Abstract: An autonomous five-dimensional (5D) system with offset boosting is constructed by modifying the well-known three-dimensional autonomous Liu and Chen system. Equilibrium points of the proposed autonomous 5D system are found and its stability is analyzed. The proposed system includes Hopf bifurcation, periodic attractors, quasi-periodic attractors, a one-scroll chaotic attractor, a double-scroll chaotic attractor, coexisting attractors, the bistability phenomenon, offset boosting with partial amplitude control, reverse period-doubling, and an intermittency route to chaos. Using a field programmable gate array (FPGA), the proposed autonomous 5D system is implemented and the phase portraits are presented to check the numerical simulation results. The chaotic attractors and coexistence of the attractors generated by the FPGA implementation of the proposed system have good qualitative agreement with those found during the numerical simulation. Finally, a sound data encryption and communication system based on the proposed autonomous 5D chaotic system is designed and illustrated through a numerical example.

偏置增强自主五维混沌系统动态分析、FPGA实现及加密应用

[1]Azarang A, Ranjbar J, Mohseni H, et al., 2017. Output feedback synchronization of a novel chaotic system and its application in secure communication. Int J Comput Sci Netw Secur, 17:72-77.

[2]Bahi JM, Fang XL, Guyeux C, et al., 2013. FPGA design for pseudorandom number generator based on chaotic iteration used in information hiding application. Appl Math Inform Sci, 7(6):2175-2188.

[3]Barakat ML, Radwan AG, Salama KN, 2011. Hardware realization of chaos based block cipher for image encryption. Int Conf on Microelectronics.

[4]Charef A, 2006. Analogue realisation of fractional-order integrator, differentiator and fractional PI^λD^µ controller. IEEE Proc Contr Theory Appl, 153(6):714-720.

[5]Chen YM, Yang QG, 2015. A new Lorenz-type hyperchaotic system with a curve of equilibria. Math Comput Simul, 112:40-55.

[6]Chen YQ, Vinagre BM, Podlubny I, 2004. Continued fraction expansion approaches to discretizing fractional order derivatives—an expository review. Nonl Dynam, 38(1-4): 155-170.

[7]Dong EZ, Liang ZH, Du SZ, et al., 2016. Topological horseshoe analysis on a four-wing chaotic attractor and its FPGA implement. Nonl Dynam, 83(1-2):623-630.

[8]Hou YY, Chen HC, Chang JF, et al., 2012. Design and implementation of the Sprott chaotic secure digital communication systems. Appl Math Comput, 218(24): 11799-11805.

[9]Hu G, 2009. Generating hyperchaotic attractors with three positive Lyapunov exponents via state feedback control. Int J Bifurc Chaos, 19(2):651-660.

[10]Ismail SM, Said LA, Rezk AA, et al., 2017. Generalized fractional logistic map encryption system based on FPGA. Int J Electron Commun, 80:114-126.

[11]Jia Q, 2007. Projective synchronization of a new hyperchaotic Lorenz system. Phys Lett A, 370(1):40-45.

[12]Jiang CX, Carletta JE, Hartley TT, 2007. Implementation of fractional-order operators on field programmable gate arrays. In: Sabatier J, Agrawal OP, Tenreiro JA (Eds.), Machado Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht, the Netherlands, p.333-346.

[13]Li CQ, Lin DD, Feng BB, et al., 2018a. Cryptanalysis of a chaotic image encryption algorithm based on information entropy. IEEE Access, 6:75834-75841.

[14]Li CQ, Lin DD, Lü JH, et al., 2018b. Cryptanalyzing an image encryption algorithm based on autoblocking and electrocardiography. IEEE Multim, 25(4):46-56.

[15]Li CQ, Feng BB, Li SJ, et al., 2019. Dynamic analysis of digital chaotic maps via state-mapping networks. IEEE Trans Circ Syst I, 66(6):2322-2335.

[16]Li QD, Zeng HZ, Li J, 2015. Hyperchaos in a 4D memristive circuit with infinitely many stable equilibria. Nonl Dynam, 79(4):2295-2308.

[17]Li X, 2009. Modified projective synchronization of a new hyperchaotic system via nonlinear control. Commun Theor Phys, 52(2):274-278.

[18]Li XW, Wanga Y, Wang QH, et al., 2019. Modified integral imaging reconstruction and encryption using an improved SR reconstruction algorithm. Opt Laser Eng, 112:162- 169.

[19]Li YX, Chen GR, Tang WKS, 2005. Controlling a unified chaotic system to hyperchaotic. IEEE Trans Circ Syst II, 52(4):204-207.

[20]Liu WB, Chen GR, 2004. Dynamical analysis of a chaotic system with two double-scroll chaotic attractors. Int J Bifurc Chaos, 14(3):971-998.

[21]Lorenz EN, 1963. Deterministic nonperiodic flow. J Atmos Sci, 20(2):130-141.

