Secure Communication Based on Hyperchaotic Underactuated Projective Synchronization

This paper proposes a scheme for secure telecommunication based on the projective synchronization of hyperchaotic systems. The design of the synchronizer considers the presence of disturbances to increase the robustness of the method. The main advantage of the proposed approach lies in that only two control inputs are required to synchronize the master and slave systems. Hence, the control structure is simple, which, in turn, simplifies the applications. Based on Lyapunov theory, the proposed approach ensures the finite-time convergence of the synchronization error to a small neighborhood of the origin, even in the presence of disturbances in all states. Simulations to both show the advantages and applications of the proposed approach were also accomplished to validate the theory.


I. INTRODUCTION
In the past few years, significant progress has been made in the study of chaotic systems. Chaotic systems are deterministic nonlinear systems that show sensitive dependence on initial conditions and have an aperiodic behavior [1]. A necessary condition for a system to be chaotic lies in that at least one Lyapunov exponent be positive [2]. The chaotic motion on a strange attractor was first discovered in the 60s by Lorenz [3]. In the following years, some relevant chaotic models, such as those by Rössler [4], Chen [5], Sprott [6], and Lü [7], were introduced. More recently, many works have been proposed in the literature. See, for example, [8]- [11].
On the other hand, a system can exhibit hyperchaos when at least two of their associated Lyapunov exponents are positive, and its dimension is higher than three [12]. Hyperchaos was initially introduced in 1979 by Rössler [13]. Since then, other important models have been proposed. Refer to [2], [14]- [26] and the references therein to quote a few.
Synchronization lies in adjusting the dynamic behavior of two dynamic systems, known as master (drive) and slave (response), so that their trajectories converge in time. The synchronization of chaotic systems was first introduced in 1990 [56]. Several classes of synchronization have been proposed since then: antisynchronization (AS) [57], lag synchronization (LS) [58], projective synchronization (PS) [16], [59]- [63], modified projective synchronization (MPS) [61], and function projective synchronization FPS [64]. In general, the synchronization type depends on a scaling factor. For instance, PS is characterized by a constant proportional synchronization between the master and slave systems. Hence, identical and AS are particular cases of this kind of synchronization with scaling factors 1 or -1. Matrix and functional scaling factors define MPS and FPS, respectively. However, most of the synchronization works above are only valid under either fully-actuated control or matching condition [65].
In summary, the synchronization of either chaos or hyperchaos is characterized by two main hypotheses: the control dimension and system order are equal [9], [17], [19], [20], [39], [61], [63], [66], [69], [70] and unknowns are not considered in the stability analysis [11], [17], [19], [20], [23], [26], [60], [66]- [68]. The former is related to the complexity of the synchronization scheme and the latter to the robustness of the method. Synchronization of underactuated hyperchaotic systems is rarely found in the literature. Also, to the best of our knowledge, the robust projective synchronization of underactuated hyperchaotic systems is not present in the literature. It should be noted that underactuation is a condition defined by a higher number of independent variables than control signals [71]. Thus, underactuated systems have fewer actuators than degrees of freedom [72]- [75], i.e., the control dimension is lesser than the state dimension. The main significance of the underactuated projective synchronization is the reduction of actuators in diverse applications.
Motivated by the previous facts, this work presents a robust scheme for projective synchronization of a hyperchaotic system to overcome the aforementioned drawbacks. Hence the proposed approach is based on both Lyapunov theory (to ensure boundedness and finite-time convergence) and underactuated control law (to simplify the application). More specifically, this paper presents the following contributions.
It should be noted that a simple structure leads to easier implementations. The consideration of disturbances in the stability analysis aim at robustness against disturbances which are inevitable in actual applications. To the best of our knowledge, this is the first time that a robust underactuated projective synchronization method for hyperchaotic systems is proposed in the literature.
The work is organized as follows. In Section II, the problem and main assumptions are introduced. The synchronization error is presented in Section III. In Section IV, a control law, which ensures that the synchronization errors are bounded and finite-time convergent, is proposed. Section V is concerned with application of the proposed method in secure telecommunication. In Section VI, the development of an electronic circuit for the implementation of the proposed method is accomplished, and a comparison study with another work in the literature is performed. Finally, the conclusions of the paper are drawn in Section VII.

