flows with hyperbolic sinusoidal function, we study the form of a three-dimensional chaotic structure as: . x = a 1 x + a 2 y + a 3 z + a 4 xy + a 5 xz + a 6 yz + b, . y = a 7 xy + a 8 xz + a 9 yz + a 10 sinh(y) + c, . z = a 11 x + a 12 y + a 13 z + a 14 xy + a 15 xz + a 16 yz (1) where x, y and z denote the system states; a 1 , . . . , a 16 indicate the coefficients of the terms; b and c are two scalars which define the chaos behavior.
A computer examination is executed investigating millions of combinations of different forms, various initial states and different constant values, looking for dissipative cases for which the largest Lyapunov exponent is bigger than 0.001. The system is in chaos state if the largest Lyapunov exponent is bigger than zero, and the system is in steady period state if the largest Lyapunov exponent is smaller Symmetry 2020, 12,2047 3 of 18 than zero [38,39]. Therefore, in the present work, a three-dimensional chaotic flow is reported which is specified by: where the parameter a 6 in (1) has been denoted as parameter a in system (2).
Next, three different scenarios depending on the values of system's (2) parameters b, c are discussed in details.

Scenario A: Line of Equilibria
If b = c = 0, the chaotic flow (2) will have a line of equilibria, i.e., E A = [0, 0, z * ] T , where z * is the equilibrium point value in z axis and T means the transpose of the vector. To analyze the state trajectories in the vicinity of the equilibrium point, the Jacobian matrix is obtained from (2) as: For equilibrium point E A = [0, 0, z * ] T , the Jacobian matrix is defined as Therefore, the eigenvalues of the linearized system are achieved as The equilibrium is a saddle node for − 1 √ a < z * < 1 √ a . The equilibrium point is an unstable node for z * = ± 1 √ a . For the values z * > 1 √ a and z * < − 1 √ a , since one eigenvalue is zero and two eigenvalues are imaginary, the stability of the equilibrium point cannot be determined by this method; the equilibrium point may be stable, unstable or marginally stable.
If one design parameter is varied and the norm of the state variables vector is plotted for finding the fixed points of the system versus the changing parameter, finally the bifurcation diagram is obtained [40]. In the bifurcation diagrams, the fixed points maybe disappear, appear, or change their stability nature when the design parameter is changed. Those variations may occur even for infinitesimal changes in the parameter. Bifurcation diagram is used for the stability analysis of a dynamical system [41,42]. Moreover, the Lyapunov exponents spectrum makes it possible to qualitatively quantify a local property with respect to the attractor's stability. The positive/negative values of the Lyapunov exponents can be observed as a measure of the averaged exponential divergence/convergence of neighborhood trajectories [43,44]. The bifurcation diagram for y, when the states cut the plane z = 0 with dz/dt < 0, as well as the spectrum of system's Lyapunov exponents (LE i , i = 1, 2, 3), by varying the value of a to explore the dynamical form of system (2), while keeping the initial states as [x 0 , y 0 , z 0 ] = [2, 0.2, 1], are depicted in Figure 1. Therefore, the suggested structure (2) is integrated via the classical Runge-Kutta integration algorithm [45], numerically. For all of the parameters, the simulation calculations are executed via the parameters and variables in extended precision mode. In addition, the spectrum of the Lyapunov exponents are found via the Wolf's algorithm [46].  The dynamics (2) shows a chaotic attractor, for a = 2 (Figure 2), and a limit cycle of Period-1, for a = 2.35 ( Figure 3). The spectrum of Lyanpunov exponents (Figure 1b) approves the dynamic behavior of the system as it has been revealed via bifurcation diagram.

