Finite-time generalized synchronization of non-identical fractional order chaotic systems and its application in speech secure communication.

This paper focuses on the finite-time generalized synchronization problem of non-identical fractional order chaotic (or hyper-chaotic) systems by a designing adaptive sliding mode controller and its application to secure communication. The effects of both disturbances and model uncertainties are taken into account. A novel fractional order integral sliding mode surface is designed and its stability to the origin is proved in a given finite time. By the aid of the fractional Lyapunov stability theory, a robust controller with adaptive update laws is proposed and its finite-time stability for generalized synchronization between two non-identical fractional-order chaotic systems in the presence of model uncertainties and external disturbances is derived. Numerical simulations are provided to demonstrate the effectiveness and robustness of the presented approach. All simulation results obtained are in good agreement with the theoretical analysis. According to the proposed generalized finite-time synchronization criterion, a novel speech cryptosystem is proposed to send or share voice messages privately via secure channel. Security and performance analyses are given to show the practical effect of the proposed theories.

1 Introduction

Chaos synchronization between two identical or non-identical systems is a fascinating problem in nonlinear sciences. Since the pioneering work of Pecora and Carroll [1], the synchronization problem has been widely studied in various fields of science and engineering such as finance system, mechanical systems, power system, encryption, secure communications, and etc. In recent years, the synchronization problem between fractional-order chaotic systems has raised great attentions for its potential applications, especially in cryptography and secure communication. At present, there are various types of fractional-order synchronization, for example, complete synchronization [2], lag synchronization [3], anti-synchronization [4], impulsive synchronization [5], projective synchronization [6], and generalized synchronization [7]. Among all kinds of synchronizations, the generalized synchronization [8] between the drive system and the response system characterized by two optional functions could obtain desired types in practice applications. Particularly, it can be used to extend the coexistence of different synchronization types. Very recently, the generalized synchronization between two dynamical systems with different dimensions has been studied in [9-11]. Ouannas et al. [12] explored the coexistence of different synchronization types of fractional-order chaotic systems with different dimensions. Wang et al. [13] reported the synchronization between non-identical fractional-order chaotic and hyper-chaotic systems with different orders. Golmankhaneh et al. [14] reported the study of synchronization in non-identical fractional-order chaotic systems.

Note that, the synchronization in non-identical fractional-order chaotic systems can obtain more flexible response mechanism. In practical applications, the mismatched parameters and the uncertainties of master system and slave system are unavoidable. Thus, it is essential to consider and analyze the uncertainties and disturbances. Furthermore, lots of scholars have studied the synchronization method for chaotic systems with different uncertainties, such as the linear feedback method [9], adaptive-feedback scheme [10], adaptive fuzzy approach [11], back-stepping strategy [15], sliding mode control(SMC) [16,17] and adaptive sliding mode control. Amongst these methods, SMC achieved a fast convergence performance and high robustness against the system uncertainties and external disturbances. The generalized robust synchronization approach for mismatched fractional order dynamical systems with different dimensions via sliding mode control was investigated in [13,18]. As we all known, SMC usually assumes the upper bound of the system uncertainties in advance. Nevertheless, in practice, the upper bound may not be exactly known because of the complexity of uncertainties. Therefore, an adaptive mechanism combining the superiority of SMC has been proposed to estimate the unknown bounds of the system uncertainties. In [19], the authors investigated sliding mode synchronization of multiple uncoupled integer order chaotic systems with uncertainties and disturbances, and more general cases were not established on multiple coupled chaotic systems with unknown parameters and disturbances. Further, Chen et al.[20] proposed adaptive sliding mode synchronization for multiple chaotic systems with unknown parameters and disturbances, and the appropriate adaptive laws were given to estimate unknown parameters. In addition, the adaptive sliding mode synchronization of fractional order chaotic systems have been also discussed by researchers [21,22].

