A new 4D chaotic system with hidden attractor and its engineering applications: Analog circuit design and field programmable gate array implementation

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1 Pramana J. Phs. (2018) 90:70 Indian Academ of Sciences A new 4D chaotic sstem with hidden attractor and its engineering applications: Analog circuit design and field programmable gate arra implementation HAMID REZA ABDOLMOHAMMADI 1, ABDUL JALIL M KHALAF 2, SHIRIN PANAHI 3, KARTHIKEYAN RAJAGOPAL 4, VIET-THANH PHAM 5 and SAJAD JAFARI 3, 1 Department of Electrical Engineering, Golpaegan Universit of Technolog, Golpaegan, Iran 2 Department of Mathematics, Facult of Computer Science and Mathematics, Universit of Kufa, Najaf, Iraq 3 Biomedical Engineering Department, Amirkabir Universit of Technolog, Tehran , Iran 4 Department of Electrical and Communication Engineering, The Papua New Guinea Universit of Technolog, Lae, Papua New Guinea 5 School of Electronics and Telecommunications, Hanoi Universit of Science and Technolog, 01 Dai Co Viet, Hanoi, Vietnam Corresponding author. sajadjafari@aut.ac.ir MS received 19 November 2017; revised 15 December 2017; accepted 19 December 2017; published online 30 April 2018 Abstract. Nowadas, designing chaotic sstems with hidden attractor is one of the most interesting topics in nonlinear dnamics and chaos. In this paper, a new 4D chaotic sstem is proposed. This new chaotic sstem has no equilibria, and so it belongs to the categor of sstems with hidden attractors. Dnamical features of this sstem are investigated with the help of its state-space portraits, bifurcation diagram, Lapunov exponents diagram, and basin of attraction. Also a hardware realisation of this sstem is proposed b using field programmable gate arras (FPGA). In addition, an electronic circuit design for the chaotic sstem is introduced. Kewords. PACS Nos Chaotic sstem; no equilibrium; hidden attractors; field programmable gate arras a; Ac; Pq 1. Introduction Chaos, a phenomenon with some unusual features, was introduced to the world b Lorenz during his investigations on weather dnamics [1]. After his important exploration, a considerable amount of literatures has been published about chaos and chaotic sstems [2 4]. The emphasis of man studies on chaotic sstems has been on equilibrium points which were supposed to pla important roles in our general understanding of chaos [5 8]. But all these lasted till the introduction of chaotic sstems with no equilibria [9 12]. Then a new interesting branch was opened to the world of chaos. Before the exploration of these new sstems, there was a general belief that unstable equilibria are helpful to locate conventional chaotic attractors. It means that we need a point on the unstable manifold which should be close enough to the unstable fixed point to send the trajector directl to the attractors. If this situation cannot be satisfied, the attractor(s) of the sstem are called hidden attractors [13 16]. One of the most important features of sstems with hidden attractors is the unexpected and the afflictive response to the perturbation in the structure or external noise [17 20]. Hidden attractors have been found to be important even in real dnamical sstems [21,22]. This topic has been a ver hot topic in recent ears [23 25], which is interesting for scientists who work on engineering applications of chaotic sstems such as control [26], biomedical engineering [27], communication [28], optimisation [29], quantum [30], fuzz [31], encrption [32] and random number generation [33,34]. Man engineering applications use FPGA because of its effective and eas implementation as it can be configured for an nonlinear function implementation [35 39]. In this paper we introduce a novel no-equilibrium chaotic jerk sstem. Such sstems are in close relation with other rare chaotic sstems which have been introduced recentl. From them we can point out sstems with surfaces of equilibria [40,41], curves