Scenario B: No Equilibrium Point
If b 0, c = 0 and by keeping a = 2, for obtaining the equilibrium point, we solve .
Consequently, the chaotic flow has no equilibrium point in this case. Therefore, it belongs to the category of chaotic systems containing hidden attractors.
Taking the bifurcation diagram of y (Figure 4a), along with the Lyapunov exponents spectrum (Figure 4b) by changing b for 0 < b < 0.005, in order to explore the dynamics (2), for initial conditions [x 0 , y 0 , z 0 ] = [2, 0.2, 1], interesting dynamical behavior has been investigated. As it is obtained from bifurcation diagram (Figure 4a), the system passes from a chaotic region, for b ∈ [0, 0.075), to a periodic one as the parameter b increases.

Scenario B: No Equilibrium Point
If ≠ 0, = 0 and by keeping a = 2, for obtaining the equilibrium point, we solve = 0, = 0 and = 0, that is

Scenario B: No Equilibrium Point
If ≠ 0, = 0 and by keeping a = 2, for obtaining the equilibrium point, we solve = 0, = 0 and = 0, that is  The strange attractors of the system (2) are displayed for a = 2, = 0.005 and = 0 in Figure  5. In this case, the Lyapunov exponents are LE1 = 0.14107, LE2 = 0, LE3 = −1.33835, which confirmed the chaotic behavior of the system (2). The Kaplan-York dimension of the chaotic flow is DKY = 2.1054. Besides, the Poincaré map in x-y plane presents the folding properties of chaos when z = 0 with dz/dt < 0 ( Figure 6).    The strange attractors of the system (2) are displayed for a = 2, b = 0.005 and c = 0 in Figure 5. In this case, the Lyapunov exponents are LE 1 = 0.14107, LE 2 = 0, LE 3 = −1.33835, which confirmed the chaotic behavior of the system (2). The Kaplan-York dimension of the chaotic flow is D KY = 2.1054. Besides, the Poincaré map in x-y plane presents the folding properties of chaos when z = 0 with dz/dt < 0 ( Figure 6).
The strange attractors of the system (2) are displayed for a = 2, = 0.005 and = 0 in Figure  5. In this case, the Lyapunov exponents are LE1 = 0.14107, LE2 = 0, LE3 = −1.33835, which confirmed the chaotic behavior of the system (2). The Kaplan-York dimension of the chaotic flow is DKY = 2.1054. Besides, the Poincaré map in x-y plane presents the folding properties of chaos when z = 0 with dz/dt < 0 ( Figure 6).

Scenario C: Self-Excited Attractor
If = 0, ≠ 0 and a = 2 this chaotic flow has only one equilibrium = [0, ℎ ( ),0] . For the equilibrium point , the Jacobian matrix is found as: Then, the eigenvalues of the linearized chaotic flow are obtained as:

Scenario C: Self-Excited Attractor
If b = 0, c 0 and a = 2 this chaotic flow has only one equilibrium For the equilibrium point E C , the Jacobian matrix is found as: Then, the eigenvalues of the linearized chaotic flow are obtained as: For c > 0.1253, the eigenvalues of the chaotic flow are found as λ = − √ 1 + c 2 , 1±iω 2 , and the equilibrium point is a saddle focus. For c < 0.1253, the eigenvalues of the chaotic flow are obtained as √ ∆ 2 , and the equilibrium point is a saddle node. Figure 7 depicts the bifurcation diagram of variable y as well as the spectrum of Lyapunov exponents by varying c, for 0 < c < 0.05, to explore the dynamics (2), for initial states [x 0 , y 0 , It is shown from bifurcation diagram (Figure 7a) that the system passes from a chaotic region for c ∈ [0, 0.0285) to a periodic one as the parameter c increases. The respective spectrum of Lyapunov exponents to parameter c displays the aforementioned system's (2) dynamical behavior for a = 2 and b = 0.