However, most of previous studies focused only on the asymptotical synchronization [23]. In practice, it is more valuable and preponderant to study the synchronization in a given finite time other than that in an unpredictable infinite time. For the finite-time stability methods [24], Cai et al. [25] studied the generalized synchronization in finite time among chaotic systems with different order. Further, Zhao et al. [26] investigated the generalized synchronization of integer-order coupled chaotic systems within finite time. Zhang et al. [27]also implemented global synchronization of two integer-order chaotic systems with different dimensions. Chen et al. [28] investigated finite-time multi-switching synchronization of multiple uncertain complex chaotic systems with network transmission mode, and the unknown parameters and disturbances were considered. Furthermore, based on fractional-order the finite-time stability methods, Wu et al. [29] investigated the global synchronization in finite time between non-identical fractional order neural networks (FNNs). With respect to finite-time synchronization, some more results have also been found in [30,31]. Whereas, most of those results focused on the synchronization of identical (non-identical) fractional order dynamical systems either the generalized synchronization in infinite time or without considering the effects of the system uncertainties. It is known that the application of synchronization in secure communication process, chaos-based cryptography can offer a fast and secure way for information protection [32]. The minimum synchronization error and time can be required to recover or send the encoded message. Besides, for in the chaotic masking [33,34], digital sound encryption techniques in fractional order chaotic systems with a higher level of security are desired, and it can give a powerful solution along with algorithms. Then, it is necessary to study finite-time chaos synchronization problems of fractional order dynamical systems with uncertainties. More importantly, considering the actual situation [34], state variables and uncertainties in the error system have crucial influence on encryption and decryption process of the message signal. Therefore, how to design a controller, which can efficiently reduce the synchronization errors in finite-time and to maximally ensure information transmission security, is a significant and challenging topic.

Inspired by aforementioned previous works, the contributions of this paper can be summarized on four aspects.

Based on the definition of the generalized synchronization, the synchronization schemes of two non-identical fractional order chaotic systems are proposed to achieve finite-time generalized synchronization with considering the uncertainties and external disturbances.

To study generalized synchronization, a novel fractional order integral sliding mode surface is designed and its stability to the origin is proved in a given finite time.

According to the fractional Lyapunov stability theory, an appropriate sliding mode controller with adaptive update laws is proposed under external disturbances and model uncertainties, and the stability conditions for achieving the generalized synchronization are explicitly derived in finite time.

Numerical simulation results further highlight the validity, the novelty and applicability of the proposed approach for non-identical fractional order chaotic systems. For the application of secure communication process, a new speech encryption system is introduced to share voice messages secretly via secure channel in terms of the proposed synchronization criterion. Meanwhile, the security of the proposed theories is also analyzed and discussed.

The remainder framework of this study is arranged as follows. In Section 2, the preliminary definition and lemma knowledge necessary are reviewed throughout the paper. The generalized synchronization scheme is introduced in Section 3. Then, numerical simulations are carried out to highlight the effectiveness and applicability of the proposed approach in Section 4. The application in speech secure communication is described in Section 5. Finally, the conclusion is given in Section 6.

2. Definitions

In this section, some remarkable definitions of fractional calculus and some helpful lemmas are recalled in the following.

Definition 1 [35] The αth-order Caputo fractional integral of a function f(t) is described by

Where, 1>α>0, Γ(⋅) denotes the gamma function and αΓ(α) = Γ(α+1).

Definition 2 [35] The αth-order Caputo fractional derivative of a function f(t) is defined as:

Where, 1>α>0 and m is the smallest integer number.

Lemma 1 [35] When the fractional-order derivative is integrable, let Ω = [a,b] be an interval on the real axis ℝ, and let n = [α]+1 for α∉N or n = α for α∈N. If x(t)∈C[a,b], one can obtain:

Especially, if 0<α≤1 and x(t)∈C1[a,b], then

Lemma 2 [35] Assume α∈(0,1), p∈R, then

Lemma 3 [29,36]Suppose α∈(0,1), and x(t) denotes a continuous and differentiable function, then it satisfies the following inequality

Lemma 4 [27] If d∈ℝ, i = 1,2⋯n and ξ∈(0,1) are arbitrary real numbers, the following inequalities satisfy:

Lemma 5 [37] Consider the following n-dimensional fractional-order dynamical system

Where, α∈(0,1), x(t)∈ℝ is the system state and f:[0,∞)×ℝ→ℝ is a continuous nonlinear function. Assume that there exists a continuously differential Lyapunov function V(t,x(t)) and strictly increasing class-K functions b1, b2 and b3 satisfying

Where, β∈(0,1), Then the equilibrium point x = 0 of the fractional-order system (7) is asymptotically stable.

Remark 1 For simplicity, the Caputo fractional calculus of order α as and are substituted for D and I, respectively.

3. The generalized synchronization scheme

In this section, the main goal is to develop the generalized synchronization between two non-identical fractional order chaotic/hyper-chaotic systems in finite-time, which play an important role to acquire the main results via applying sliding mode technique.

Consider the following n-dimensional fractional order master system

Where, 0<α≤1 is the fractional order of the system, x = [x1, x2,⋯,x] is the system state vector; denote unknown model uncertainties and external disturbances, respectively. F(x) = [F1(x), F2(x),⋯F(x)] is a nonlinear function.

