Exponential Chaotic Model for Generating Robust Chaos

Robust chaos is defined as the inexistence of periodic windows and coexisting attractors in the neighborhood of parameter space. This characteristic is desired because a chaotic system with robust chaos can overcome the chaos disappearance caused by parameter disturbance in practical applications. However, many existing chaotic systems fail to consider the robust chaos. This article introduces an exponential chaotic model (ECM) to produce new one-dimensional (1-D) chaotic maps with robust chaos. ECM is a universal framework and can produce many new chaotic maps employing any two 1-D chaotic maps as base and exponent maps. As examples, we present nine chaotic maps produced by ECM, discuss their bifurcation diagrams and prove their robust chaos. Performance evaluations also show that these nine chaotic maps of ECM can obtain robust chaos in a large parameter space. To show the practical applications of ECM, we employ these nine chaotic maps of ECM in secure communication. Simulation results show their superior performance against various channel noise during data transmission.

When studying chaos theory and applying chaotic systems to real applications, researchers have found that existing chaotic systems may have disadvantages in several aspects [18]. First, chaos degradation may occur when chaotic systems are digitized, because the precision of all the software and hardware platforms is finite [19]. Theoretically, the approaching states of a chaotic system locally oscillate in its phase plane. However, they may overlap when being digitized in real platforms. Then the chaotic behaviors may degrade to regular behaviors [20]. The chaos degradation brings seriously negative influences to the chaos-based applications [21]. Second, with the quick increment of computer ability and fast development of chaos theory, many chaotic behaviors with low degree of complexity can be estimated [22], [23]. The commonly used strategies of estimating chaotic behaviors are to predict their chaotic signals [24] and to identify their initial states [25]. In addition, existing chaotic systems may not have robust chaos [26]. The robust chaos is extremely important for many chaos-based applications, because a chaotic system owning robust chaos can avoid the chaos degradation caused by parameter perturbations [27], [28].
At present, many research works have focused on improving the chaos complexity of chaotic systems [29], [30]. These efforts can be classified into two types: 1) perturbing the chaotic signals of chaotic systems and 2) designing new chaotic systems. The first type of efforts can significantly delay the chaos degradation. Deng et al. [31] proposed a feedback control method to enhance the complexity of chaotic signals. Li et al. [32] presented a reseeding mixing technology to enlarge the periods of chaotic signals. The reseeding concept is to remove the chaotic signals with short periods from the system while the mixing concept is to enlarge the periods of chaotic signals. The second type of efforts can obtain new chaotic systems with higher complexity [33], [34]. Huang et al. [35] introduced a grid multiwing butterfly chaotic attractor, which is generated by replacing the state variables in the Lorenz system using piecewise hysteresis functions. In [36], a unified approach was proposed to design hyperchaotic systems with desired property. Shen and Jia [37] first investigated the conditions of generating one-dimensional (1-D) discrete chaotic systems, and This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ then constructed several new 1-D chaotic systems according to these conditions. This article significantly contributes to the development of discrete chaos. Although these efforts can reduce the negative effects of chaotic systems in practical applications, they still have some limitations. When using different technologies to perturb chaotic signals, the mathematical equations and integrality of chaotic systems may be broken. This leads the chaotic systems to lose some natural properties. For previous works of designing new chaotic systems, they fail to consider the robust chaos and thus the constructed chaotic systems cannot generate robust chaotic behaviors [33], [37]- [39]. The robust chaos is a desired property for many real applications. As a result, designing chaotic systems with robust chaos is a meaningful and attractive research topic.
To obtain 1-D chaotic maps with robust chaos, this article introduces an exponential chaotic model (ECM). Motivated by the concept of exponential operation in arithmetic, ECM can produce many new chaotic maps using 1-D chaotic maps as base and exponent maps. Theoretical analysis, experimental results, and real application are provided to prove the robust chaos and complex chaotic behaviors of the new chaotic maps generated by ECM. The contributions and novelty of this article are summarized as follows.
1) We propose ECM as a universal framework. Employing two 1-D chaotic maps as base and exponent maps, ECM can produce many new 1-D chaotic maps with robust chaos. 2) As examples, we present nine new chaotic maps generated by ECM. Theoretical analysis demonstrates their robust chaos. 3) We quantitatively evaluate the chaotic maps of ECM using Lyapunov exponent (LE), correlation dimension (CD), and initial state sensitivity. Experiment results show that these chaotic maps are sensitive to their initial states, and have robust chaotic behaviors in a large parameter space. 4) To show the practical application of ECM, we apply the nine chaotic maps of ECM to secure communication.
Simulation results indicate that these maps exhibit better performance in resisting various channel noise than their base and exponent maps. The rest of this article is organized as follows. Section II introduces three 1-D chaotic maps and the definition of robust chaos. Section III presents the proposed ECM and analyzes its dynamic properties. Section IV generates nine 1-D chaotic maps of ECM as examples, analyzes their bifurcation diagrams and proves their robust chaos. Section V evaluates the chaos complexity and performance of these chaotic maps of ECM and Section VI investigates their application in secure communication. Section VII concludes this article.

