Abstract
Multiscroll chaotic attractors generated by irregular saturated nonlinear functions with optimized positive Lyapunov exponent are designed and implemented. The saturated nonlinear functions are designed in an irregular way by modifying their parameters such as slopes, delays between slopes, and breakpoints. Then, the positive Lyapunov exponent is optimized using the differential evolution algorithm to obtain chaotic attractors with 2 to 5 scrolls. We observed that the resulting chaotic attractors present more complex dynamics when different patterns of irregular saturated nonlinear functions are considered. After that, the optimized chaotic oscillators are physically implemented with an analog discrete circuit to validate the use of proposed irregular saturated functions. Experimental results are consistent with MATLAB™ and SPICE circuit simulator. Finally, the synchronization between optimized and nonoptimized chaotic oscillators is demonstrated.
1. Introduction
The chaotic behavior has attracted a lot of attention for scientific community due to extreme sensitivity to its initial conditions and the broadband nature of its chaotic signals [1–28]. Therefore, in the last years, literature is vast in papers oriented to study new chaotic systems [1–5], propose novel applications [6–9], increase the degree of chaos (hyperchaotic systems [8, 10, 11]), get fractional order chaotic systems [7, 12, 13], synchronize the chaotic behavior [14–18], optimize chaotic systems [19–22], and implement chaotic oscillators using electronic circuits [23–28]. In all these studies, chaos behavior is analyzed and verified by using different approaches, for example, frequency spectrum, Poincaré maps, bifurcation diagrams, Lyapunov exponents, and stability of equilibrium points. Among them, Lyapunov exponents provide a direct measure of the sensitive dependence on initial conditions by quantifying the exponential rates at which neighboring orbits on an attractor diverge as the system evolves in time [29–31].
For an -dimensional nonlinear system, if the system has at least one positive Lyapunov exponent (LE) and is purely deterministic, then it is chaotic. Indeed, a tool commonly used to determine the presence of chaos in several numerical and experimental results is to compute only the positive LE [31].
Besides, the positive LE can be very useful to determine the unpredictability grade of the chaotic oscillator because its magnitude specifies the maximum average exponential rate corresponding to divergence of trajectories on an attractor and thus the maximum amount of instability along any direction [29–31]. That is, a high value of the positive LE can be taken as an indication of a high degree of chaos in the dynamical system [19–22, 32–35].
For instance, in [32] the speed effects and leg amputations on the dynamic stability of running are analyzed by computing the largest LE. The results revealed that the value of the largest LE is positive not only for unaffected patients but also for the one-leg affected patients. However, the positive LE of embedded time-series data from the affected leg was higher than for the unaffected leg indicating a more rich dynamics. In [33] a numerical scheme based on the Lattice Boltzmann method for the flow in complex mixer geometries to compute trajectories of passive tracers for the quantification of chaotic mixing was reported. They reported a better efficiency in chaotic micromixers when the value of positive LE was higher. Moreover, chaotic systems with a high value of the positive LE have also been used to improve the performance of chaos-based applications, for example, in [34] it was demonstrated that a high value of the positive LE in an optimization algorithm based on a particle swarm implies that the particles are inclined to explore different regions and find better fitness values. Therefore, the particle swarm with just a little variation in the value of positive LE usually achieved a better performance, especially for multimodal functions. In [35] the efficiency of hybrid chaotic optimization algorithms was studied by revealing effects on the search speed as a function of chaotic sequences from different chaotic maps. It was found that the higher the magnitude of positive LE, the faster the search speed in whole optimization space. Accordingly, the efficiency of global optimization was directly proportional to the value of positive LE.