Circuit Design of the Proposed Chaotic Flow
In recent years, the physical realizations of theoretical chaos forms have been investigated extensively for approving the feasibility and employing them in practical usages [47][48][49][50]. Therefore, in this section, a circuit realization with the hyperbolic sinusoidal nonlinearity is presented. For the reason of easiness, the general design methodology is applied according to the operational amplifiers [51,52]. The circuit is designed by using the common electronic components as displayed in Figure  10. There are an inverting amplifier (U4), three integrators (U1-U3), and two analog multipliers (U7, U8) of type AD633. The circuit for simulating the hyperbolic sinusoidal nonlinearity, in the dotted frame, includes three resistors (RS1-RS3), two operational amplifiers (U5, U6) and two diodes (D1, D2).
Based on Figure 8, via the Kirchhoff's laws, the circuital equation of the circuit is found as where IS, n and VT are diode's reverse bias saturation current, the diode's ideality factor, and the thermal voltage, correspondingly. Normalizing the Equation (9) with τ = t/RC, the dimensionless structure can be designated by The variables (X, Y, Z) are equivalent to output voltages of integrators (U1-U3), when the power supply is ±15 VDC. The system (10) corresponds to the suggested system with the hyperbolic sinusoidal nonlinear function (2). The electronic components are selected for a = 2, and b = c = 0; then we have R = R4 = 30 kΩ, R1 = 15 kΩ, R2 = 10 kΩ, R3 = 90 kΩ, RS1 = 100 kΩ, RS2 = 50.66 kΩ, RS3 = 18.65 MΩ and C = 10 nF. The planned circuit of Figure 10 has been executed in Multisim, and some PSpice results are presented in Figure 11. One can obviously confirm the consistency of the simulations ( Figure 11) and numerical outcomes (Figure 2).

Circuit Design of the Proposed Chaotic Flow
In recent years, the physical realizations of theoretical chaos forms have been investigated extensively for approving the feasibility and employing them in practical usages [47][48][49][50]. Therefore, in this section, a circuit realization with the hyperbolic sinusoidal nonlinearity is presented. For the reason of easiness, the general design methodology is applied according to the operational amplifiers [51,52]. The circuit is designed by using the common electronic components as displayed in Figure 10. There are an inverting amplifier (U4), three integrators (U1-U3), and two analog multipliers (U7, U8) of type AD633. The circuit for simulating the hyperbolic sinusoidal nonlinearity, in the dotted frame, includes three resistors (R S1 -R S3 ), two operational amplifiers (U5, U6) and two diodes (D 1 , D 2 ).
Based on Figure 8, via the Kirchhoff's laws, the circuital equation of the circuit is found as where I S , n and V T are diode's reverse bias saturation current, the diode's ideality factor, and the thermal voltage, correspondingly. Normalizing the Equation (9) with τ = t/RC, the dimensionless structure can be designated by The variables (X, Y, Z) are equivalent to output voltages of integrators (U1-U3), when the power supply is ±15 V DC . The system (10) corresponds to the suggested system with the hyperbolic sinusoidal nonlinear function (2). The electronic components are selected for a = 2, and b = c = 0; then we have R = R 4 = 30 kΩ, R 1 = 15 kΩ, R 2 = 10 kΩ, R 3 = 90 kΩ, R S1 = 100 kΩ, R S2 = 50.66 kΩ, R S3 = 18.65 MΩ and C = 10 nF. The planned circuit of Figure 10 has been executed in Multisim, and some PSpice results are presented in Figure 11. One can obviously confirm the consistency of the simulations ( Figure 11) and numerical outcomes (Figure 2). Symmetry 2020, 12, x FOR PEER REVIEW 10 of 18

Voice Encryption Algorithm Design and Its Analysis
Chaos-based cryptography is one of the topics that has been intensively studied in recent years because of the randomness and rich dynamics of chaotic systems [53][54][55][56][57][58]. These works usually focus on image encryption. A new RNG algorithm design is performed by using the developed chaotic form and National Institute of Standards and Technology (NIST) 800-22 [59] randomness tests are employed to study the randomness of the obtained random numbers. It is noted that the NIST statistical test suit is used for the evaluation of the advanced encryption standard candidate algorithms. At first, a voice encryption algorithm is proposed using the obtained random number. Then, the voice encryption is executed by using the proposed algorithm and frequency spectrum analysis of the encryption procedure is executed.