Consider the corresponding m- dimensional fractional order slave system

Where, 0<β≤1 is the fractional order of the system, y = [y1, y2,⋯,y] is the system state vector; denote unknown model uncertainties and disturbances or perturbations, respectively. And G(y) = [G1(y), G2(y),⋯,G(y)] is a nonlinear function; the control input is U(t) = [u1(t), u2(t),⋯,u(t)].

Now, the definition of generalized synchronization between fractional-order chaotic systems is given in the following expression.

Definition 3 Consider the above systems (9) and (10) with different initial values denoted by x0 and y0. Assume that there exist an open neighborhood Θ⊂ℝr of the origin, two continuously differentiable functions ϕ: ℝn→ℝr and φ: ℝm→ℝr, i.e., e0 = φ(y0)−ϕ(x0)∈Θ, and a constant T = T(e(0))∈(0,∞), one can get

Where, e(t)∈ℝ denotes the synchronization error of the master system (15) and slave system (16). Then, ‖e(t)‖≡0, t≥T, that is, the synchronization error can be achieved to be zero within a finite time.

Remark 2 Fractional-order chaotic systems with same dimensions when m = n, that is ϕ(x(t)) = x(t), φ(y(t)) = y(t), if the synchronization error is e(t) = y(t)−x(t), it can be transformed into globally complete synchronization; if the synchronization error is e(t) = x(t)−y(t), it can accomplish globally anti-synchronization; if the synchronization error is e(t) = y(t)−px(t), and p is the projective coefficient, it can become globally projective synchronization; if the synchronization error is e(t) = x(t), the above Eq (11) can be inferred as the following form , it will be transformed into the stabilization of the master system. Obviously, these are special cases of our proposed methods. Zhang et al. [26] proposed the finite-time synchronization of the error system e(t) = φ(y)−ϕ(x), but the author did not consider disturbances and model uncertainties, and the error system should be also integer-order system. Nevertheless, this idea is appreciated.

Defining e(t) = φ(y)−ϕ(x), from system (9) and system (10), in order to introduce conveniently, the control signals U(t) are considered as U′(t) and U″(t), i.e., U(t) = U′(t)+U″(t). The compensation controller U″(t) can be preset as , and the separated controller U′(t) will be designed later. Then, we have

Where, J(x) and J(y) are the Jacobin matrices of the functions ϕ(x) and φ(y), respectively, i.e.

Remark 3 r≤min{m, n}, the Jacobin matrix J is row full-rank, it is well known that J is a square matrix, the inverse matrix exists. In terms of the generalized inverse matrix definition, the right inverse matrix exists when it is not a square matrix. To simplify the symbol, denotes the inverse or right inverse matrix of J in this study.

Assumption 1 It is assumed that disturbances d(t) and model uncertainties Δf(x) of the master system (15) are all bounded, there exist unknown positive constants γ, i.e., , i = 1,2,…,r.

Assumption 2 The disturbances d(t) and model uncertainties Δg(x) of the slave system (10) are assumed to be all bounded, there also exist unknown positive constants ε, i.e., |J(Δg(y)+d(t))|≤ε, i = 1,2,…,r.

Remark 4 The objective of this study can be formulated as designing an appropriate control law U′(t) for any different dimensional systems (9) and (10) with disturbances and model uncertainties, the finite-time stability for the error system (12) can be accomplished in the light of Definition 3.

To further study generalized synchronization of two chaotic systems (9) and (10) with different dimensions, it can be transformed into the globally stability of equilibrium point for error system (12). Here, a sliding mode technique will be used to solve the generalized synchronization problem. In general, the design process of sliding mode control includes the following two major steps. The first step is to determine a suitable sliding surface with some required system dynamic characteristics. Second, the appropriate sliding mode control laws are arranged to guarantee the state trajectories onto the sliding surface and subsequently stay on it forever. Therefore, a novel fractional integral siding surface is constructed as follows:

Where s(t) = [s1, s2,…,s]∈ℝ is the sliding surface, e(t) = [e1, e2,…e]∈ℝ is the synchronization error state, 0<δ<β<1, k1>0 is the gain coefficient.

Based on the sliding mode control strategy, the sliding surface and its derivative should satisfy: s(t) = 0 and . We know , then, implies Ds(t) = 0. Therefore, from (13), one has

Based on Lemma 5, the following finite time convergence theorem of the fractional terminal sliding surface (13) is analytically proved.