In this framework, optimized chaotic systems with a high value of the positive LE can be extremely suitable to enhance the existing chaos-based applications. In electronics, a great variety of multiscroll chaotic oscillators has been implemented with commercially available electronic devices, as well as with integrated circuits technology [5, 10, 23, 26–28, 36, 37]. However, those experimental realizations are not optimized to provide a high value of the positive Lyapunov exponent (PLE). Although some authors have already used optimization algorithms based on evolutionary computation to get optimized multiscrolls chaotic systems [19, 22], they were obtained by using piecewise-linear (PWL) functions in the form of saturated nonlinear functions (SNLF) with symmetry properties. In addition, the experimental verification of those approaches is lacking.
This paper is motivated by the aforementioned discussion. In that scenario, we design irregular SNLF to obtain multiscroll chaotic attractors with optimized values of the positive LE. Additionally, we also demonstrate its practical feasibility by the physical implementation of the resulting multiscroll chaotic oscillators. Two cases were considered to design the irregular SNLFs. The first one consists of changing the breakpoints of SNLF to get different slopes, whereas the second one modifies the delay between slopes in different sections of SNLF. In both cases, once the parameters of SNLF are defined, we apply the evolutionary algorithms reported in [19, 22] to find the optimal value for system’s parameters which maximizes the magnitude of positive LE. As a result, multiscroll chaotic oscillators with a more complex dynamics are generated. Experimental results for 2-, 3-, 4-, and 5-scroll chaotic attractors were obtained with the aim of an analog discrete circuit based on commercial operational amplifiers (OpAmps). Further, we show the synchronization of those optimized chaotic oscillators by using generalized Hamiltonian forms because they can enhance the synchronization and realization of secure communication systems, for instance.
The paper is organized as follows. Section 2 describes the multiscroll chaotic oscillator under study; Section 3 outlines the steps to obtain optimized values of positive LE as well as the electronic design. Sections 4 and 5 present the experimental confirmation of the proposed approach for the two cases: slopes varying and different delays between slopes, respectively. Section 6 demonstrates the synchronization between optimized and nonoptimized chaotic oscillators. Finally, conclusions are given in Section 7.
2. SNLF-Based Multiscroll Chaotic Oscillator
The case of study in this work is the chaotic oscillator described bywhere is the state variables, is the SNLF, and is the system’s parameters. To maximize the value of positive LE requires varying the coefficients of chaotic oscillator, leading to a huge number of combinations. Herein, system’s parameters are varied within the range . So, we define four variables where each one can have = possible combinations. This result justifies the application of heuristics like the ones already introduced in [19, 22]. For all cases analyzed in this work, the phase-space plots show the state variables versus .
The first step consists of manipulating with the goal of incrementing the complexity of multiscroll chaotic oscillator. To generate 2 scrolls, the SNFL description is given bywhere represents the value of saturated regions, is the slope between two saturated regions, is a break point connecting a saturated region with a slope, and is the delay, as shown in Figure 1. In general, the number of scrolls to be generated equals the number of saturated regions . Therefore, (2) can be augmented to generate -scrolls as shown in [38]. It means that by augmenting segments in a symmetric way as shown in Figures 1(a) and 1(b), respectively, even and odd number of scrolls are generated. However, in this work we show how to use irregular SNLF functions, that is, nonsymmetric, to obtain multiscrolls as the ones shown in Figure 2. The irregular SNLF functions are herein designed using different values for breakpoints with where is the number of breakpoints, slopes , and saturated levels .
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3. Methodology to Maximize the Positive LE of Multiscrolls Oscillators and Its Circuit Design
The solution to (1) by using symmetric SNLF described by (2) has been performed by applying evolutionary algorithms in [19, 22], where the positive LE was optimized. By applying the differential evolution (DE) algorithm already introduced in [22], we obtained several feasible solutions for the coefficients , and , which provide higher values of the positive LE as the number of scrolls is incremented. In all cases, the value of positive LE computed by DE is higher than when using traditional values (TV) for system’s parameters as TV = [19]. Details of applying DE algorithm to optimize the positive LE can be found in [22].