II. PROBLEM STATEMENT
Consider the following master hyperchaotic system [20]: Based on (1), a perturbed slave hyperchaotic system can be defined as where x m , y m , z m , and w m are the state variables of the master system (1); x s , y s , z s , and w s are the state variables of the slave system (2); h 1 , h 2 , h 3 , and h 4 are the disturbances; u 1 and u 2 are the control signals. The system parameters satisfy a ∈ [10,25] The aim of this work lies in the synchronization of (1) and (2) by using an underactuated control scheme, for any initial condition, even in the presence of unmatched disturbances.
Assumption 1: It is assumed that the disturbances in (2) are bounded. More specifically, Remark 1: It is noteworthy that systems (1) and (2) are, in general, different, due to the presence of disturbances. Besides, our approach assumes that all these disturbances are not in the control span. Hence, the control signal can not be used straightforwardly to tackle with them. However, we will exploit the master boundedness and the particular structure of (1)-(2), in the next sections, to devise a simple and robust control law.
Remark 2: Note that Equation (1) does not have a disturbance term. That is no loss of generality because any disturbance in (1) would join with those in (2) in the stability analysis.

III. PROJECTIVE SYNCHRONIZATION ERROR
In this section, the main error associated with the synchronization problem is defined.
The projective synchronization error is defined as where δ is a nonzero constant defined by the user. Based on (1)-(2), the time-derivative of (5) resultṡ

IV. LYAPUNOV STABILITY ANALYSIS
After the formulation of the synchronization error equations, the next step is selecting a control law. In what follows, we consider a standard Lyapunov function candidate V , which hinges upon the synchronization error, and choose a control law to makeV lower than zero outside a compact region at the origin Ω. The key drivers for the design are then the peculiar structure of (2), boundedness of (1), and an enlargement process for constrainingV to be negative definite outside Ω. Theorem 1: Consider the master and slave systems (1)-(2), Assumption 1, and the control laws  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Figure 5). U1 and U4 are generated by the CONTROL block ( Figure 2). X_M and X_S are the X state blocks ( Figure 6). Y_M and Y_S are the Y state blocks (Figure 7). Z_M and Z_S are the Z state blocks (Figure 8). W_M and W_S are the W state blocks (Figure 9). +XZ_M and +XZ_S are the inversion blocks ( Figure 3). -XZ_M, -XZ_S,-XY_M, and -XY_S are the multiplication blocks ( Figure 4). V1 to V12 are voltage to signal converters and the S1 and S2 are signal to voltage converters of the Simulink.
If VOLUME X, 2020 Then, the synchronization error is ultimately bounded. It converges in finite-time to the compact set where the convergence time holds and 1 + e 2 2 + e 2 3 + e 2 4 , and σ i , i = 1, ..., 11 are positive constants. Proof: Consider the following Lyapunov function candidate The time-derivative of (11) resultṡ V = e 1ė1 + e 2ė2 + e 3ė3 + e 4ė4 By replacing (6) in (12), we havė Also, by employing (7) in (13) On the other hand, from Fact 1, Assumption 1, and Young's inequality, it follows that Based on (15), (14) implieṡ (16) By using (8), (16) can be written aṡ Then, one can see thatV < 0 in (17) when e ∈ Ω T . Since Ω is a compact set, the errors starting inside Ω will remain there forever. In case that the errors start outside Ω, it can be seen thatV < 0, then V and, consequently, e will decrease monotonically until the errors enter Ω at some finite time t max .
To determine t max , it should be noted that (17) which further yields Therefore, (19) shows that the synchronization error converges to the compact set Ω at least in a finite-time t max and, hence ( [79]), that the this error is ultimately bounded.

Remark 3:
The main idea of the proposed method lies in the usage of the system structure and Lyapunov theory to design an adequate control law. Based on a trial-and-error procedure, we considered all possibilities of underactuated control in the analysis and the simplest one was chosen. Young's inequality was used in the stability analysis in the process to makeV < 0 outside of a small compact set, whose size can be decreased by increasing the control gains, even in the presence of bounded unmatched disturbances.
Remark 4: Integrator backstepping [80], [81] and sliding mode [26], [82] are also used in the control of underactuated systems. However, most works based on backstepping suffer from "the explosion of complexity," and, in general, the presence of matched disturbances is assumed. See, for example, [80], [81]. On the other hand, sliding mode control suffers from chattering, and the disturbances are also supposed to be matched [81]. Besides, most of synchronizers based on hyperchaotic systems found in the literature employ complete actuation [8], [53], [83], [84]. Then, the main peculiarity of our work, in contrast to the literature, thus lies in that neither matching condition nor fully-actuated control is assumed.
Remark 5: Note that the performance of the proposed method, as far as the residual synchronization error is considered, is affected by the control gains ψ 1 and ψ 3 , scaling factor δ, disturbances, and upper bounds for the states of the master system, as can be seen from (8) and the definitions below (10). In general, the performance for the actuated states can be arbitrarily enhanced by increasing ψ 1 and ψ 3 . For non-actuated states, although there is a complex relationship between β and ρ in (9), the performance can also be indirectly controlled through ψ 1 and ψ 3 since higher control gains can allow for higher ρ in (9), as can be seen from the stability analysis.