RNG Algorithm Design and NIST 800-22 Test Results
In this subsection, an RNG algorithm design is developed via the newly introduced chaos in order to obtain the random numbers to be employed in the algorithm. The design process of the RNG algorithm is carried out as exposed in Algorithm 1 (see Appendix A). In the design process of the RNG algorithm, firstly the initial states and parameters of the chaotic system are defined. Then, the sampling interval of the system is determined and the chaotic system is considered by using fourthorder Runge-Kutta (RK-4) integration algorithm using this sampling value. As an outcome of the system analysis, float values are found for each cycle from each phase. On the float values obtained Figure 11. PSpice chaotic attractors of system with hyperbolic sinusoidal nonlinearity in (a) X-Y plane, (b) X-Z plane, and (c) Y-Z plane (x: 2 V/Div, y: 2 V/Div).

Voice Encryption Algorithm Design and Its Analysis
Chaos-based cryptography is one of the topics that has been intensively studied in recent years because of the randomness and rich dynamics of chaotic systems [53][54][55][56][57][58]. These works usually focus on image encryption. A new RNG algorithm design is performed by using the developed chaotic form and National Institute of Standards and Technology (NIST) 800-22 [59] randomness tests are employed to study the randomness of the obtained random numbers. It is noted that the NIST statistical test suit is used for the evaluation of the advanced encryption standard candidate algorithms. At first, a voice encryption algorithm is proposed using the obtained random number. Then, the voice encryption is executed by using the proposed algorithm and frequency spectrum analysis of the encryption procedure is executed.

RNG Algorithm Design and NIST 800-22 Test Results
In this subsection, an RNG algorithm design is developed via the newly introduced chaos in order to obtain the random numbers to be employed in the algorithm. The design process of the RNG algorithm is carried out as exposed in Algorithm 1 (see Appendix A). In the design process of the RNG algorithm, firstly the initial states and parameters of the chaotic system are defined. Then, the sampling interval of the system is determined and the chaotic system is considered by using fourth-order Runge-Kutta (RK-4) integration algorithm using this sampling value. As an outcome of the system analysis, float values are found for each cycle from each phase. On the float values obtained from each phase, the step values of the decimal parts after the comma are subjected to the mode 2 operation. As a result, 15 bits are generated from each phase in the each cycle. Further, these obtained values are added to the number sequences for each phase (rngx, rngy, rngz). This process continues for each number sequence until the 1 M. bit is generated for the NIST 800-22 randomness examinations. Because at least 1 M. bit is needed for NIST 800-22 tests. After 1 M bits are generated from each phase, the phases are subjected to Exclusive Or (XOR) processing in binary form and new random number sequences are generated as named rngxy, rngxz, rngyz, rngxyz in the Algorithm 1. The generated values from the x and y phases are subjected to XOR processing to obtain a rngxy random number sequence. Similarly, generated from the phases y and z for rngyz, the x and z phases for rngyz and the x, y, z phases for rngxyz are subjected to XOR processing. Finally, NIST 800-22 tests are employed to all obtained random bit sequences. When random bit sequences are tested singularly, they cannot pass some tests. For this reason, random bit sequences generated are subjected to 2 or 3 XOR operations.
For the safe use of random numbers, they must have an appropriate randomness. The NIST 800-22 tests are a set of internationally accepted and frequently used tests in the literature that define the numbers' randomness via a variety of different tests. The NIST 800-22 test outcomes for the random number sequences originated from the developed RNG algorithm are displayed in Table 1. According to the test outcomes, it is seen that all the random numbers created passed all the examinations.

