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Abstract

This introductory chapter summarizes recent advances on the simulation and electronic implementations of integer and fractional-order chaotic oscillators. It highlights the mathematical modeling and special methods to perform time simulation of chaotic systems and the associated issues for electronic realization.

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References

  1. I. Petráš, Fractional-Order Chaotic Systems (Springer, Berlin, 2011), pp. 103–184

    Book  Google Scholar 

  2. V.-T. Pham, S. Vaidyanathan, C. Volos, T. Kapitaniak, Nonlinear Dynamical Systems with Self-excited and Hidden Attractors, vol. 133 (Springer, Berlin, 2018)

    Book  MATH  Google Scholar 

  3. H.K. Khalil, Nonlinear Systems (Prentice Hall, Englewood Cliffs, 1996)

    Google Scholar 

  4. P.A. Cook, Nonlinear Dynamical Systems (Prentice Hall, Englewood Cliffs, 1994)

    Google Scholar 

  5. H. Degn, A.V. Holden, L.F. Olsen, Chaos in Biological Systems, vol. 138 (Springer, New York, 2013)

    Google Scholar 

  6. V.H. Carbajal-Gomez, E. Tlelo-Cuautle, J.M. Muñoz-Pacheco, L.G. de la Fraga, C. Sanchez-Lopez, F.V. Fernandez-Fernandez, Optimization and CMOS design of chaotic oscillators robust to PVT variations. Integration 65, 32–42 (2018)

    Article  Google Scholar 

  7. A.D. Pano-Azucena, J. de Jesus Rangel-Magdaleno, E. Tlelo-Cuautle, A. de Jesus Quintas-Valles, Arduino-based chaotic secure communication system using multi-directional multi-scroll chaotic oscillators. Nonlinear Dynam. 87(4), 2203–2217 (2017)

    Article  Google Scholar 

  8. A.D. Pano-Azucena, E. Tlelo-Cuautle, J.M. Muñoz-Pacheco, L.G. de la Fraga, FPGA-based implementation of different families of fractional-order chaotic oscillators applying Grünwald–Letnikov method. Commun. Nonlinear Sci. Numer. Simul. 72, 516–527 (2019)

    Article  MathSciNet  Google Scholar 

  9. A.A. Rezk, A.H. Madian, A.G. Radwan, A.M. Soliman, Reconfigurable chaotic pseudo random number generator based on FPGA. AEU-Int. J. Electron. Commun. 98, 174–180 (2019)

    Article  Google Scholar 

  10. O. Guillén-Fernández, A. Meléndez-Cano, E. Tlelo-Cuautle, J.C. Núñez-Pérez, J. de Jesus Rangel-Magdaleno, On the synchronization techniques of chaotic oscillators and their FPGA-based implementation for secure image transmission. PloS One 14(2), e0209618 (2019)

    Article  Google Scholar 

  11. C.K. Volos, D.A. Prousalis, S. Vaidyanathan, V.-T. Pham, J.M. Munoz-Pacheco, E. Tlelo-Cuautle, Kinematic control of a robot by using a non-autonomous chaotic system, in Advances and Applications in Nonlinear Control Systems (Springer, Berlin, 2016), pp. 1–17

    MATH  Google Scholar 

  12. T.S. Parker, L. Chua, Practical Numerical Algorithms for Chaotic Systems (Springer, New York, 2012)

    MATH  Google Scholar 

  13. E. Tlelo-Cuautle, L.G. de la Fraga, J. Rangel-Magdaleno, Engineering Applications of FPGAs (Springer, Berlin, 2016)

    Book  Google Scholar 

  14. J.D. Lambert, Computational Methods in Ordinary Differential Equations (Wiley, Hoboken, 1973)

    MATH  Google Scholar 

  15. R.M. Corless, What good are numerical simulations of chaotic dynamical systems? Comput. Math. Appl. 28(10–12), 107–121 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  16. C. Varsakelis, P. Anagnostidis, On the susceptibility of numerical methods to computational chaos and superstability. Commun. Nonlinear Sci. Numer. Simul. 33, 118–132 (2016)

