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IEEE/CAA Journal of Automatica Sinica
Citation: | Xiaoyuan Wang, Chenxi Jin, Xiaotao Min, Dongsheng Yu and Herbert Ho Ching Iu, "An Exponential Chaotic Oscillator Design and Its Dynamic Analysis," IEEE/CAA J. Autom. Sinica, vol. 7, no. 4, pp. 1081-1086, July 2020. doi: 10.1109/JAS.2020.1003252 |
IN nature, chaos is a ubiquitous phenomenon. In the 1960s, the famous meteorologist Lorenz found that the solution of a completely determined set of third-order ordinary differential equations, when selecting a specific range of parameters, becomes very irregular and uncertain [1]. Subsequently, many scientists and scholars have conducted extensive and in-depth research on the Lorenz system, and successively discovered many systems with chaotic properties. In the late 1990s, Chen et al.built the Chen system [2]. Later, Lü et al. built the Lü system on the basis of the Lorenz system and Chen system [3], and showed the evolution between these two systems. In order to enrich the type of chaotic systems and meet the needs of their application, it is necessary to improve the existing chaotic systems to get the systems with higher nonlinearity and complexity. In a chaotic system, the nonlinear term can affect the complexity of the system [4]. The optimizations of chaotic systems are usually made by the following two modifications to the nonlinear term of the existing system: one method is to make simple adjustments to the nonlinear terms without affecting its order [5], [6], another way is to modify the nonlinear term to a higher order nonlinear term, such as changing the product term to an exponential or logarithmic function [7]–[10].
Since chaotic systems are aperiodic, unpredictable and extremely sensitive to the initial conditions, they can produce high-performance pseudo-random sequences and can be widely used in the fields such as secure communication [11]–[14], image encryption [15]–[18], and random number generators [19], [20]. The randomness of the sequence generated by the chaotic system is mainly related to the complexity and nonlinearity of the system equations. The higher the complexity and nonlinearity of the system are, the more random the resulting sequence is. Therefore, it is valuable to design a chaotic system with stronger nonlinearity (such as an exponential chaotic system).
In this paper, the Lü system is improved to obtain a more complex chaotic system based on an exponential function. In Section II, the novel exponential chaotic system is proposed and verified by drawing the phase diagrams and calculation of the Lyapunov exponents. In Section III, dynamic characteristics analyses of the new system are carried out, including equilibrium points, stabilities of the system, Poincaré mappings, Lyapunov exponent spectrums, and bifurcation diagrams. Especially in Section III-E, through the NIST test, the randomness of the sequence generated by the exponential chaotic system is quantitatively analyzed. In Section IV, the mathematical model of this chaotic system is transformed into a circuit model and the Multisim circuit simulation is carried out. Finally, some conclusions are drawn in Section V.
The product nonlinear term xy of the third equation of the Lü system is replaced by an exponential nonlinear term exy to construct a new exponential chaotic system. Its mathematical equations can be described as
{˙x=a(y−x)˙y=cy−xz˙z=exy−bz |
(1) |
where x, y, and z are system state variables, a, b, and c are parameters within a certain change region.
When the parameters of the system are set to a= 20.75, b= 6.05, and c= 12.2, the new exponential chaotic system is in chaos state. By setting the initial values of the chaotic system as x0= 0.01, y0= 0.01, z0= 0.01, the chaotic attractors of the system obtained by the MATLAB simulation are shown in Fig. 1.
The Lyapunov exponents of the exponential chaotic system are calculated as LE1= 2.4241, LE2= 0.00025537, and LE3= –17.0243 by Jacobi method. The Lyapunov dimension D is 2.1424 calculated by (2), which indicates that the system is chaotic under the appropriate parameters.
D=j+1|LEj+1|j∑i=1LEi=2+(LE1+LE2)|LE3|=2+(2.4241+0.00025537)|−17.0243|=2.1424 |
(2) |
where j is the largest integer satisfying LE1 + ··· +LEj≥ 0.