Voice Encryption Algorithm Design and Its Application
A new voice encryption algorithm is developed via RNG algorithm introduced in the previous section, voice encryption usage and its analysis are performed. The block diagram of the encryption process is presented in Figure 12. In the encryption algorithm, after entering initial states and parameters of chaotic system, these values are transmitted to the receiving side as a key for generation of the random number sequences to be employed in the decryption process. In order to realize the bit-based encryption, the values are obtained from the voice file consisting of float values with appropriate sampling step and converted into binary form. With random bit sequences past all NIST 800-22 randomness tests obtained from the RNG, the voice file in the binary form is encrypted. XOR operation is used in encryption process. After the encryption operation, the encrypted binary bit array is changed to the float form to generate the encrypted voice file. After the encrypted voice data is sent to the receiver side in this way, the decryption process is performed by applying the reverse operations in the encryption. Thus, the original voice file is attained on the receiving side. Symmetry 2020, 12, x FOR PEER REVIEW 13 of 18 The voice files in encryption procedure are shown in Figure 13. The original, encrypted and decrypted voice file is demonstrated in Figure 13a-c, respectively. When comparing the original and encrypted voice file in Figure 13a,b; it is seen that a very different file is gotten than the original and the encryption process is successful. When Figure 13a,c are examined, it is observed that the decryption procedure is successful. Figure 14 shows the frequency spectrum analysis results of the encryption process. Frequency spectrum analysis is carried out to determine the frequency range of voice files. To determine the success of the encryption process, frequency spectrum analyzes are performed on original and encrypted voice files. The spectrum analysis results of original and encrypted voice files are illustrated in Figure 14a,b. If we compare these two graphs, it seems that the encrypted voice file has a rather wide frequency spectrum range than the original. When these results are evaluated, it shows the success of the encryption process.  The voice files in encryption procedure are shown in Figure 13. The original, encrypted and decrypted voice file is demonstrated in Figure 13a-c, respectively. When comparing the original and encrypted voice file in Figure 13a,b; it is seen that a very different file is gotten than the original and the encryption process is successful. When Figure 13a,c are examined, it is observed that the decryption procedure is successful. Figure 14 shows the frequency spectrum analysis results of the encryption process. Frequency spectrum analysis is carried out to determine the frequency range of voice files.
To determine the success of the encryption process, frequency spectrum analyzes are performed on original and encrypted voice files. The spectrum analysis results of original and encrypted voice files are illustrated in Figure 14a,b. If we compare these two graphs, it seems that the encrypted voice file has a rather wide frequency spectrum range than the original. When these results are evaluated, it shows the success of the encryption process. The voice files in encryption procedure are shown in Figure 13. The original, encrypted and decrypted voice file is demonstrated in Figure 13a-c, respectively. When comparing the original and encrypted voice file in Figure 13a,b; it is seen that a very different file is gotten than the original and the encryption process is successful. When Figure 13a,c are examined, it is observed that the decryption procedure is successful. Figure 14 shows the frequency spectrum analysis results of the encryption process. Frequency spectrum analysis is carried out to determine the frequency range of voice files. To determine the success of the encryption process, frequency spectrum analyzes are performed on original and encrypted voice files. The spectrum analysis results of original and encrypted voice files are illustrated in Figure 14a,b. If we compare these two graphs, it seems that the encrypted voice file has a rather wide frequency spectrum range than the original. When these results are evaluated, it shows the success of the encryption process.