Theorem 1 Consider the sliding mode dynamics (14). The error system will be global asymptotically stable and converge to the equilibrium e(t) = 0 within finite time upper bounded by:

Proof Select the following Lyapunov function candidate:

By applying Lemma 3, one has

Substituting De(t), i = 1,2,…,r from (14) into (16), and sign(e)×sign(e) = 1, one obtains

Using Lemma 4 the following inequality , one gets

Based on Lemma 5, the dynamic error e(t), i = 1,2,…,r will converge to zero asymptotically. In terms of Lemma 2, one has

From (19), it can be easy to derive the following form:

Taking fractional-order integral of (20) from t0 to t by Lemma 1, one obtains

According to Definition 1, one gets

Combining (21) and (22), one can get

From (23), one can obtain that , such that V1(t) = 0 for arbitrary , and the sliding-mode dynamic error e(t), i = 1,2,…,r will converge to zero in finite time, i.e., is the upper bound of convergence time, given by . This completes the proof.

In what follows, in order to satisfy the sliding condition under disturbances and model uncertainties, the adaptive sliding control law is proposed as follows:

Where, i = 1,2,…,r, 0<σ<β<1, k2>0 is the gain coefficient. and are denoted as estimates of εi and γi, respectively. In this subsection, the adaptive update laws are designed by the following algorithm:

Theorem 2 Under Assumption 1 and 2, consider the synchronization error system (18) with uncertainties and external disturbances. Based on the control law (24) with the adaptive laws (25), then its trajectories will globally reach the sliding surface s(t) = 0 within finite time upper bounded by:

Proof We choose the following Lyapunov function candidate

Based on Lemma 3, taking the fractional-order derivative of V2(t) as follows:

Combining (12), (14), (24) and (25), one obtains

On the basis of Assumptions 1 and 2, one yields,

In terms of Lemma 4, it is easy to obtain that

On the basis of Lemma 5, the state trajectories of the error system will converge to s(t) = 0 asymptotically. By Lemma 2, one has

Then, it can be easy to derive the following form:

Taking fractional-order integral of (32) from t0 to t by Lemma 1, one obtains

According to Definition 1, one gets

From (33) and (34), one can obtain

Hence, it implies that the state trajectories of the error system (12) will reach the predefined sliding surface s(t) = 0 in a given finite time under the controller (30). One can obtain that

Therefore, this completes the proof.

Remark 5 Theorem 1 has proved that the synchronization error (11) can be achieved to be zero within a finite time, and Theorem 2 has proved that the state trajectories of the error system can converge to zero within a given finite time. Therefore, according to Theorems 1 and 2, the upper bound of total convergence time can be estimated as .

Remark 6 In the proposed synchronization method, all potentialities: dimension, fractional order derivative, identical or non-identical and with or without disturbances and model uncertainties, are included in the dynamical systems. Consequently, the so-called ‘generalized synchronization approach’ is entirely adequate for synchronizing any dynamical systems in finite time.

Remark 7 From the perspective of control engineering, the main problems that greatly limits the control performance in practical applications are: a) the model parameters or the upper bound of the dynamical system, b) the presence of uncertainties and external disturbances in the system, c) The feasibility of the control inputs. In the proposed control scheme, it is not necessary to give prior knowledge of the upper bounds. Furthermore, due to the efficiency of the adaptive sliding mode control, it has good robustness against uncertainties and disturbances. It’s also noteworthy that for real implementations of the adaptive update laws in Eq (25), the disturbances and model uncertainties need to satisfy the Assumption 1 and 2. Further, the control inputs are feasible in real applications. Hence, the proposed controller is very suitable for practical applications.

4. Numerical simulations

In this section, some numerical simulations are provided to highlight the validity and effectiveness of our proposed methods obtained in the previous section. Two cases are discussed by applying the method to two non-identical fractional order chaotic systems with and without commensurate orders.

In the following, let us consider the hyper-chaotic fractional order Lorenz system (36) as a master system, and chaotic fractional order reverse butterfly-shape system (37) as a slave system. These systems with disturbances and model uncertainties are taken from [18].

Master system:

Slave system:

The disturbances and model uncertainties of the systems are chosen as follows:

By using Assumption 1 and 2, one can obtain ε = (0.1,0.8,0.8), γ = (0.35,0.6,0.55)(i = 1,2,3).The following examples are illustrated by using two varieties of cases for fractional order derivatives: one is α≠β and another one is α = β.