2 70 Page 2 of 7 Pramana J. Phs. (2018) 90:70 of equilibria [42 44], stable equilibria [45,46], nonhperbolic equilibria [47,48], multiscroll attractors [49 51], multistabilit [52 54] and extreme multistabilit [55 58]. The rest of this paper is organised as follows: In 2 we introduce the proposed sstem and describe its dnamical properties. Section 3 is about FPGA implementation of the proposed sstem. In 4, we present an electronic design for the sstem. Finall 5 gives the conclusion. Different projections of the strange attractors are plotted in figure 1, when the parameters are set as a = 1, b = Now we investigate the dnamical behaviour of the proposed sstem with respect to changing parameters a and b. Figure 2a represents the bifurcation diagram and figure 2b represents Lapunov exponents of sstem (1) with respect to change in parameter a, while figure 3 is plotted for parameter b. The well-known period doubling route to chaos can be observed from both figures. The basin of attraction for sstem (1) is plotted in figure 4. As mentioned before, there are no equilibria which can intersect with the basin of strange attractor. Therefore, the strange attractor is hidden. 2. New no-equilibrium chaotic sstem The new 4D chaotic flow, which has no equilibria, can be written as follows: x z w = a =z =w = 0.06x x 0.56x z 0.12xw +bw FPGA implementation Most of the hardware realisations of dnamical sstems use field programmable gate arras (FPGA) because of their effective and eas implementations. FPGAs are attractive tools in realising differential equations. The parallel architectural feature of FPGA along with the hardware functionalit can be appropriatel configured to our desired designs. FPGAs can be applied for an nonlinear function implementation which makes it the best choice for nonlinear ODEs. Hence, these platforms can serve as engineering solutions for conducting scientific experiments [59]. (1) This sstem has no equilibria which can be investigated as follows: a = 0 z = x = 0 w = x x 0.56x z 0.12xw + bw 0.71 = 0. (2) z x w x w w z x z Figure 1. Different projections of the strange attractor of sstem (1) for a = 1 and b = The initial conditions are ( 0.45, 1.2, 3.31, 0.2).

3 Pramana J. Phs. (2018) 90:70 Page 3 of 7 70 (a) (b) λ 1 λ 2 λ 3 λ a Figure 2. (a) Bifurcation diagram of sstem (1) with respect to parameter a (for b = 0.49) and (b) Lapunov exponents of sstem (1) with respect to parameter a. Figure 4. Basin of attractions for sstem (1) in the plane z = 0. Initial conditions in the light blue region lead to the chaotic attractor and unbounded regions are shown in ellow. dx i dt [x i (k + 1) x i (k)] = lim. (3) h 0 h The discrete state equations for implementing in FPGA can be derived b using (3) in the sstem, (a) x(k + 1) = [(k)]h + x(k) (k + 1) = [z(k)]h + (k) z(k + 1) = [w(k)]h + z(k) w(k + 1) [ 0.06x(k) = 2 ] 0.38x(k)(k) 0.56x(k)z(k) h 0.12x(k)w(k) 0.49(k)w(k) w(k). (4) λ (b) Figure 3. (a) Bifurcation diagram of sstem (1) with respect to parameter b (a = 1) and (b) Lapunov exponents of sstem (1) with respect to parameter b. The challenge in implementing the proposed chaotic attractors is the integrator block configuration and design which is not a readil available block in the sstem generator. Hence, we implement the integrators using the mathematical relation, a The value of h is taken as and the initial conditions are fed into the forward register. The sstem generator model is then cross-compiled to generate Vivado project files which are finall implemented in Kintex 7 (xc7k160tfbg484-1). RTL schematics are generated and presented to represent the better definition of the design. Static and dnamic power consumption in watts is also presented. We utilise the hardware software co-simulation [59] to implement the chaotic hperjerk sstems. For this, we use the sstem generator toolbox of Matlab for Xilinx Vivado 2016 version and Simulink is used just to see the output waveforms. The communication between the hardware and software is established with Ethernet-like protocols and we have designed the hardware co-simulation for execution on Kintex 7-XC7 chipset. For implementing integrator, we use the Xilinx register transfer level (RTL) schematic diagram shown in figure 5. Figure 6 shows the overall RTL schematics and figure 7 shows the state level RTL schematics of the chaotic hperjerk sstem.

4 70 Page 4 of 7 Pramana J. Phs. (2018) 90:70 Figure 5. RTL schematic block diagram of the integrator. Figure 6. Complete RTL schematic block diagram of chaotic hperjerk sstem with clocks. Figure 7. RTL schematic block diagram of sstem (1). Table 1. Resources utilized b the chaotic sstem (1). Resource Utilisation Available Utilization (%) LUT FF DSP IO BUFG Figure 8. Power utilised b the FPGA-implemented sstem (1). Figure 8 shows the power utilisation chart and table 1 represents the resources utilised b the FPGAimplemented chaotic hperjerk sstem. Figure 9 shows the 2D phase portraits of the chaotic hperjerk sstem using hardware and software co-simulation with Simulink used for seeing the phase portraits and