The exponential chaotic system proposed in this paper is symmetrical about the z-axis, which can be proved by transforming (x, y, z) to (–x, –y, z) as shown in
{˙x=a(y−x)˙y=cy−xz˙z=exy−bz⇔{−˙x=−a(y−x)−˙y=−(cy−xz˙z=exy−bz.) |
(3) |
The dissipative property of the exponential chaotic system is analyzed first. By adding the partial differentiation of each equation in (1) with respect to x, y, and z, the expression of divergence can be obtained as follows:
ΔV=∂˙x∂x+∂˙y∂y+∂˙z∂z=−a−b+c |
(4) |
when the parameters are set to a = 20.75, b = 6.05, and c = 12.2, the divergence of the system is: ΔV = –a – b + c = –20.75 – 6.05 + 12.2 = –14.6 < 0. It indicates that the system is dissipated and may generate chaos. Under the condition of these parameters, the system converges in exponential form with the convergence rate as
dVdt=e−(a+b−c)t. |
(5) |
The above analyses illustrate that the trajectory of the system will eventually be confined to a zero-volume area and the progressive movement of the system will be fixed to the attractor, which indicate the existence of the attractor in this system.
By setting
{˙x=a(y−x)=0˙y=cy−xz=0˙z=exy−bz=0. |
(6) |
The three equilibrium points of the system areS1 = (0, 0, 0.1653), S2 = (2.0912, 2.0912, 12.2), and S3 = (–2.0912, –2.0912, 12.2). And the Jacobi matrix of the system is
J=[−aa0−zc−xyexyxexy−b]. |
(7) |
Let the characteristic equation |J–λE| = 0, where E is the identity matrix, then the corresponding eigenvalues of the three equilibrium points can be obtained: the eigenvalues corresponding to the equilibrium point S1 = (0, 0, 0.1653) are λ1 = –20.6456, λ2 = 12.0956, and λ3 = –6.0500. The two negative real roots of the characteristic equation correspond to two stable solutions, one positive real root corresponds to one unstable solution. Since the two eigenvalues are negative real numbers and one eigenvalue is a positive real number, the equilibrium point S1 is the confluence of an unstable manifold and two stable manifolds. The three characteristic values corresponding to equilibrium points S2= (2.0912, 2.0912, 12.2) and S3 = (–2.0912, –2.0912, 12.2) are
Time domain waveform is also an important method to analyze whether the system is in a chaotic state. From the time domain waveform, the changing trajectories of the state variable x, y, z with time can be clearly and intuitively observed. When parameters are given as: a = 20.75, b = 6.05, c = 12.2 and the initial value of the system is chosen as x0 = 0.01, y0 = 0.01, z0 = 0.01, the time series diagrams of the exponential system are as shown in Fig. 2, which are quasi-random, non-periodicand consistent with the characteristics of the chaotic state.
In addition, the Poincaré mappings are also drawn under the above parameters, as shown in Fig. 3. The cross planes in Figs. 3(a)–3(c) are selected as z = 20, y = 0, and x = 0, respectively. It can be observed from Fig. 3 that the Poincaré map is shown as dense points with layer structures, which indicates that the system is in the chaotic state.
Since the change of parameters will directly affect the equilibrium points of the system and its corresponding stability, changing parameters will certainly cause the system to be in different states. Therefore, it is necessary to discuss the effect of changing parameters on the system. The steps of the entire Lyapunov exponent spectrum shown below are set to 0.05. And each bifurcation diagram contains 1000 points.
By setting the initial conditions of the system to x0 = 0.01, y0 = 0.01, and z0 = 0.01, fixing the parameters b = 6.05, c = 12.2, and changing parameter a among the interval of [17.5, 27], the Lyapunov exponent spectrum of the system for each different acan be shown in Fig. 4(a). Also, the bifurcation diagram of the state variable x is shown in Fig. 4(b).
For three-dimensional systems, there are three Lyapunov exponents, which are recorded as LE1, LE2, and LE3 from large to small. When the parameter a ∈ [19.1,23.65]∩[24.45, 25.15], in most casesLE1> 0,LE2 = 0, and LE3< 0, which means that the system is in the chaotic state. And when the parametera is in the range of [17.5, 19.1]∩[23.65, 24.45]LE1 = 0 > LE2>LE3, which signifies the system is in the periodic state.