4.1. Non-commensurate order

In our simulation, the fractional derivatives are selected as α = 0.98, β = 0.99 respectively. The initial values of the systems (36) and (37) are given as (x1,x2,x3,x4) = (2,−2,4,1), (y1,y2,y3) = (2,−1,1). By taking those parameters, the dynamics of master system (36) and slave system (37) without the controller exhibits a chaotic behavior as illustrated in Figs 1 and 2.The corresponding time series of the master and slave systems are shown in Figs 3 and 4.

Fig 1

Phase portraits of master system when α = 0.98.

Fig 2

Phase portraits of the uncontrolled slave system when β = 0.99.

Fig 3

Time series of master system when α = 0.98.

Fig 4

Time series of the uncontrolled slave system when β = 0.99.

Assume the continuous differentiable functions of the systems (36) and (37) are

Then . Further, the corresponding Jacobian matrix and the generalized inverse matrix is obtained

By (19) and (27), the sliding surface and control input are provided as follows:

And the control signals U(t) = U′(t)+U″(t), compensation controllers From (24), adaptive controllers U′i(t) are determined by the following

Where, and are obtained by the adaptive update laws (25)

In control scheme (42), The initial conditions of and are chosen as 0.5, the control parameters are chosen as k1 = k2 = 10, both δ and σ are set equal to 0.8. It’s noted that the controller is performed at t0 = 20s and start from initial state , thus e(20) = φ(y(20))−ϕ(x(20)) = (−0.06,−17.28,−99.85).

The synchronization errors of the master system (36) and slave system (37) are plotted in Fig 5. Obviously, the synchronization errors converge to zero rapidly, and the stabilized time is estimated within 1.00 s, which indicates that the global synchronization is successfully realized, as depicted in Fig 6. Furthermore, the corresponding time response of the sliding surface (13) is plotted in Fig 7.

Fig 5

State trajectories of the synchronization errors between master and slave system when α = 0.98 and β = 0.99.

Fig 6

State trajectories of master system and slave system whenα = 0.98 and β = 0.99.

Fig 7

Time response of the sliding surfaces.

According to Theorem 1, the states of the sliding mode dynamics system (14) will converge within a given time

In Fig 6, it is seen that the error system (12) can stabilize to zero with the reaching time of t1≈0.8s and satisfy the estimated reaching time .

Besides, by (19) based on Definition 1 and Theorem 2, the estimated time is calculated as

From Fig 7, it is obvious that the simulation time t2≈0.8s satisfies the estimated time .

4.2. Commensurate order

Assume that the commensurate order α = β = 0.96 and the initial values of the systems (36) and (37) are set as (x1, x2, x3, x4) = (3,1,−2,1), (y1, y2, y3) = (1,2,3). And the control signals U(t) = U′(t)+U″(t), compensation controllers i = 1,2,3. Then, adaptive controllers are determined by the form of (42). The other parameter values are the same as Case I. It is noteworthy that the controller is implemented at t0 = 20s. The results are depicted in Figs 8–10. As shown in Figs 8 and 9, the generalized synchronization is successful accomplished and the synchronization errors tend to zero within 1.0s. From Fig 10, it is clear that the sliding surfaces converge to zero within 1.0s. The calculation method of convergence time is similar to Theorem 1 and 2, the upper bound of those can be estimated as respectively. The results verify the presented control strategy can synchronize two different dimensional fractional order chaotic systems within finite-time.

Fig 8

State trajectories of the synchronization errors between master and slave system when α = β = 0.96.

Fig 10

Time response of the sliding surfaces.

Fig 9

State trajectories between master system and slave system when α = β = 0.96.

Remark 8 For the proposed synchronization approach, the error state trajectories with both cases will converge to zero within 1.0s according to Figs 5 and 10. But the existing method in reference [18], the synchronization errors converge to zero at the reaching time t ≥1.6s. Therefore, the convergence rate of synchronization errors for fractional order chaotic systems is faster than that of the existent approach in [18], which implies the superiority and effectiveness of the presented scheme in our study. In addition, integer order dynamical system is also appropriate.

Remark 9 According to Remark 8, it is obvious that the novelty and superiority of the model is highlighted by the proposed approach. The finite-time generalized synchronization can be applied to both commensurate and incommensurate systems. Moreover, the fact that it acquires the coexistence of several kinds of synchronization types, as see in [12], the coexistence of three different synchronization types, that is, identical synchronization (IS), anti-phase synchronization (AS), and inverse full state hybrid projective synchronization (IFSHPS). But the fact that the proposed approach is more general than that in [12], since it guarantees the combination of only three specific synchronization types. Furthermore, the proposed synchronization approach gives a deeper insight into the synchronization phenomena between fractional order chaotic systems.