    Article  MathSciNet  Google Scholar 

  17. E.N. Lorenz, Deterministic nonperiodic flow. J. Atmos. Sci. 20(2), 130–141 (1963)

    Article  MATH  Google Scholar 

  18. O.E. Rössler, An equation for continuous chaos. Phys. Lett. A 57(5), 397–398 (1976)

    Article  MATH  Google Scholar 

  19. G. Chen, T. Ueta, Yet another chaotic attractor. Int. J. Bifur. Chaos 9(7), 1465–1466 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  20. J. Lü, G. Chen, S. Zhang, Dynamical analysis of a new chaotic attractor. Int. J. Bifur. Chaos 12(5), 1001–1015 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. C. Liu, T. Liu, L. Liu, K. Liu, A new chaotic attractor. Chaos Solitons Fractals 22(5), 1031–1038 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  22. M.A. Zidan, A.G. Radwan, K.N. Salama, Controllable v-shape multiscroll butterfly attractor: system and circuit implementation. Int. J. Bifur. Chaos 22(6), 1250143 (2012)

    Article  MATH  Google Scholar 

  23. J.C. Sprott, Some simple chaotic flows. Phys. Rev. E 50(2), R647 (1994)

    Article  MathSciNet  Google Scholar 

  24. M.W. Hirsch, S. Smale, R.L. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos (Academic, Cambridge, 2012)

    MATH  Google Scholar 

  25. A.D. Pano-Azucena, E. Tlelo-Cuautle, G. Rodriguez-Gomez, L.G. De la Fraga, FPGA-based implementation of chaotic oscillators by applying the numerical method based on trigonometric polynomials. AIP Adv. 8(7), 075217 (2018)

    Article  Google Scholar 

  26. D. Schleicher, Hausdorff dimension, its properties, and its surprises. Am. Math. Mon. 114(6), 509–528 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. A. Wolf, J.B. Swift, H.L. Swinney, J.A. Vastano, Determining Lyapunov exponents from a time series. Phys. D Nonlinear Phenomena 16(3), 285–317 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  28. V.H. Carbajal-Gómez, E. Tlelo-Cuautle, F.V. Fernández, L.G. de la Fraga, C. Sánchez-López, Maximizing Lyapunov exponents in a chaotic oscillator by applying differential evolution. Int. J. Nonlinear Sci. Numer. Simul. 15(1), 11–17 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. A. Silva-Juarez, G. Rodriguez-Gomez, L.G. de la Fraga, O. Guillen-Fernandez, E. Tlelo-Cuautle, Optimizing the Kaplan–Yorke dimension of chaotic oscillators applying de and PSO. Technologies 7(2), 38 (2019)

    Article  Google Scholar 

  30. G. Cardano, T.R. Witmer, Ars Magna or the Rules of Algebra. Dover Books on Advanced Mathematics (Dover, New York, 1968)

    Google Scholar 

  31. I. Petráš, Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation (Springer, New York, 2011)

    Book  MATH  Google Scholar 

  32. A. Oustaloup, Fractional order sinusoidal oscillators: optimization and their use in highly linear FM modulation. IEEE Trans. Circuits Syst. 28(10), 1007–1009 (1981)

    Article  Google Scholar 

  33. A. Arenta, R. Caponetto, L. Fortuna, D. Porto, Nonlinear Non-integer Order Circuits and Systems. World Scientific Series on Nonlinear Science, Series A, vol. 38 (World Scientific, Singapore, 2002)

    Google Scholar 

  34. W.M. Ahmad, J.C. Sprott, Chaos in fractional-order autonomous nonlinear systems. Chaos, Solitons Fractals 16(2), 339–351 (2003)

    Article  MATH  Google Scholar 

  35. A.T. Azar, A.G. Radwan, S. Vaidyanathan, Fractional Order Systems: Optimization, Control, Circuit Realizations and Applications (Academic, Cambridge, 2018)

    Google Scholar 

  36. K. Rajagopal, S. Çiçek, A.J.M. Khalaf, V.-T. Pham, S. Jafari, A. Karthikeyan, P. Duraisamy, A novel class of chaotic flows with infinite equilibriums and their application in chaos-based communication design using DCSK. Z. Naturforsch. A 73(7), 609–617 (2018)

    Article  Google Scholar 

  37. C.K. Volos, S. Jafari, J. Kengne, J.M. Munoz-Pacheco, K. Rajagopal, Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-excited Attractors (MDPI, Basel, 2019)