V. CHAOS-BASED SECURE COMMUNICATION
In addition to the projective synchronization case, we also consider the application of the proposed method to secure telecommunication. To have a well-posed problem, the following assumption must be imposed. VOLUME X, 2020    Assumption 2: It is assumed that the messages are bounded. More specifically, Further, motivated by [38], it can be defined wherem 1 ,m 2 ,m 3 , andm 4 are the decoded messages.
On the other hand, by using (22) and definingm i = m i − δm i , i = 1, ..., 4, wherem 1 ,m 2 ,m 3 , andm 4 are the message errors, it can be concluded that Remark 6: It is worth noticing that the quality of the message reconstruction is the same as the synchronization, as shown in (23). Furthermore, the boundedness of the message error is assured when the synchronization error is bounded. Figure 10 depicts a typical performance of the synchronization using electronic circuits. The synchronization errors are small even when underactuated control and unmatched perturbations resulting from the non-ideal behavior of the devices are considered. Based on the simulations, we can conclude that the proposed method is easily implementable by analog electronics, and has good performance, as far as the synchronization error is considered. Figure 11 shows the performance of the encryption of two message signals using the proposed synchronization method. As expected from theoretical analysis, the recovered signals converge over time close to the original ones, even in the presence of internal or external perturbations, which are due to the usage of non-ideal devices in a realistic scenario in our simulations.

VI. SIMULATIONS
Simulations were performed using Matlab/Simulink 2020b on a Windows 10 platform, with AMD Ryzen 7 1700 processor. All the scripts to reproduce the results of this paper are available with the corresponding author under request. In all simulations, it was considered that δ = 1.

A. IMPLEMENTATION EXAMPLE
For implementation purposes, we consider a = 10, b = 47, c = 1, d = 1, and k = 2. It was also necessary to scale both the frequency and amplitude of the hyperchaotic systems (1)- (2) to decrease the transient and hold the operating voltage of the devices. The system amplitude was then decreased up to 20 times, and the system rate was increased by a factor of 1000. Hence, the scaled systems were rewritten in a VOLUME X, 2020 condensed form as where I * = {0, 1}, u 1 and u 4 are defined as in (7) being ψ 1 = ψ 3 = 100 and ψ 2 = 10000. Figure 1 shows the circuit diagram designed by using Simscape/Simulink being their blocks detailed in Figures 2 -9.
For the sake of conciseness, (24) was considered as being the transmitter when I * = 0 (X, Y , Z, and W must be substituted by X m , Y m , Z m , and W m ) and the receiver when I * = 1 (in which case X, Y , Z, and W must be substituted by X s , Y s , Z s , and W s ).
The encrypted signals are defined as s 2 = m 2 + Y m , s 3 = m 3 + Z m and the decrypted signals asm 2 = s 2 − Y s , m 3 = s 3 − Z s . This definition was adopted to use the same circuital structure for simple encryption and decryption. The considered messages were m 2 as being a sequence of bits and Comparison between the proposed method and that in [20].
were also considered. Further, all capacitors and resistors used here were non-ideal with a tolerance of 0.1% to better reproduce a more realistic scenario.
Also, an unmatched disturbance was introduced at t = 3s to check the robustness of the proposed approach when it is compared with that in [20]. The used disturbance was defined as h 1 = 0.3 · exp(5 · 10 −5 ·x 2 s ) h 2 = 0.2 · exp(10 −5 ·y 2 s ) h 3 = 3sin(3t) + cos(20t) h 4 = 5sin(10t) + 10cos(t) (25) Three systems were simulated: the master (1), slave (2), using an underactuated control, and the slave system in [20], in which a full-actuated control is employed. From simulations shown in Figure 12, it can be seen that the performance of the proposed scheme is similar to that in [20]. However, the proposed approach is simpler than that in [20] since the proposed control is only used in the x s and w s channels.

VII. CONCLUSION
An underactuated scheme for projective synchronization of hyperchaotic systems based on Lyapunov theory has been proposed in this work. The main advantages of the proposed synchronization method are its simplicity, low cost of implementation, and its ability to tackle unmatched disturbances. It has been further employed in chaos-based secure communication to depict the performance of a typical application. With the use of electronic circuits, the ease of implementation has been shown as well. It should be noted that the decoded signal is close to the original, as expected, even in the presence of unmatched disturbances and underactuated control.
Potential drawbacks of the work must mention the need to use two control signals instead of just one. This is due to the topology of the considered hyperchaotic systems.
Future works will include the underactuated synchronization of hyperchaotic systems of high dimension by using online approximators such as neural networks and fuzzy systems.

COMPLIANCE WITH ETHICAL STANDARDS
Conflict of interest: The authors declare that there are no conflict of interest present in this study.