II. PRELIMINARIES
This section reviews the logistic, tent and sine maps that are used as the base and exponent maps in the proposed ECM in Section IV. The definition of robust chaos is also presented as preliminary knowledge.

A. Existing 1-D Chaotic Maps
The logistic map is designed to describe the change of population. By inputting a value within interval [0, 1], the output of the logistic map can oscillate in this range. The mathematical definition of the logistic map is described as The variable r is a control parameter and it is on the interval [0, 1]. The tent map obtains an output on the interval [0, 1]. It stretches an input if the input is smaller than 0.5; otherwise, it folds it. The mathematical equation of the tent map is written as The variable r is a control parameter and it is also on the interval [0, 1]. The sine map is developed from the sine transform. By scaling an input with π , the sine map transforms an input on the interval [0, 1] into the same interval. It has the definition as where variable r is a control parameter and it is also on the interval r ∈ [0, 1]. Bifurcation diagram shows the occupied or approaching states by a dynamical system with the increment of its control parameter(s). The top row of Fig. 1 shows the bifurcation diagrams of the logistic, tent, and sine maps. The LE is an indicator of chaos [40] and the bottom row of Fig. 1 shows the LEs of the three chaotic maps. As can be seen, the logistic, tent, and sine maps show chaos properties in the parameter ranges [0.8925, 1], (0.5.1), and [0.8656, 1], respectively. The logistic and sine maps have similar bifurcation diagrams, because the Taylor series of sin(x) is a combination of polynomials. However, the logistic and sine maps are totally different chaotic maps with different definitions and behaviors.