    Google Scholar 

  38. D. Baleanu, J.A.T. Machado, A.C.J. Luo, Fractional Dynamics and Control (Springer, New York, 2011)

    Google Scholar 

  39. C. Li, X. Liao, J. Yu, Synchronization of fractional order chaotic systems. Phys. Rev. E 68(6), 067203 (2003)

    Google Scholar 

  40. R. Martínez-Guerra, C.A. Pérez-Pinacho, Advances in Synchronization of Coupled Fractional Order Systems: Fundamentals and Methods (Springer, Berlin, 2018)

    Book  MATH  Google Scholar 

  41. A.T. Azar, S. Vaidyanathan, A. Ouannas, Fractional Order Control and Synchronization of Chaotic Systems, vol. 688 (Springer, Berlin, 2017)

    Book  MATH  Google Scholar 

  42. A. Tepljakov, Fractional-Order Modeling and Control of Dynamic Systems (Springer, Berlin, 2017)

    Book  MATH  Google Scholar 

  43. K. Rajagopal, S. Jafari, S. Kacar, A. Karthikeyan, A. Akgül, Fractional order simple chaotic oscillator with saturable reactors and its engineering applications. Inf. Technol. Control 48(1), 115–128 (2019)

    Google Scholar 

  44. L.F. Ávalos-Ruiz, C.J. Zúñiga-Aguilar, J.F. Gómez-Aguilar, R.F. Escobar-Jiménez, H.M. Romero-Ugalde, FPGA implementation and control of chaotic systems involving the variable-order fractional operator with Mittag–Leffler law. Chaos Solitons Fractals 115, 177–189 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  45. K. Rajagopal, F. Nazarimehr, A. Karthikeyan, A. Srinivasan, S. Jafari, Fractional order synchronous reluctance motor: analysis, chaos control and FPGA implementation. Asian J. Control 20(5), 1979–1993 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  46. Z. Wei, A. Akgul, U.E. Kocamaz, I. Moroz, W. Zhang, Control, electronic circuit application and fractional-order analysis of hidden chaotic attractors in the self-exciting homopolar disc dynamo. Chaos Solitons Fractals 111, 157–168 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  47. E.-Z. Dong, Z. Wang, X. Yu, Z.-Q. Chen, Z.-H. Wang, Topological horseshoe analysis and field-programmable gate array implementation of a fractional-order four-wing chaotic attractor. Chin. Phys. B 27(1), 010503 (2018)

    Article  Google Scholar 

  48. K. Rajagopal, G. Laarem, A. Karthikeyan, A. Srinivasan, FPGA implementation of adaptive sliding mode control and genetically optimized PID control for fractional-order induction motor system with uncertain load. Adv. Differ. Equ. 2017(1), 273 (2017)

    Google Scholar 

  49. K. Rajagopal, A. Karthikeyan, P. Duraisamy, Bifurcation analysis and chaos control of a fractional order portal frame with nonideal loading using adaptive sliding mode control. Shock. Vib. 2017, Article ID 2321060, 14 (2017)

    Google Scholar 

  50. D.K. Shah, R.B. Chaurasiya, V.A. Vyawahare, K. Pichhode, M.D. Patil, FPGA implementation of fractional-order chaotic systems. AEU-Int. J. Electron. Commun. 78, 245–257 (2017)

    Article  Google Scholar 

  51. A. Karthikeyan, K. Rajagopal, Chaos control in fractional order smart grid with adaptive sliding mode control and genetically optimized PID control and its FPGA implementation. Complexity 2017, Article ID 3815146, 18 (2017)

    Google Scholar 

  52. K. Oldham, J. Spanier, The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order, vol. 111 (Elsevier, Amsterdam, 1974)

    MATH  Google Scholar 

  53. S.S. Ray, Fractional Calculus with Applications for Nuclear Reactor Dynamics (CRC Press, Boca Raton, 2015)

    Book  Google Scholar 

  54. O.M. Duarte, Fractional Calculus for Scientists and Engineers (Springer, Berlin, 2011), 114 pp.

    MATH  Google Scholar 

  55. F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models (World Scientific, Singapore, 2010)

    Book  MATH  Google Scholar 

  56. V.E. Tarasov, Fractional Dynamics; Applications of the Fractional Calculus to Dynamics of Particles, Fields and Media (Springer, Berlin, 2010), 522 pp.