5 Pramana J. Phs. (2018) 90:70 Page 5 of 7 70 Figure 9. 2D phase portraits of sstem (1) using hardware and software co-simulation with Simulink used for seeing the phase portraits and Kintex 7-XC7 FPGA with Xilinx sstem generator used for processing. (a) x, (b) z, (c) zw and (d) wx planes. Figure 10. Circuit design for the proposed hperjerk sstem (1). Kintex 7-XC7 FPGA with Xilinx sstem generator used for processing. 4. Electronic circuit design of the sstem Electronic circuit is another approach to realise the theoretical hperjerk sstem (1). As shown in figure 10, we have designed a circuit for sstem (1) with electronic components [60 66]. B denoting the voltages at the operational amplifiers (U1, U2, U3, U4 ) as x,, z, w, the circuital equation is described b x = 1 R1 C

6 70 Page 6 of 7 Pramana J. Phs. (2018) 90:70 Figure 11. PSpice phase portraits of the electronic circuit. ẏ = 1 R 2 C z ż = 1 R 3 C w ẇ = 1 10R 4 C x2 1 10R 5 C x 1 10R 6 C xz 1 10R 7 C xw 1 10R 8 C w 1 R 9 C V 1. (5) The circuit has been implemented b using PSpice for R 1 = R 2 = R 3 = R 9 = R = 100 k, R 4 = k, R 5 = k, R 6 = k, R 7 = k, R 8 = k,c = 1nF, and V 1 = 0.71 V DC. PSpice results obtained in figure 11 indicate that the circuit exhibits chaotic attractors. 5. Conclusion In this paper, a new 4D chaotic flow is presented. Dnamical analsis was done for this new chaotic sstem which showed that the new sstem has no equilibria and so it has hidden attractors. The basin of attraction of the sstem is plotted which is another proof for having hidden attractor because its chaotic area does not have an intersection with an unstable fixed point. Then, bifurcation diagram and Lapanov exponent were plotted with respect to the parameters a and b, showing a period doubling route to chaos. The FPGA implementation, which is one of the most proper and useful hardware realisations of chaotic sstem, was proposed for this new chaotic sstem. In addition, we have presented an electronic circuit of the hperjerk sstem, which provides another wa to implement the theoretical model. References [1] E N Lorenz, J. Atmos. Sci. 20(2), 130 (1963) [2] O E Rössler, Phs. Lett. A 57(5), 397 (1976) [3] G Chen and T Ueta, Int. J. Bifurc. Chaos 9, 1465 (1999) [4] J C Sprott, Phs. Rev. E 50, R647 (1994) [5] X Wang and G Chen, Commun. Nonlinear Sci. 17, 1264 (2012) [6] X Wang and G Chen, Nonlinear Dnam. 71(3), 429 (2013) [7] L P Shil Nikov, Sov. Math. Dokl. 6, 163 (1965) [8] L P Shil nikov, Methods of qualitative theor in nonlinear dnamics (World Scientific, 2001) Vol. 5 [9] Z Wei, Phs. Lett. A 376(2), 102 (2011) [10] S Jafari, J Sprott and S M R H Golpaegani, Phs. Lett. A 377, 699 (2013) [11] Y Lin, C Wang, H He and L L Zhou, Pramana J. Phs. 86, 801 (2016) [12] Y Feng and W Pan, Pramana J. Phs. 88, 62 (2017) [13] D Dudkowski, S Jafari, T Kapitaniak, N V Kuznetsov, G A Leonov and A Prasad, Phs. Rep. 637, 1 (2016) [14] G A Leonov, N Kuznetsov, O Kuznetsova, S Seledzhi and V Vagaitsev, Trans. Sst. Contr. 6, 54 (2011) [15] G Leonov, N Kuznetsov and V Vagaitsev, Phs. Lett. A 375, 2230 (2011) [16] W Pan and L Li, Pramana J. Phs. 88, 87 (2017) [17] G Leonov, N Kuznetsov and T Mokaev, Commun. Nonlinear Sci. Numer. Simul. 28, 166 [18] G Leonov, N Kuznetsov and T Mokaev, Eur. Phs. J. Spec. Top. 224, 1421 [19] P Sharma, M Shrimali, A Prasad, N Kuznetsov and G Leonov, Eur. Phs. J. Spec. Top. 224, 1485 [20] P R Sharma, M D Shrimali, A Prasad, N V Kuznetsov and G A Leonov, Int. J. Bifurc. Chaos 25, [21] Z Wei, I Moroz, Z Wang, J C Sprott and T Kapitaniak, Int. J. Bifurc. Chaos 26, (2016) [22] Z Wei, I Moroz, J C Sprott, Z Wang and W Zhang, Int. J. Bifurc. Chaos 27, (2017)

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