[22]Ojoniyi OS, Njah AN, 2016. A 5D hyperchaotic Sprott B system with coexisting hidden attractors. Chaos Sol Fract, 87:172-181.

[23]Qi GY, van Wyk MA, van Wyk BJ, et al., 2008. On a new hyperchaotic system. Phys Lett A, 372(2):124-136.

[24]Rajagopal K, Guessas L, Vaidyanathan S, et al., 2017a. Dynamical analysis and FPGA implementation of a novel hyperchaotic system and its synchronization using adaptive sliding mode control and genetically optimized PID control. Math Probl Eng, 2017:7307452.

[25]Rajagopal K, Karthikeyan A, Srinivasan AK, 2017b. FPGA implementation of novel fractional-order chaotic systems with two equilibriums and no equilibrium and its adaptive sliding mode synchronization. Nonl Dynam, 87(4):2281- 2304.

[26]Rajagopal K, Guessas L, Karthikeyan A, et al., 2017c. Fractional order memristor no equilibrium chaotic system with its adaptive sliding mode synchronization and genetically optimized fractional order PID synchronization. Complexity, 2017:1892618.

[27]Rajagopal K, Karthikeyan A, Duraisamy P, 2017d. Hyperchaotic chameleon: fractional order FPGA implementation. Complexity, 2017:8979408.

[28]Rajagopal K, Kingni ST, Kuiate GF, et al., 2018. Autonomous Jerk oscillator with cosine hyperbolic nonlinearity: analysis, FPGA implementation, and synchronization. Adv Math Phys, 2018:7273531.

[29]Rech PC, 2014. Delimiting hyperchaotic regions in parameter planes of a 5D continuous-time dynamical system. Appl Math Comput, 247:13-17.

[30]Rössler OE, 1979. An equation for hyperchaos. Phys Lett A, 71(2-3):155-157.

[31]Shen CW, Yu SM, Lü JH, et al., 2014. A systematic methodology for constructing hyperchaotic systems with multiple positive Lyapunov exponents and circuit implementation. IEEE Trans Circ Syst I, 61(3):854-864.

[32]Singh JP, Rajagopal K, Roy BK, 2018. A new 5D hyperchaotic system with stable equilibrium point, transient chaotic behaviour and its fractional-order form. Pramana, 91(3): 33.

[33]Thamilmaran K, Lakshmanan M, Venkatesan A, 2004. Hyperchaos in a modified canonical Chua’s circuit. Int J Bifurc Chaos, 14(1):221-243.

[34]Tlelo-Cuautle E, Rangel-Magdaleno JJ, Pano-Azucena AD, et al., 2015. FPGA realization of multi-scroll chaotic oscillators. Commun Nonl Sci Numer Simul, 27(1-3):66-80.

[35]Vaidyanathan S, 2013. A ten-term novel 4-D hyperchaotic system with three quadratic nonlinearities and its control. Int J Contr Theory Appl, 6(2):97-109.

[36]Wang JH, Chen ZQ, Chen GR, et al., 2008. A novel hyperchaotic system and its complex dynamics. Int J Bifurc Chaos, 18(11):3309-3324.

[37]Wang QX, Yu SM, Li CQ, et al., 2016. Theoretical design and FPGA-based implementation of higher-dimensional digital chaotic systems. IEEE Trans Circ Syst I, 63(3):401- 412.

[38]Wang XY, Wang MJ, 2008. A hyperchaos generated from Lorenz system. Phys A, 387(14):3751-3758.

[39]Wei ZC, Moroz I, Sprott JC, et al., 2017. Hidden hyperchaos and electronic circuit application in a 5D self-exciting homopolar disc dynamo. Chaos, 27:033101.

[40]Wei ZC, Rajagopal K, Zhang W, et al., 2018. Synchronisation, electronic circuit implementation, and fractional-order analysis of 5D ordinary differential equations with hidden hyperchaotic attractors. Pramana, 90(4):50.

[41]Woods R, McAllister J, Yi Y, et al., 2017. FPGA-Based Implementation of Signal Processing Systems. Wiley, Chichester, UK.

[42]Xu YM, Wang LD, Duan SK, 2016. A memristor-based chaotic system and its field programmable gate array implementation. Acta Phys Sin, 65(12):120503 (in Chinese).

[43]Yang QG, Chen CT, 2013. A 5D hyperchaotic system with three positive Lyapunov exponents coined. Int J Bifurc Chaos, 23(6):1350109.

[44]Yang QG, Liu YJ, 2009. A hyperchaotic system from a chaotic system with one saddle and two stable node-foci. J Math Anal Appl, 360(1):293-306.

[45]Yang QG, Zhang KM, Chen GR, 2009. Hyperchaotic attractors from a linearly controlled Lorenz system. Nonl Anal Real World Appl, 10(3):1601-1617.