5. Applications in speech secure communication

In this section, due to the technology of finite-time synchronization and uncertainties of the synchronous system can enhance the security of communication. A new speech cryptosystem is proposed to send or share voice messages privately according to generalized finite-time synchronization criterion of non-identical fractional order chaotic drive system and response system improving the level of security. Before signal transmission, based on synchronization theory among fractional-order chaotic systems, the generalized synchronization errors of the systems (36) and (37) will converge to zero under the given control inputs (42) and a time t>t1+t2. Then, the sender A records an audio message and generates him/her own key of chaotic sequence K. The receiver B obtains the secret key of chaotic sequence K by means of the proposed synchronization criterion and then decrypts the original speech signal. It is noted that encryption-decryption keys K, K are the same. Furthermore, the overall diagram of speech encryption–decryption process is depicted in Fig 11.

Fig 11

The overall diagram of speech secure communication.

5.1. Description of encryption and decryption scheme

Now the speech encryption and decryption algorithm is designed based on generalized synchronization of fractional order chaotic system. Chaotic sequences can be generated by using the systems (36) and (37), and the arbitrary two functions of ϕ(x) from the systems (36) are chosen to generate the key of encryption algorithm. The original sound is then encrypted by employing a simple XOR operation with the chaotic sequence generated. The decryption algorithm is basically similar to the encryption algorithm. As shown in Fig 12, the complete algorithm can be described in the following steps:

Fig 12

Encryption and decryption scheme of the speech signal based on synchronization criterion.

Step 1: The preliminary speech signal preprocessing. Selecting any channel signal of the recorded two-channel speech signals. The chosen signal will realize the conversion from decimal to binary. And its amplitude will be magnified 1000 times.

Step 2: The encryption sequence generated. The arbitrary two functions of ϕ(x) from the systems (36) are chosen to generate the chaotic sequence a1, b1. Then there will get the difference of a hundred numbers from a1 and b1. If a1(i)−b1(i)>0, i = 1,2,3⋯100, the sequence of encryption J(i) = 1, In contrast, J(i) = 0, that is to say, the key of encryption K has been generated.

Step 3: Encryption. The sequence of encryption J(i) will be transformed into binary number and then be encrypted by using a simple XOR operation.

Step 4: Decryption. The corresponding two functions of φ(y) from the systems (37) are obtained to generate the tracking sequence a2, b2 in terms of synchronization criterion. Additionally, the decryption algorithm is the same as the encryption algorithm. Then the sequence of decryption T(i) could be generated. Likewise, the corresponding XOR operation will also been performed. The decrypted speech signal can be acquired.

Step 5: Decrypted speech signal post-processing. The decrypted speech signal will realize the conversion from binary to decimal, and its amplitude will be reduced by 1000 times.

Step 6: Get the original speech.

5.2. Demonstration of the experimental results

In this experiment, the generalized finite-time synchronization between the systems (36) and (37) is implemented in Case I, while the controller is applied at t0 = 5s. Synchronous trajectories are depicted in Fig 13. Further, Assume that A records a speech message and wishes to send it to B secretly. Both A and B should be agreeing on a time t>t0. The message has been saved by A as the audio format of s.mp3.The original speech signal and corresponding FFT spectrum is displayed in Fig 14. It has 7.2 seconds long with 44100 samples. For encryption, the preprocessing for original speech signal is completed in advance. As shown in Fig 15, the original speech signal preprocessing and FFT spectrum are represented graphically. From Fig 16, the arbitrary 40 points chosen from the original speech are converted to the binary bits. According to encryption scheme in the previous section, encrypted speech signal and its corresponding FFT spectrum are illustrated in Fig 17. It is clear that the speech signal is entirely covered by the chaotic secret key sequence, and the original profile could not be seen at all. For decryption, the decrypted speech signal and its corresponding FFT spectrum are depicted in Fig 18. Finally, B will obtain an original speech message without any loss of information, because the decrypted audio signal is fully restored. Therefore, B can play decrypt speech and hear an original voice. It is well known that the secret key sequence generated by the drive system is utilized to encrypt the speech signal and that generated by the response system operates in recovering the encryption signal.

Fig 13

Synchronous trajectories of drive system and response system when α = 0.98 and β = 0.99.

Fig 14

The original speech signal and FFT spectrum.

Fig 15

Original speech signal preprocessing and FFT spectrum.

Fig 16

The binary number of the original speech signal.

Fig 17

Encrypted speech signal and FFT spectrum.

Fig 18

Decrypted speech signal and FFT spectrum.