On the other hand, the SNLF can be implemented as shown in Figure 3, where the number of saturated levels (SL) determines the number of OpAmps to be used considering OpAmp = SL-1, as shown in [23]. In that reference the whole electronic realization of (1) is given by using OpAmps. That realization is redrawn in Figure 4, where the SNLF function is embedded in block PWL, and system’s parameters , and are implemented by the ratio of resistors , , , and , respectively.
Otherwise, this article shows how realizing irregular SNLFs by modifying Figure 3, such as asymmetric variation of the slopes, modifying the breakpoints , and asymmetric variation of the saturated levels . The experiments are performed considering the following steps:(1)Modify the SNLFs to obtain an irregular one using MATLAB.(2)Optimize the value of positive LE by computing the optimal values for , and with the DE algorithm given in [22].(3)Implement physically the SNLFs with OpAmps.(4)Validate experimentally the optimized chaotic oscillator by generating the required number of scrolls.
For instance, the MATLAB, SPICE, and experimental results of a 5-scrolls chaotic attractor obtained with a irregular SNLF are shown in Figure 5. More details are given in the following sections.
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4. Experimental Results by Varying the Slopes of SNLFs
The slope in SNLF is described herein by , as shown in Figure 1, which is modified asymmetrically to generate irregular SNLFs for obtaining 3, 4, and 5 scrolls. The first step consists of determining the minimum and maximum slopes values using MATLAB, so that the results to attain from 2 to 7 scrolls are given in Table 1. To generate more scrolls, the minimum and maximum values are the same as for 6 and 7 scrolls. Next, the values of circuit elements to get different slopes in SNLF are found. This is done by using commercially available OpAmps in Figure 3. Table 2 lists the calculated values to obtain slopes of 5, 10, 20, 30, 40, 50, and 100. In all cases it is assumed a saturation voltage = 15 V for the OpAmps. Using Table 2, one can combine different slopes to realize irregular SNLFs. In this manner, Table 3 shows the combinations of slopes to generate 3, 4, and 5 scrolls. In the experiments, those values have been selected randomly from Table 2. Each combination of slopes is used to implement the irregular SNLF described by (2) and then to implement the dynamical system described in (1). All cases in Table 3 for generating 3-, 4-, and 5-scroll chaotic attractors with optimized positive LE are described below.
4.1. Optimized 3-Scrolls Chaotic Oscillator with Different Slopes
Using the irregular SNLF with the slopes listed in Table 3 and setting traditional values of , the positive LE is listed in Table 4. This is positive for the three cases indicating chaotic behavior. Afterwards, by applying DE algorithm [22] the optimized positive LE provides new coefficient values that are also listed in Table 4. As supposed the value of positive LE is higher than those values without optimization. The optimization of positive LE applying [22] was executed using a population of 50 individuals and 50 iterations.
According to case 1 from Table 3 and from Figure 1, the generation of 3 scrolls with optimized positive LE implies slopes being , which are located in the third and first quadrants, respectively. Cases 2 and 3 require the slopes being and , respectively. The circuit element values for realizing these slopes were already listed in Table 2. Figure 6 shows the experimental results for the irregular SNLF for each case and the associated attractor showing 3 scrolls with optimized positive LE.
(a) Slopes: = 5 and 10. Ch1:1V/Div, Ch2:1V/Div
(b) Slopes: =10 and 20. Ch1:1V/Div, Ch2:1V/Div
(c) Slopes: = 20 and 50. Ch1:1V/Div, Ch2:1V/Div
4.2. Optimized 4-Scroll Chaotic Oscillator with Different Slopes
For generating 4 scrolls with optimized positive LE, the values of the slopes are also listed in Table 3. Again, by setting traditional values , positive LE is listed in Table 4. As a result, it is more positive for the three cases than those values computed for 3 scrolls in Table 4. This confirms that positive LE increases by augmenting the number of scrolls.