B. Definition of Robust Chaos
The robust chaos is defined as the inexistence of periodic windows and coexisting attractors in the whole chaotic range [26]. The existence of the periodic windows in chaotic range indicates that a small change to the parameter will destroy the chaos, implying frail chaos. A chaotic system owning robust chaos can avoid the chaos degradation caused by the electromagnetic interference, noise pollution and other perturbations to the system parameters. Thus, the robust chaos is a desired property for chaos-based applications [41]. For example, when a chaotic system is used to describe a physical system, the robust chaos is required to correctly reflect whether the physical system has chaotic behaviors, because the digitized parameter is only an approximation of the real parameter value [42]. According to [26], a dynamical system can generate robust chaos if it does not have stable periodic orbits and has a unique chaotic attractor in the whole parameter space. The robust chaos of a dynamical system can be verified using strict mathematical theorem if it is differentiable.
First, we give the definition of 1-D discrete S-unimodal map, which is described as Definition 1 [26,Definition 11.7].
has a unique local maximum at (m, n); and 4) f (x) has a negative Schwarzian derivative for all x ∈ I − {y, f (y) = 0}.
Then the occurrence of robust chaos for a 1-D discrete S-unimodal map can be described in Theorem 1 [26,Th. 11.12].
According to the property of the S-unimodal map [26, Th. 11.1], the parametric S-unimodal map ϕ ν (x) for all ν ∈ (ν min , ν max ) follows that there is at most one attracting periodic orbit with the critical point in its basin of the attraction. In this case, the orbit with initial value x 0 = k maps in two iterates to the fixed point x = m, because ϕ ν (k) = n and ϕ ν (n) = m. The fixed point x = m is unstable because Thus, the map ϕ ν (x) does not have any stable periodic orbits and there is a unique chaotic attractor. This indicates that ϕ ν (x) can generate robust chaos for all ν ∈ (ν min , ν max ).
One can also see detailed analysis and proof of Theorem 1 in [26, Ch. 11.2.1].
One can easily prove that the logistic map in (1) and sine map in (3) are S-unimodal maps on the interval [0, 1] and their unique maximum values can be obtained at x = 0.5. However, there does not exist a parameter interval (ν min , ν max ) that satisfies L(ν, 0.5) = S(ν, 0.5) = 1 ∀ν ∈ (ν min , ν max ). Then, the logistic and sine maps cannot satisfy the requirements of Theorem 1 and they cannot generate robust chaos. As the tent map in (2) is an unsmooth map and has a corner at x = 0.5, Theorem 1 is not feasible to judge whether it has robust chaos.

III. EXPONENTIAL CHAOTIC MODEL
This section presents the proposed ECM and discusses its dynamic behaviors.

A. ECM
Motivated by the concept of exponential operation in arithmetic, we propose ECM to produce new 1-D chaotic maps using two 1-D chaotic maps as base and exponent maps. Fig. 2 presents the structure of ECM. As can be seen, the two maps f (a, x i ) and g(b, x i ) are the base and exponent maps, respectively, a and b are their control parameters, and c is a small bias to balance the output of g(b, x i ).
The mathematical representation of ECM is written as The outputs of ECM are generated in an iteration form. The variable x i is simultaneously fed into the inputs of the base map f (a, x i ) and exponent map g(b, x i ). The base part is the output of f (a, x i ) while the exponent part is the logarithm with base e to the combination of the output of g(b, x i ) and the bias c.
Using one 1-D chaotic map as the base map and another 1-D chaotic map as the exponent map, ECM in (4) can achieve the following properties.
1) Users have flexibility to produce a large number of 1-D chaotic maps employing different combinations of the base map f (a, x i ) and exponent map g(b, x i ).
2) The base map f (a, x i ) and exponent map g(b, x i ) can be the identical or different 1-D chaotic maps. Even only exchanging the settings of f (a, x i ) and g(b, x i ), ECM can produce totally different chaotic maps, e.g., the TEL and LET maps in Table I. 3) The chaotic maps of ECM have a larger cycle length and can be less affected by digitization than their base and exponent maps. Supposing that the cycle lengths of the digitized base map f (x) and digitized exponent map g(x) are p and q, respectively, the cycle length of the digitized ECM E(x) = f (x) ln(g(x)+c) is the lowest common multiple of p and q.

B. Fixed Point and Stability Analysis
The fixed point is important characteristic of a dynamical system and it can reflect the dynamic property of the system. Here, we calculate the fixed points of ECM and investigate their stability. The fixed points of the dynamical system The stability of a fixed point depends on the derivative J of the dynamical system at that point. The fixed point is a critical point and it is superstable when J = 0; it is stable when 0 < |J| < 1; it is neutral when |J| = 1; and it is unstable and the attractors closing to the point will escape from it when |J| > 1. The derivative of ECM in (4) can be calculated as follows: Setting the base map f (a, x i ) and exponent map g(b, x i ) in (5) and (6) as existing 1-D chaotic maps, one can calculate out the fixed points and their related derivatives of the chaotic maps generated by ECM, and then can further deduce the dynamic behaviors of these chaotic maps.