Conclusions and Discussion
In this article, a new chaotic system with hyperbolic sinusoidal nonlinearity is designed. The proposed system belongs to a new category of dynamical systems with hidden chaotic flows, which assist in further understanding of chaotic attractors and also to use them in interesting applications like cryptography and secure communication schemes. The feature of hidden chaotic attractors, such as in systems with line of equilibria or in systems with no equilibrium point, makes them more suitable for the aforementioned applications, due to the fact that using systems with hidden attractors adds complexity to the dynamical system, which is used in this kind of applications. Therefore, in this work a voice encryption scheme, which is based on the specific systems was studied. Based on the variations of parameters of the system, this flow presented two classes of hidden attractors (with line of equilibria and no equilibrium point) plus a self-excited attractor, which has been reported to the literature for the first time. Dynamical behavior of the proposed system was explored and its bifurcation diagram and spectrum of Lyapunov exponents were propounded. For the appropriate selection of the parameters, the flow could display periodic oscillations and double-scroll chaos attractors. The system's electronic simulation investigated the confirmation of the double-scroll chaos attractor in real word. Via the proposed chaotic system, a novel RNG design was realized and random number generation was performed. NIST 800-22 randomness examinations were employed to the produced numbers and it was determined that all tests passed. By using RNG design, a novel voice encryption algorithm was established and encryption process was done. Frequency spectrum analysis of the voice encryption procedure was executed. In line with the analysis outcomes, it has been found that the new RNG design produces high random numbers and that the suggested encryption algorithm effectively achieves the encryption process. Therefore, authors aim with this work to attract the interest of the research community in the use of chaotic dynamical systems with hidden attractors in encryption schemes, as the results are proved to be very promising. Finally, as a future plan, the hardware implementation of the specific approach has been planned.
Author Contributions: S.M. was responsible for formal analysis and writing-original draft preparation. C.K.V. and Ü.Ç. were responsible to conceptualization, methodology, and software. S.K. was responsible for software, supervision and editing. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.

Conflicts of Interest:
The authors of this paper declare that they have no conflicts of interest.
Date Availability: The MATLAB files data used to support the findings of this paper are available from the corresponding author upon request.

Conclusions and Discussion
In this article, a new chaotic system with hyperbolic sinusoidal nonlinearity is designed. The proposed system belongs to a new category of dynamical systems with hidden chaotic flows, which assist in further understanding of chaotic attractors and also to use them in interesting applications like cryptography and secure communication schemes. The feature of hidden chaotic attractors, such as in systems with line of equilibria or in systems with no equilibrium point, makes them more suitable for the aforementioned applications, due to the fact that using systems with hidden attractors adds complexity to the dynamical system, which is used in this kind of applications. Therefore, in this work a voice encryption scheme, which is based on the specific systems was studied. Based on the variations of parameters of the system, this flow presented two classes of hidden attractors (with line of equilibria and no equilibrium point) plus a self-excited attractor, which has been reported to the literature for the first time. Dynamical behavior of the proposed system was explored and its bifurcation diagram and spectrum of Lyapunov exponents were propounded. For the appropriate selection of the parameters, the flow could display periodic oscillations and double-scroll chaos attractors. The system's electronic simulation investigated the confirmation of the double-scroll chaos attractor in real word. Via the proposed chaotic system, a novel RNG design was realized and random number generation was performed. NIST 800-22 randomness examinations were employed to the produced numbers and it was determined that all tests passed. By using RNG design, a novel voice encryption algorithm was established and encryption process was done. Frequency spectrum analysis of the voice encryption procedure was executed. In line with the analysis outcomes, it has been found that the new RNG design produces high random numbers and that the suggested encryption algorithm effectively achieves the encryption process. Therefore, authors aim with this work to attract the interest of the research community in the use of chaotic dynamical systems with hidden attractors in encryption schemes, as the results are proved to be very promising. Finally, as a future plan, the hardware implementation of the specific approach has been planned.
Author Contributions: S.M. was responsible for formal analysis and writing-original draft preparation. C.K.V. and Ü.Ç. were responsible to conceptualization, methodology, and software. S.K. was responsible for software, supervision and editing. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.

Conflicts of Interest:
The authors of this paper declare that they have no conflicts of interest.
Date Availability: The MATLAB files data used to support the findings of this paper are available from the corresponding author upon request.