The optimized positive LE provides new values for system’s parameters that are listed in Table 4. As can be appreciated, the optimized positive LE is also higher for each case than for 3 scrolls. The optimization for positive LE applying [22] was executed using a population of 50 individuals and 50 iterations. The irregular SNLFs are realized again using the slopes listed in Table 3 and Figure 1. Again, the circuit element values for realizing the three slopes are taken from Table 2. Figure 7 shows the experimental results for the irregular SNLF and the associated chaotic attractor showing 4 scrolls. As one can infer, according to Table 1, the first case for 4 scrolls uses the minimum slope value of 5, which is also appreciated in the irregular SNLF shown in Figure 7(a).
(a) Slopes: = 10, 10, and 5. Ch1:1V/Div, Ch2:1V/Div
(b) Slopes: = 50, 10, and 20. Ch1:1V/Div, Ch2:1V/Div
(c) Slopes: = 30, 50, and 70. Ch1:1V/Div, Ch2:1V/Div
4.3. Optimized 5-Scroll Chaotic Oscillator with Different Slopes
Generating 5 scrolls with optimized positive LE implies using four slopes, which are listed in Table 3. For those cases and by setting traditional values of , the positive LE is listed in Table 4, where their magnitudes are slightly more higher than those ones for 3 and 4 scrolls in Table 4. Again, positive LE increases by augmenting the number of scrolls from 4 to 5.
The optimized positive LE is listed in Table 4, which is also higher than for generating 4 scrolls. The irregular SNLFs are realized again using the slopes listed in Table 3 and Figure 1. Four slopes are used because the SNLF shown in Figure 1(b) is increased. Once again, the circuit element values are taken from Table 2. Figure 8 shows the experimental results for the irregular SNLF and the associated chaotic attractor showing 5 scrolls.
(a) Slopes: = 10, 10, 20, and 20. Ch1:1V/Div, Ch2:1V/Div
(b) Slopes: = 10, 30, 30, and 10. Ch1:1V/Div, Ch2:1V/Div
(c) Slopes: = 10, 100, 10, and 100. Ch1:1V/Div, Ch2:1V/Div
In this section the irregular SNLFs was realized by modifying the values of slopes . The experiments confirmed the generation of 3, 4, and 5 scrolls that have an optimized positive LE, which was computed by applying DE algorithm in [22].
5. Experimental Results by Varying the Delay of Slopes in SNLFs
This section shows the experimental verification of optimized multiscroll chaotic oscillators by varying asymmetrically the delay (see Figure 1) of slopes , that is, the distance separating the center of slope with respect to the horizontal axis. Contrary to the previous section, the value of slopes is keept as a constant for all cases, for example, . Also, the saturation levels are the same.
The physical realization of irregular SNLFs using commercially available OpAmps requires multiple voltage dividers to get the required voltages in Figure 3. Table 5 lists the values of for each slope, according to the number of scrolls to be generated.
5.1. Optimized 2-Scroll Chaotic Oscillator with Different Delays
Table 6 lists the positive LEs of three cases from Table 5 by setting traditional values of . In this case, only case 1 shows chaotic regime because cases 2 and 3 do not have a positive LE. However, after applying the optimization algorithm from [22], all cases have a positive LE, as listed in Table 6.
The experimental realization is performed by using system’s parameters listed in Table 6 and the irregular SNLF with delays listed in Table 5. In case 1, = −0.5 V for in Figure 3, so that the saturation region on the right is wider that the left one. This gives as a result a larger scroll in the region of larger saturation width, as shown in Figure 9(a). The center of slope defined by delay is the connecting point of neighbouring scrolls, that is, at −0.5 V. The other two cases are shown in Figures 9(b) and 9(c), respectively.
(a) Case 1: = −0.5. Ch1:1V/Div, Ch2:1V/Div
(b) Case 2: = 1. Ch1:2V/Div, Ch2:2V/Div
(c) Case 3: = 1.5. Ch1:2V/Div, Ch2:2V/Div
5.2. Optimized 3-Scroll Chaotic Oscillator with Different Delays
Table 6 lists the positive LE for generating 3 scrolls and by setting traditional values of . All cases have a positive LE. After applying the optimization algorithm from [22], the value of positive LE is increased as listed in Table 6. Again, as for the previous section, the chaotic complexity is being increased by augmenting the number of scrolls. From Table 5, the three cases for obtaining 3 scrolls and by varying the delay are shown in Figure 10.