    MATH  Google Scholar 

  57. D. Baleanu, Z.B. Günvec, M.J.A. Tenreiro, New Trends in Nanotechnology and Fractional Calculus Applications (Springer, Berlin, 2010), 544 pp.

    Book  Google Scholar 

  58. C.-B. Fu, A.-H. Tian, Y.-C. Li, H.-T. Yau, Fractional order chaos synchronization for real-time intelligent diagnosis of islanding in solar power grid systems. Energies 11(5), 1183 (2018)

    Article  Google Scholar 

  59. Z. Gan, X. Chai, K. Yuan, Y. Lu, A novel image encryption algorithm based on LFT based S-boxes and chaos. Multimed. Tools Appl. 77(7), 8759–8783 (2018)

    Article  Google Scholar 

  60. V.P. Latha, F.A. Rihan, R. Rakkiyappan, G. Velmurugan, A fractional-order model for Ebola virus infection with delayed immune response on heterogeneous complex networks. J. Comput. Appl. Math. 339, 134–146 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  61. X. Lin, S. Zhou, H. Li, H. Tang, Y. Qi, Rhythm oscillation in fractional-order relaxation oscillator and its application in image enhancement. J. Comput. Appl. Math. 339, 69–84 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  62. K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations (Wiley, Hoboken, 1993)

    MATH  Google Scholar 

  63. I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Mathematics in Science and Engineering (Elsevier, Amsterdam, 1999)

    Google Scholar 

  64. M. Caputo, Linear models of dissipation whose Q is almost frequency independent-II. Geophys. J. Int. 13(5), 529–539 (1967)

    Article  Google Scholar 

  65. L. Dorcak, J. Prokop, I. Kostial, Investigation of the properties of fractional-order dynamical systems, in Proceedings of 11th International Conference on Process Control (1994), pp. 19–20

    Google Scholar 

  66. I. Pan, S. Das, Intelligent Fractional Order Systems and Control: An Introduction, vol. 438 (Springer, Berlin, 2012)

    MATH  Google Scholar 

  67. W. Deng, J. Lü, Generating multi-directional multi-scroll chaotic attractors via a fractional differential hysteresis system. Phys. Lett. A 369(5–6), 438–443 (2007)

    Article  MATH  Google Scholar 

  68. N.J. Ford, A.C. Simpson, The numerical solution of fractional differential equations: speed versus accuracy. Numer. Algorithms 26(4), 333–346 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  69. Y. Chen, I. Petras, D. Xue, Fractional order control - a tutorial, in 2009 American Control Conference (2009), pp. 1397–1411

    Google Scholar 

  70. I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and some of Their applications, vol. 198 (Elsevier, Amsterdam, 1998)

    MATH  Google Scholar 

  71. D. Cafagna, G. Grassi, On the simplest fractional-order memristor-based chaotic system. Nonlinear Dynam. 70(2), 1185–1197 (2012)

    Article  MathSciNet  Google Scholar 

  72. R. Garrappa, Short tutorial: solving fractional differential equations by Matlab codes. Department of Mathematics, University of Bari (2014)

    Google Scholar 

  73. M.-F. Danca, N. Kuznetsov, Matlab code for Lyapunov exponents of fractional-order systems. Int. J. Bifurcation Chaos 28(5), 1850067 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  74. K. Diethelm, N.J. Ford, A.D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dynam. 29(1–4), 3–22 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  75. J.M. Muñoz-Pacheco, E. Zambrano-Serrano, O. Félix-Beltrán, L.C. Gómez-Pavón, A. Luis-Ramos, Synchronization of PWL function-based 2d and 3d multi-scroll chaotic systems. Nonlinear Dynam. 70(2), 1633–1643 (2012)

    Article  MathSciNet  Google Scholar 

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Tlelo-Cuautle, E., Dalia Pano-Azucena, A., Guillén-Fernández, O., Silva-Juárez, A. (2020). Integer and Fractional-Order Chaotic Circuits and Systems. In: Analog/Digital Implementation of Fractional Order Chaotic Circuits and Applications. Springer, Cham. https://doi.org/10.1007/978-3-030-31250-3_1

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