IV. GENERATION OF NEW CHAOTIC MAPS
Setting the base and exponent maps as different 1-D chaotic maps, ECM can produce many new chaotic maps. This section produces nine chaotic maps of ECM employing the three existing chaotic maps introduced in Section II-A, and analyzes their bifurcation diagrams and robust chaos. Table I lists the nine produced chaotic maps using ECM. When the base map f (a, x i ) and exponent map g(b, x i ) are chosen as an identical chaotic map, the chaotic map performs exponential operation with itself to obtain a new chaotic map, including the tent-exponent-tent (TET) map, logisticexponent-logistic (LEL) map, and sine-exponent-sine (SES) map. When the base and exponent maps are selected as two chaotic maps, swapping the base and exponent maps can result in two completely different chaotic maps. For example, the tent-exponent-logistic (TEL) and logistic-exponent-tent (LET) maps. In addition, when two chaotic maps with similar behaviors are used in ECM, exchanging the settings of the base and exponent maps, ECM can also produce new chaotic maps with totally different chaotic behaviors. For example, the logisticexponent-sine (LES) and sine-exponent-logistic (SEL) maps. All the chaotic maps of ECM have three control parameters a, b, and c. The parameters a and b are from their corresponding base and exponent maps, and they are on the interval [0, 1], while the parameter c is a small bias of the system and it is on the interval [2, 2.8]. This article investigates the behaviors of these new chaotic maps in the parameter (b, c) space by setting a = 1.

A. Bifurcation Diagram
The bifurcation diagram of a dynamical system can occupy a portion of its phase space. A lower degree of density in the bifurcation diagram indicates more uniform chaotic signals of the corresponding dynamical system. Thus, the bifurcation diagram can visually reflect the behavior of a dynamical system. Fig. 3 shows the bifurcation diagrams of these nine chaotic maps of ECM on the parameter (b, c) space, where b ∈ [0, 1] and c ∈ [2, 2.8]. One can observe that these nine chaotic maps display complex chaotic behaviors in all the parameter settings. Their output values uniformly distribute in the whole output ranges. On the other hand, their base and exponent maps have chaotic behaviors only in small parameter ranges and their outputs cannot uniformly distribute, which can be observed from their bifurcation diagrams in Fig. 1. With uniform-distribution outputs, these new chaotic maps are suitable for many applications such as pseudo-random number generator.
A bifurcation structure can indicate how a dynamical system leads to chaos. As can be observed from Fig. 1, the existing logistic and sine maps have period-doubling bifurcation structures while the tent map has double-sided bifurcation structure leading to chaos. Because the proposed ECM is an efficient chaotification structure, the new chaotic maps produced by ECM have robust chaotic behaviors in the entire parameter space and they may not have obvious bifurcation structures that lead to chaos. Their robust chaos will be theoretically proved in the next section and experimentally analyzed using the LE and CD in Figs. 5 and 6.