Remark 9 Notice that from the aforementioned result, spectrogram analysis is a random test tool that divides the signal in the time domain into slots to calculate Fast Fourier Transform (FFT) for each slot. The magnitude square of FFT is plotted versus each slot to indicate the energy of sound. There are 44100 samples of the input speech signal with7.2 seconds long. Then, the spectrogram of original and encrypted signals is made. The energy of sound appears to be small as a result of low speech signal’s amplitude, so its amplitude will be magnified 1000 times. For encrypting the speech file, spectrogram appears random which indicates the randomness distribution of sound components’ energy. In the decryption process, the waveforms of the decrypted speech signal are shown by the decryption key. Encryption and decryption scheme of the speech signal based on synchronization criterion is straightforward with low-level hardware complexity. Furthermore, the proposed method also effectively enhances the security of signal transmission. Image encryption is one of the most significant and common applications of synchronization between fractional order chaotic systems. A future direction of investigation is to recast the methodology adopted in this paper to the image encryption.

5.3. Security analysis

In the proposed synchronization criterion, the fractional order chaotic systems (36) and (37) are used to generate the secret key of encryption and decryption sequences K, K. Due to some secret elements of fractional-order chaotic systems, such as the parameters and initial conditions of the system, fractional orders α and β and the convergence time, will add the total number of different secret keys directly. Hence, the new algorithm has a large enough key space to resist brute-force attacks. The key sequences generated are totally uncorrelated and random. When such sequences are utilized, experimental results have revealed that the proposed algorithm possesses some secret elements of traditional cryptography, such as complex chaotic behavior, time-varying nonlinearity, disturbances and model uncertainties. Therefore, the reverse recover of the original speech message is totally hopeless except by the receiver. Meanwhile, the demonstration and analysis of the speech cryptosystem based the fractional order dynamical systems have shown that, the proposed encryption and decryption scheme is more secure and appropriate for sending and receiving messages secretly. Furthermore, these have basically addressed all existing security disadvantages with respect to chaos based audio encryption methods and have provided a new idea for the ever-increasing practical applications.

6. Conclusion

This paper is concerned with the generalized finite-time synchronization between two non-identical fractional order chaotic (or hyper-chaotic) systems based on adaptive sliding mode controller and its application in secure communication. First, the definition of generalized finite-time synchronization is given. Second, a novel fractional order integral sliding surface is presented and its finite-time convergence theorem is analytically proved. Then, according to the fractional Lyapunov stability theory, a robust controller with adaptive update laws is proposed to ensure the occurrence of the sliding motion. Meanwhile, its finite-time stability condition is derived by considering model uncertainties and external disturbances. The results of theoretical analysis show that the finite-time stability and the robustness of the proposed control scheme are mathematically proved, and the upper bound of the convergence time is explicitly evaluated. Finally, numerical simulations illustrate the effectiveness and robustness of the presented approach, which are in good agreement with the results of theoretical analysis. What’s more, in order to demonstrate the practical effect of generalized synchronization with application to the speech secure communication, a novel sound encryption mechanism is proposed and a successful case is given to show the applicability of the proposed theories. It is worthwhile to note that the proposed synchronization approach not only can be extended to a wide range of nonlinear fractional-order chaotic systems and time-delayed chaotic systems, but also can be further applied to create a new encryption mechanism or a new way guaranteeing information safety. A future direction of investigation is to recast the methodology adopted in this paper to the image encryption.

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Additional Editor Comments:

Based on the two reviewers' suggestions, the paper needs to be revised largely from the technique to the writing.

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Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Author

1. Is the manuscript technically sound, and do the data support the conclusions?

The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented.

Reviewer #1: Yes

Reviewer #2: Yes

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2. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: Yes

Reviewer #2: N/A

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3. Have the authors made all data underlying the findings in their manuscript fully available?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: Yes

Reviewer #2: No

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4. Is the manuscript presented in an intelligible fashion and written in standard English?

PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.

Reviewer #1: Yes

Reviewer #2: Yes

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5. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #1: In my opinion, the article is innovative and suitable for publication in this journal after the following  revisions can be finished.

1 This article needs further careful examination to avoid language errors that should not occur, for example at the beginning of the Introduction, there are two Chaos Chaos; Title of Chapter II：“2. Basic definition and lemma”I think “definitions”should be more suitable.

2.The reference format of the article is very chaotic, and the author needs to revise it uniformly according to the requirements of the journal.

3.The relationship between theorem 1 and theorem 2 is not well explained, which is easy to cause confusion in understanding. Please explain in detail the relationship between theorem 1 and theorem 2.