(a) Case 1: = −1 and 1.5. Ch1:2V/Div, Ch2:2V/Div
(b) Case 2: = −2 and 1. Plotting state variables versus , instead of versus . On the left, Ch1:2V/Div, Ch2:2V/Div; on the right Ch1:1V/Div, Ch2:1V/Div
(c) Case 3: = −1.5 and 2. Ch1:2V/Div, Ch2:2V/Div
5.3. Optimized 4-Scroll Chaotic Oscillator with Different Delays
The three cases for getting 4 scrolls and by setting traditional values of have the positive LE listed in Table 6. Not all the cases have a positive LE; however, after applying the optimization DE algorithm from [22], all cases have a greater positive LE than for 3 scrolls, as listed in Table 6, thus confirming again that the chaotic complexity is being increased by augmenting the number of scrolls. From Table 5, the three cases for obtaining 4 scrolls and by varying the delay are shown in Figure 11.
(a) Case 1: = −2, 0.5 and 2. Ch1:2V/Div, Ch2:2V/Div
(b) Case 2: = −2.5, 0 and 1.5. Ch1:1V/Div, Ch2:1V/Div
(c) Case 3: = −1.5, 0 and 2. On the left Ch1:2V/Div, Ch2:2V/Div, and on the right Ch1:1V/Div, Ch2:1V/Div
5.4. Optimized 5-Scroll Chaotic Oscillator with Different Delays
Finally, the three cases for getting 5 scrolls and by setting traditional values of have the positive LE listed in Table 6. All the cases have a positive LE, and they are optimized by applying the optimization DE algorithm [22], where only case 3 has a positive LE greater than for generating 4 scrolls, as listed in Table 6. Anyway, the chaotic complexity is being increased by augmenting the number of scrolls. From Table 5, the three cases for attaining 5 scrolls and by varying the delay are shown in Figure 12.
(a) Case 1: = −3, −1, 1.3 and 2.5. On the left Ch1:2V/Div, Ch2:2V/Div, and on the right Ch1:1V/Div, Ch2:1V/Div
(b) Case 2: = −3, −1.5, 0.5 and 2.5. On the left Ch1:2V/Div, Ch2:2V/Div, and on the right Ch1:1V/Div, Ch2:1V/Div
(c) Case 3: = −2.5, −1, 1 and 2.5. On the left Ch1:2V/Div, Ch2:2V/Div, and on the right Ch1:1V/Div, Ch2:1V/Div
6. Synchronization between Optimized and Nonoptimized Multiscroll Chaotic Attractors
In this section the synchronization between optimized and nonoptimized multiscroll chaotic attractors is shown by applying the proposed approach given in [39], that is, generalized Hamiltonian forms. Besides, it is demonstrated that, by using an optimized 4-scroll chaotic oscillator as master system, the slave system that is not optimized can show a similar behavior. This is an important result since it is not necessary to optimize each chaotic oscillator independently.
Then, let us take the multiscroll chaotic system (1) as the master system as follows:By using a Hamilton energy function defined by , we can obtain the slave system defined bywhich synchronizes with master system (4) when the synchronization error tends to zero by selecting appropriate synchronization gains . The synchronization gains are selected as explained in detail in [39].
6.1. Synchronization of Nonoptimized 4-Scroll with Case 1 in Table 4: Different Slopes
By using different slopes for SNLF (irregular) as given in case 1, Table 4, the resulting 4-scroll chaotic attractor can be synchronized with another 4-scroll chaotic attractor that preserves traditional values for system’s parameters as well as a regular SNLF. For master system, we use the slopes = (case 1, Table 4) for irregular SNLF, system’s parameters , , , , synchronization gains , and initial conditions . On the other hand, for slave system traditional values , , , are considered; also a symmetrical SNLF with slopes = are used, and . Figure 13 shows the simulation results of synchronization. In Figure 13(a) the phase portraits for master system and slave system without and with synchronization are given. Figures 13(b) and 13(c) show the and planes for the same 4-scroll attractor. Synchronization error is represented as a straight line in phase planes , , and as shown in Figure 13(d).