B. Robust Chaos
According to Section II-B, the robust chaos is explained as the absence of periodic windows and coexisting attractors in the neighborhood of parameter space. This section proves that the nine chaotic maps of ECM in Table I can satisfy the requirements of robust chaos in parameter b ∈ (0, 1) when parameters a = 1 and c = 2.8. As the LEL, LES, SEL, and SES maps are differentiable, we first use Theorem 1 to prove their robust chaos.
According to the calculation in (6), the derivative of ECM at x = 0.5 can be obtained as One can obtain that f (1, 0. This is because f (1, x) and g(b, x) are L(r, x) or S(r, x), and 0 < L(r, x), S(r, x) < 1. Then the sign of J(x) in (6) is the same with that of the following equation: (1, x)).
Let the function H(x) = x ln(x). One can get that  f (1, x), and g(b, x) satisfy that 0 < f (1, x), g(b, x) < 1, because they are the L(r, x) or S(r, x) and 0 < L(r, x), S(r, x) < 1. Then one can obtain that From (8), one can get that f (1, x), g (b, x) > 0 when 0 < x < 0.5. Then in (10) satisfies that As the base map f (1, x) and exponent map g(b, x) are logistic map L(r, x) or sine map S(r, x), the δ(x) for the LEL, LES, SEL, and SES maps when 0 < x < 0.5 are obtained as The derivative of δ SEL (x) can be calculated as One can easily calculate out that when 0 < x < 0.5, only one rootx exists in δ SEL (x) = 0 and δ SEL (0) > 0. Thus, δ SEL (x) strictly increases on x ∈ (0,x], and strictly decreases on x ∈ [x, 0.5). Then Thus, we can obtain that when 0 < x < 0.5, the of the LEL, LES, SEL, and SES maps have = δ(x) > 0. Because the sign of J(x) is the same with that of the , one can obtain that When 0.5 < x < 1, one can get that g (b, x), f (1, x) < 0 from (8). Using (11), the in (10) satisfies that Using the same analysis with 0 < x < 0.5, we can obtain J(x) < 0 for 0.5 < x < 1.
The Schwarzian derivative of ECM is defined as According to the definitions of the LEL, LES, SEL, and SES maps in Table I,  As the LET, SET, TEL, TES, and TET maps are piecewise maps and are not differentiable, one cannot use Theorem 1 to analyze their robust chaos. However, their robust chaotic behaviors can be analyzed using the stability of their fixed points. Using (5) and (6), one can calculate out all fixed points and the related derivatives of these chaotic maps listed in Table I. Fig. 4 plots the calculated fixed points and the related derivatives of these chaotic maps on parameter space b ∈ (0, 1) when a = 1 and c = 2.8. As can be seen, these nine chaotic maps have unique fixed points on (0, 1) and their related absolute derivatives |J| > 1 for the whole parameter space b ∈ (0, 1). Thus, all the nine chaotic maps do not have stable periodic orbits and they own unique chaotic attractors in the whole parameter space b, indicating that they have robust chaotic behaviors.

V. PERFORMANCE EVALUATIONS
This section quantitatively evaluates the nine chaotic maps of ECM in the parameter (b, c) space. The chaotic behaviors are evaluated using the LE [40], CD [43], and initial state sensitivity.

A. Lyapunov Exponent
The LE is a commonly used criterion to test the existence of chaos [40]. It can test the chaos of a dynamical system from the deterministic equation(s). For two trajectories of a dynamical system starting from extremely close initial states, the LE measures their average separation rate in every unit time. Mathematically, the LE of the dynamical system where ε indicates a very small positive quantity. If F(·) is differentiable, its LE can also be defined as A positive LE demonstrates that small changes in the dynamical system's initial states can lead to completely different behaviors. Thus, one positive LE is an indicator of chaotic behaviors if some other conditions (e.g., phase space compactness) also meet [44]. A bigger positive LE means a faster separation rate. Our experiment uses the definition of LE to estimate the LEs of different chaotic maps. When estimating the LEs, the first 100 iteration values of the chaotic maps are discarded. Fig. 5 plots the computed LEs of the chaotic maps in Table I

B. Correlation Dimension
The CD is a type of fractal dimension. It is a metric to measure the space dimensionality occupied by a time series [43]. Different from the LE that tests the chaos from the deterministic equation(s), the CD can test the existence of chaos from experimental data. For an embedding dimension e and a time series {S i |i = 1, 2, . . . , N} generated by a chaotic system, the CD is defined as log r (16) where C e (r) is the correlation integral that is defined by where θ(α) is a step function. θ(α) = 1 if α ≥ 0 and θ(α) = 0 if α < 0. is the time delay unit and it usually equals to 1 for discrete-time system. If it exists, the CD can also be presented as the gradient of log C e (r) against log r, namely A bigger CD indicates that a dynamical system can generate outputs occupying a larger dimensionality and thus its attractors are more irregular. Our experiments use the Grassberger-Procacciai algorithm [45] to compute the CDs of time series generated by different chaotic maps and Fig. 6 shows the computed CDs of the nine chaotic maps with respect to parameters b ∈ [0, 1] and c ∈ [2, 2.8] when a = 1. The step sizes for b and c are 0.032 and 0.04, respectively. One can see that the chaotic maps of ECM have positive CDs in all the parameter space of b and c. The experimental results are consistent with the LE results. This means that the chaotic maps of ECM have robust chaotic behaviors from the viewpoints of experimental data.