Next, by keeping a = 20.75 and c = 12.2 and changing the value of parameter b within [0, 11], we can get the Lyapunov exponent spectrum and the bifurcation diagram of the state variable x, as shown in Fig. 5. From the figure, we can find that the system is in different states with the change of parameter b. When the parameter b exceeds 2.65, the system is in chaotic state for most of the time until b reaches 10.4.
Finally, by fixing the parameters a = 20.75, b = 6.05 and changing the parameter c in the interval [8, 16], the Lyapunov exponent spectrum and the bifurcation diagram of the state variable x can be obtained, which is shown in Fig. 6.
As can be seen from Fig. 6, with parameter c increasing, the system gradually changes from periodic state to chaotic state, and finally returns to periodic state. When c∈[8.3, 13.45]∩[13.95, 14.55], the system changes from quasi-periodic state to chaotic state. Finally, afterc is greater than 16.75, the system returns to the periodic state.
In order to further study the influence of parameters, taking the parameter a as an example to illustrate how the system goes from period-1 to chaotic state by period-doubling bifurcations phase diagrams. When a is equal to 17.6, the system is in period-1 as shown in Fig. 7(a). As the parameter a gradually increases, the system goes through period-2, period-n, and finally be in chaotic state.
In order to further analyze the performance of the exponential chaotic system in practical applications, the Statistical Test Suite provided by the National Institute of Standards and Technology of the United States was used for analysis. The version of the Statistical Test Suite used in this analysis is 2.1.2. Due to the data generated by the chaotic system needs to be discretized into a binary sequence in this test. Here, the variable x is selected for generating the binary sequence. So after calculating the sequence of the variable x by the MATLAB simulation, we generate the binary sequence through the following equation:
xb=round(arcsin(√x−xminxmax−xmin)×108/π)mod2 |
(8) |
where xmax and xmin are the maximum and minimum values of the variable x, round represents the rounding operation, and mod means modular arithmetic. Finally, 1000 sets of binary sequences with a length of 1 000 000 were obtained through the MATLAB. The test results are shown in Table I.
Statistical test | Exponential chaotic system | Lü system | |||
P-value | Proportion | P-value | Proportion | ||
Frequency | 0.672470 | 0.992 | 0.560545 | 0.991 | |
Block frequency | 0.307077 | 0.992 | 0.137282 | 0.992 | |
Cumulative sums | 0.641284 | 0.991 | 0.862883 | 0.989 | |
Runs | 0.836048 | 0.995 | 0.080027 | 0.997 | |
Longest run | 0.108791 | 0.988 | 0.858002 | 0.989 | |
Rank | 0.077131 | 0.988 | 0.660012 | 0.987 | |
FFT | 0.478839 | 0.993 | 0.004981 | 0.983 | |
Non overlapping Template | 0.996839 | 0.988 | 0.996677 | 0.991 | |
Overlapping template | 0.164425 | 0.985 | 0.083018 | 0.991 | |
Universal | 0.437274 | 0.990 | 0.087692 | 0.988 | |
Approximate entrop | 0.894918 | 0.988 | 0.721777 | 0.997 | |
Random excursions | 0.902994 | 0.988 | 0.974555 | 0.981 | |
Random excursions Variant | 0.877948 | 0.989 | 0.924461 | 0.992 | |
Serial | 0.169044 | 0.989 | 0.000000 | 0.984 | |
Linear complexity | 0.334538 | 0.992 | 0.455937 | 0.992 |
The P-value is the probability that a perfect random number generator would have produced a sequence less random than the sequence that was tested. If the P-value is greater than 0.01 and Proportion is greater than 0.98, it is acceptable that the sequence is random. As can be seen from Table I, the exponential chaotic system passed all fifteen tests, but the Lü system passed only fourteen of them. Also the exponential chaotic system has 9 tests with P-values greater than those of the Lü system in all 15 tests. It shows that the sequence generated by the exponential chaotic system is more random than that produced by the Lü system.