4.Figure 1-4 only list the behavior evolution of system state and output, without further detailed comparative analysis. Please give a detailed comparative analysis of status and output signals.

Reviewer #2: The problem of the finite-time generalized synchronization for non-identical fractional order chaotic (or hyper-chaotic) systems by a designing adaptive sliding mode controller and its application to secure communication has been focused on in this manuscript. And some simulation results have been provided to demonstrate the effectiveness and robustness of the presented approach. However, the following comments need to be considered in the revision and require major revisions.

(1) What are the advantages of the fractional order system considered in this paper over the integer order system? What are the differences and difficulties in handling?

(2) The contributions of this paper need to be improved.

(3) The background in the introduction part is not enough, please refer to the following reference:

IEEE Transactions on Neural Networks and Learning Systems, DOI: 10.1109/TNNLS.2019.2952410.

(4) In the Definitions 1-2, Lemmas 2, 3, 5 in the paper, they require \\alpha \\in (0,1), while Lemma 1 requires \\alpha \\in (0,1], which makes the definitions 1-2, Lemmas 2, 3, 5 no longer hold when Lemma 1 holds. Please explain this contradiction issue in detail.

(5) The symbol “\\varepsilon_{i}” given in Lemma 4 is not reflected in the following inequality (6), and the symbol \\xi in (6) does not give a corresponding definition. In addition, there are many symbols in the full paper that appear for the first time without giving corresponding definitions, such as the symbol \\tau, the symbol before the function f(t) in definition 1, etc.

(6) The authors mentioned in Remark 4 that “designing an appropriate control law ′() for any different dimensional systems (9) and (10)”? Is this designed control law applicable to any different dimensional systems?

(7) What is the prior knowledge of the upper bound of the system mentioned by the authors in Remark 6? In the control scheme proposed in this paper, why is it not necessary to give a priori knowledge of the upper bound? What method was used to deal with it?

(8) The expression of the finite time T calculated in the paper contains the initial values of some variables. How are these initial values determined?

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Reviewer #1: No

Reviewer #2: No

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Submitted filename: PONE-D-21-24308 comments.pdf

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2 Dec 2021

We thank the innominate reviewers for their valuable comments and suggestions in improving the quality of the manuscript. We have modified the new manuscript following your advice. Thank you for your suggestion again.

Submitted filename: Response to Reviewers_11.26.docx

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11 Jan 2022

Finite-time generalized synchronization of non-identical fractional order chaotic systems and its application in speech secure communication

PONE-D-21-24308R1

Dear Dr. Yang,

We’re pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it meets all outstanding technical requirements.

Within one week, you’ll receive an e-mail detailing the required amendments. When these have been addressed, you’ll receive a formal acceptance letter and your manuscript will be scheduled for publication.

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Kind regards,

Academic Editor

PLOS ONE

Additional Editor Comments (optional):

Based on the two reviewers' comments, the paper can be accepted for publication now.

Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Author

1. If the authors have adequately addressed your comments raised in a previous round of review and you feel that this manuscript is now acceptable for publication, you may indicate that here to bypass the “Comments to the Author” section, enter your conflict of interest statement in the “Confidential to Editor” section, and submit your "Accept" recommendation.

Reviewer #1: All comments have been addressed

Reviewer #2: All comments have been addressed

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2. Is the manuscript technically sound, and do the data support the conclusions?

The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented.

Reviewer #1: Yes

Reviewer #2: (No Response)

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3. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: Yes

Reviewer #2: (No Response)

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4. Have the authors made all data underlying the findings in their manuscript fully available?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: Yes

Reviewer #2: (No Response)

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5. Is the manuscript presented in an intelligible fashion and written in standard English?

PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.

Reviewer #1: Yes

Reviewer #2: (No Response)

**********

6. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #1: Authors have made good modifications and basically answered my questions. In my opinion, it's acceptable.

Reviewer #2: (No Response)

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7. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files.

If you choose “no”, your identity will remain anonymous but your review may still be made public.

Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy.

Reviewer #1: No

Reviewer #2: No

Submitted filename: Comments to PONE-D-21-24308R1.pdf

Click here for additional data file.

14 Mar 2022

PONE-D-21-24308R1

Finite-time generalized synchronization of non-identical fractional order chaotic systems and its application in speech secure communication

Dear Dr. Yang:

I'm pleased to inform you that your manuscript has been deemed suitable for publication in PLOS ONE. Congratulations! Your manuscript is now with our production department.

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on behalf of

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