C. Initial State Sensitivity
The most important and straightforward property of the "chaos" is the initial state sensitivity [46], which means that a dynamical system owing chaotic behaviors is sensitive to the change of its control parameter(s) and initial value. The initial state sensitivity can be measured using the correlation coefficient (CC), which is defined as where X t and Y t are two sequences of data, μ is the mean value, σ is the standard deviation, and E[.] is the expectation function. The absolute CC closing to 1 means that the data sequences X t and Y t have high relationship, and vice versa. For each of the nine chaotic maps produced by ECM, our experiments are designed as follows.  Fig. 8 plots the differences between S 3 and S 1 , and between S 4 and S 1 in the first 40 states. As can be seen, when the iteration increases, the small change in the initial value or control parameters can result in significantly different outputs. Table II shows the CCs of the nine chaotic maps produced by ECM. One can see that all the CCs approach to 0. These prove that the chaotic maps of ECM are quite sensitive to their initial states.

VI. APPLICATION TO SECURE COMMUNICATION
Chaos is a natural candidate for secure communication, due to its significant properties, such as the unpredictability and ergodicity [47]. When chaotic maps are applied to secure communication, the distribution of their chaotic signals greatly affects the performance of resisting channel noise. As the chaotic maps produced by ECM have uniformly distributed chaotic signals, they are suitable for secure communication.  Here, we use the reference-shifted differential chaos shift keying (RS-DCSK) [48] to demonstrate the application of the chaotic maps generated by ECM in secure communication.

A. RS-DCSK
The RS-DCSK is a kind of noncoherent frameworks [47] and it is able to recover the information bits using the inner correlation of the received signal. It can be divided into two  parts: 1) the transmitter and 2) receiver. The transmitter is to encode the information bits using chaotic sequences and sends the transmission signal to the receiver, while the receiver decodes the received signal to recover the information bits.
1) Transmitter: Fig. 9 shows the structure of the transmitter in RS-DCSK. The signal is transmitted frame by frame. A frame consists of two parts: 1) reference signal and 2) information signal. Suppose the spread factor is M, the reference signal is an M-length chaotic sequence generated by a chaotic generator, while the information signal is obtained by modifying the reference signal using the information bits. As can be seen from Fig. 9, the kth frame contains two information bits, b 2k and b 2k+1 . The information signal for b 2k is {x 2kM+1 , . . . , x 2(k+1)M }, while the information signal for b 2k+1 is {x (2k+1/2)M+1 , x (2k+1/2)M+2 , . . . , x 2(k+1)M , x 2kM+1 , . . . , x (2k+1/2)M }, which is obtained by circularly shifting the reference signal of b 2k by M/2. Subsequently, the reference signal and information signal are modulated to obtain the transmission signal s i . In summary, the kth frame of the transmission signal s i (k) can be obtained by the following equation: 2) Receiver: The receiver is to recover the information bits and it is shown in Fig. 10. Because a signal may be blurred by channel noise during transmission, the received signal is different from the original transmission signal and it can be described as r i = s i + ξ i , where ξ i is the noise. To recover the information bits, the received signal r i is multiplied with its delayed signal r i−M and r * i−M . The signal r * i−M is obtained by circularly shifting r i−M for M/2. Then two correlators can be where γ and η are noise components. As the energy of noise component is far smaller than the energy of the transmission signal s i , the symbols of Z 2k and Z 2k+1 are determined by the information components, namely respectively. The obtained Z i is a positive number if the information bit is "1"; and Z i is a negative number if the information bit is "−1". Thus, despite the noise components, the two information bits b 2k and b 2k+1 can be recovered bŷ