In order to further verify the effectiveness of the designed system in actual circuits, the corresponding analog circuit is built. Because the maximum value of the system state variable z is greater than 40 in Fig. 2, which exceeds the linear dynamic range of the operational amplifier LF347 powered by ±13.5 V. So it is necessary to perform proportional compression transformation on each state variable, so we set x→ mx, y → my, z → mz, where m is the state variable proportional compression factor, and take m = 10, the following equation can be obtained:
{dxdt=20.75y−20.75xdydt=12.2y−10xzdzdt=110e100xy−6.05z. |
(9) |
Then, in order to make the trajectories of the chaotic circuit denser, transformation on time scale should be done. Let t → τ0τ, where τ0 is the time scale transformation factor, and setting τ0 equal to 100, we can convert the system equation as follows:
{dxdτ=−2075(−y)−2075xdydτ=−1220(−y)−1000xzdzdτ=−10(−e100xy)−605z. |
(10) |
According to (10), the equivalent circuit is designed as shown in Fig. 8, where Fig. 8(a) is an inverter circuit. The circuits shown in Fig. 8(b) and 8(c) both have the functions of addition and integration. In Fig. 8(d), operation amplifier U4 and resistors R7, R8 constitute the proportional operation circuit; the diode D1, operational amplifier U5, and resistor R9 form the exponential operation circuit; the operation amplifier U6, capacitor C3, and resistors R10, R11 form the addition circuit and the integration circuit. The outputs of all circuits are as follows:
{v−y=−R2R1vyvx=−1R3C1vx−1R4C1v−yvy=−1R5C2v−y−110R6C2vxvzvz=1R11C3R9Ise−R8vxv−y10R7UT−1R10C3vz. |
(11) |
According to the chaotic circuit diagrams in Fig. 8, the Multisim simulations are done by setting suitable parameters and choosing correct components as shown in Fig. 9. The simulation results are shown in Fig. 10. It can be seen that the circuit simulation results are consistent with the numerical simulation results of the MATLAB.
In this paper, a novel exponential chaotic system is successfully constructed and the dynamic properties of the system are studied by classical dynamic analysis methods and NIST test. The results indicate that by changing the product term to its exponential form in classical Lü system, the randomness and complexity of the presented system has been improved effectively compared with the original system, which indicates that the exponential chaotic system we proposed is more suitable for real applications, such as secure communications, information security, information hiding and chaos cryptography. Moreover, the equivalent circuit of the proposed system has been built to verify the theoretical analysis, and the simulation results are consistent with the theoretical analyses.
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Statistical test | Exponential chaotic system | Lü system | |||
P-value | Proportion | P-value | Proportion | ||
Frequency | 0.672470 | 0.992 | 0.560545 | 0.991 | |
Block frequency | 0.307077 | 0.992 | 0.137282 | 0.992 | |
Cumulative sums | 0.641284 | 0.991 | 0.862883 | 0.989 | |
Runs | 0.836048 | 0.995 | 0.080027 | 0.997 | |
Longest run | 0.108791 | 0.988 | 0.858002 | 0.989 | |
Rank | 0.077131 | 0.988 | 0.660012 | 0.987 | |
FFT | 0.478839 | 0.993 | 0.004981 | 0.983 | |
Non overlapping Template | 0.996839 | 0.988 | 0.996677 | 0.991 | |
Overlapping template | 0.164425 | 0.985 | 0.083018 | 0.991 | |
Universal | 0.437274 | 0.990 | 0.087692 | 0.988 | |
Approximate entrop | 0.894918 | 0.988 | 0.721777 | 0.997 | |
Random excursions | 0.902994 | 0.988 | 0.974555 | 0.981 | |
Random excursions Variant | 0.877948 | 0.989 | 0.924461 | 0.992 | |
Serial | 0.169044 | 0.989 | 0.000000 | 0.984 | |
Linear complexity | 0.334538 | 0.992 | 0.455937 | 0.992 |