(a) Optimized 4-scroll chaotic system, typical 4-scroll chaotic system without optimization, and typical 4-scroll chaotic system after synchronization ( plane)
(b) Optimized 4-scroll chaotic system, typical 4-scroll chaotic system without optimization, and typical 4-scroll chaotic system after synchronization ( plane)
(c) Optimized 4-scroll chaotic system, typical 4-scroll chaotic system without optimization, and typical 4-scroll chaotic system after synchronization ( plane)
(d) Synchronization error for optimized and nonoptimized 4-scroll chaotic system
6.2. Synchronization of Nonoptimized 4-Scroll with Case 1 in Table 6: Different Delays
Similar to previous section, by using different delays for slopes in SNLF (irregular) as given in case 1, Table 6, the resulting 4-scroll chaotic attractor can be synchronized with another 4-scroll chaotic attractor that preserves traditional values for system’s parameters as well as a regular SNLF. For master system, we use different delays , slope = 10 (case 1, Table 6) for irregular SNLF, system’s parameters , , , , synchronization gains , and initial conditions . On the other hand, for slave system traditional values , , , are considered; also a symmetrical SNLF with slopes = and delays are used, . Figure 14 shows the simulation results of synchronization. In Figure 14(a) the phase portraits for master system and slave system without and with synchronization are given. Synchronization error between , , and is given in Figure 14(b).
(a) Optimized 4-scroll chaotic system, typical 4-scroll chaotic system without optimization, and typical 4-scroll chaotic system after synchronization ( plane)
(b) Synchronization error for optimized and nonoptimized 4-scroll chaotic system
Next, Figure 15 shows the transient response for synchronization. In both cases, the numerical simulations confirm an adequate synchronization between optimized and nonoptimized multiscroll chaotic oscillators.
As a general result, the experimental realizations of optimized multiscroll chaotic oscillators confirmed that the irregular SNLF provides a suitable mechanism to obtain a positive LE with a higher magnitude than without optimization as shown in Tables 4 and 6. In addition, the irregular SNLF proposed in this paper leads to getting a high value of the positive LE when compared to other approaches also using evolutionary algorithms to maximize the positive LE as shown in Table 7. The main characteristics, such as maximum value of the positive LE, experimental realization, and synchronization, of this work are given in Table 7.
7. Conclusions
It has been shown the experimental realization of optimized multiscrolls chaotic attractors. More specifically, 2 to 5 scrolls were obtained by using irregular SNLFs. Two cases were considered: variation of the slopes of SNLF and variation of the delays of slopes. Both cases led us to implement irregular SNLFs and, by applying an optimization evolutionary algorithm, the positive LE was maximized. From the experimental results, it can be appreciated that the higher value of positive Lyapunov exponents is obtained by varying the slopes of SNLF. Also, all the experiments listed in the last two sections confirmed that the value of positive LE increases by augmenting the number of scrolls. On the other hand, the synchronization between optimized and nonoptimized chaotic attractors was demonstrated. Further, this research could be useful to propose engineering applications based on chaos, for example, secure communications, since the obtained results have been performed at experimental circuit level.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work has been partially supported by CONACYT-SNI (Mexico). The authors thankfully acknowledge the computer resources, technical expertise, and support provided by the “Laboratorio Nacional de Supercómputo del Sureste de México (National Laboratory of High Performance Computing),” belonging to CONACYT Network of National Laboratories. J. M. Muñoz-Pacheco acknowledges CONACYT for the financial support (no. 258880: Proyecto Apoyado por el Fondo Sectorial de Investigación para la Educación).