B. Simulation Results
When data are transmitted in noisy transmission channels, the additive white Gaussian noise (AWGN) and additive random noise (ARN) are frequently encountered. Thus, we simulate the RS-DCSK in the AWGN and ARN channels under different lengths of the spread factor. The bit error rate (BER) between the original data and recovered data is investigated to demonstrate the ability of resisting noise. The chaotic generator is selected as the nine chaotic maps of ECM in Table I and their base and exponent maps, respectively. The original information in each simulation is a randomly generated sequence with a 50 000-bit length. We design two groups of experiments to test the BERs of RS-DCSK using different chaotic maps as the chaotic generator. The first group of experiments investigates the BERs against different levels of noise. The experiments for each chaotic map are set as follows.
1) Generate ten initial states. The initial value is set as x 0 = 0.2 and the ten control parameters are obtained from the chaotic range. Specifically, for a chaotic range (C 1 , C 2 ) with a step size I, the mth control parameter is C 1 + mI. Table III lists the used chaotic ranges and step sizes for all the chaotic maps. 2) Set the length of the spread factor M = 40 and then simulate the RS-DCSK ten times using the ten initial states, and obtain the BERs in different levels of AWGN and ARN. 3) Calculate the average BERs of the ten times of experiments. Fig. 11 shows the average BERs of RS-DCSK using different chaotic maps in different levels of AWGN and ARN. The second group of experiments investigates the BERs against different lengths of the spread factor M. The experiments for each chaotic map are set as follows.
1) Generate ten initial states using the same way as the first group of experiments. 2) Set the channel noise AWGN and ARN as 18 dB and then simulate the RS-DCSK ten times using the ten initial states, and obtain the BERs under different lengths of the spread factor. 3) Calculate the average BERs of the ten times of experiments. Fig. 12 shows the average BERs of RS-DCSK using different chaotic maps under different lengths of the spread factor M. Fig. 11 shows that, with the increment of SNR in both AWGN and ARN channels, the RS-DCSK using the nine chaotic maps of ECM can obtain much smaller BERs than that of using the logistic, sine, and tent maps. Fig. 12 shows that, under various lengths of the spread factor, the RS-DCSK using the nine chaotic maps of ECM can always achieve much smaller BERs than that of using the three existing chaotic maps. This is because the nine chaotic maps generated by ECM can output much more uniform-distribution chaotic signals than the logistic, sine, and tent maps, as shown in Figs. 1 and 3. The transmission signals can achieve stronger ability of resisting channel noise if the used chaotic signals distribute more uniformly. As a result, the chaotic maps of ECM exhibit high performance in secure communication.

VII. CONCLUSION
Robust chaos is a desired property for many chaos-based applications. However, existing chaotic systems may not own this property. To address this issue, this article introduced an ECM to obtain new 1-D chaotic maps with robust chaos. Motivated by the nonlinear property of exponential operation in arithmetic, ECM employs two chaotic maps as base and exponent maps to produce new chaotic maps. To show the effectiveness of ECM, we used three existing chaotic maps as the base and exponent maps to produce nine new chaotic maps as examples. Theoretical analysis verified that these new chaotic maps can generate robust chaos. Performance evaluations using LE, CD, and CC demonstrated that the nine chaotic maps of ECM are sensitive to their initial states, and have robust chaotic behaviors. To show ECM in real applications, we applied the nine chaotic maps of ECM as chaotic generators in secure communication. The simulation results showed that these new chaotic maps exhibit better performance in resisting channel noise than their base and exponent maps.
Our future work will investigate the robust chaos of highdimensional chaotic systems and the application of robust chaos in pseudo-random number generator.