Advertisement

Integer and Fractional-Order Chaotic Circuits and Systems

  • Esteban Tlelo-Cuautle
  • Ana Dalia Pano-Azucena
  • Omar Guillén-Fernández
  • Alejandro Silva-Juárez
Chapter

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.

Keywords

Chaos Equilibrium point Eigenvalue Numerical method Multistep Lyapunov exponent Kaplan-Yorke dimension Fractional calculus Fractional-order chaotic oscillator 

References

  1. 1.
    I. Petráš, Fractional-Order Chaotic Systems (Springer, Berlin, 2011), pp. 103–184CrossRefGoogle Scholar
  2. 2.
    V.-T. Pham, S. Vaidyanathan, C. Volos, T. Kapitaniak, Nonlinear Dynamical Systems with Self-excited and Hidden Attractors, vol. 133 (Springer, Berlin, 2018)zbMATHCrossRefGoogle Scholar
  3. 3.
    H.K. Khalil, Nonlinear Systems (Prentice Hall, Englewood Cliffs, 1996)Google Scholar
  4. 4.
    P.A. Cook, Nonlinear Dynamical Systems (Prentice Hall, Englewood Cliffs, 1994)Google Scholar
  5. 5.
    H. Degn, A.V. Holden, L.F. Olsen, Chaos in Biological Systems, vol. 138 (Springer, New York, 2013)Google Scholar
  6. 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)CrossRefGoogle Scholar
  7. 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)CrossRefGoogle Scholar
  8. 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)MathSciNetCrossRefGoogle Scholar
  9. 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)CrossRefGoogle Scholar
  10. 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)CrossRefGoogle Scholar
  11. 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. 1–17zbMATHGoogle Scholar
  12. 12.
    T.S. Parker, L. Chua, Practical Numerical Algorithms for Chaotic Systems (Springer, New York, 2012)zbMATHGoogle Scholar
  13. 13.
    E. Tlelo-Cuautle, L.G. de la Fraga, J. Rangel-Magdaleno, Engineering Applications of FPGAs (Springer, Berlin, 2016)CrossRefGoogle Scholar
  14. 14.
    J.D. Lambert, Computational Methods in Ordinary Differential Equations (Wiley, Hoboken, 1973)zbMATHGoogle Scholar
  15. 15.
    R.M. Corless, What good are numerical simulations of chaotic dynamical systems? Comput. Math. Appl. 28(10–12), 107–121 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 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)MathSciNetCrossRefGoogle Scholar
  17. 17.
    E.N. Lorenz, Deterministic nonperiodic flow. J. Atmos. Sci. 20(2), 130–141 (1963)zbMATHCrossRefGoogle Scholar
  18. 18.
    O.E. Rössler, An equation for continuous chaos. Phys. Lett. A 57(5), 397–398 (1976)zbMATHCrossRefGoogle Scholar
  19. 19.
    G. Chen, T. Ueta, Yet another chaotic attractor. Int. J. Bifur. Chaos 9(7), 1465–1466 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    J. Lü, G. Chen, S. Zhang, Dynamical analysis of a new chaotic attractor. Int. J. Bifur. Chaos 12(5), 1001–1015 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    C. Liu, T. Liu, L. Liu, K. Liu, A new chaotic attractor. Chaos Solitons Fractals 22(5), 1031–1038 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 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)zbMATHCrossRefGoogle Scholar
  23. 23.
    J.C. Sprott, Some simple chaotic flows. Phys. Rev. E 50(2), R647 (1994)MathSciNetCrossRefGoogle Scholar
  24. 24.
    M.W. Hirsch, S. Smale, R.L. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos (Academic, Cambridge, 2012)zbMATHGoogle Scholar
  25. 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)CrossRefGoogle Scholar
  26. 26.
    D. Schleicher, Hausdorff dimension, its properties, and its surprises. Am. Math. Mon. 114(6), 509–528 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 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)CrossRefGoogle Scholar
  30. 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. 31.
    I. Petráš, Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation (Springer, New York, 2011)zbMATHCrossRefGoogle Scholar
  32. 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)CrossRefGoogle Scholar
  33. 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. 34.
    W.M. Ahmad, J.C. Sprott, Chaos in fractional-order autonomous nonlinear systems. Chaos, Solitons Fractals 16(2), 339–351 (2003)zbMATHCrossRefGoogle Scholar
  35. 35.
    A.T. Azar, A.G. Radwan, S. Vaidyanathan, Fractional Order Systems: Optimization, Control, Circuit Realizations and Applications (Academic, Cambridge, 2018)Google Scholar
  36. 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)CrossRefGoogle Scholar
  37. 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. 38.
    D. Baleanu, J.A.T. Machado, A.C.J. Luo, Fractional Dynamics and Control (Springer, New York, 2011)Google Scholar
  39. 39.
    C. Li, X. Liao, J. Yu, Synchronization of fractional order chaotic systems. Phys. Rev. E 68(6), 067203 (2003)Google Scholar
  40. 40.
    R. Martínez-Guerra, C.A. Pérez-Pinacho, Advances in Synchronization of Coupled Fractional Order Systems: Fundamentals and Methods (Springer, Berlin, 2018)zbMATHCrossRefGoogle Scholar
  41. 41.
    A.T. Azar, S. Vaidyanathan, A. Ouannas, Fractional Order Control and Synchronization of Chaotic Systems, vol. 688 (Springer, Berlin, 2017)zbMATHCrossRefGoogle Scholar
  42. 42.
    A. Tepljakov, Fractional-Order Modeling and Control of Dynamic Systems (Springer, Berlin, 2017)zbMATHCrossRefGoogle Scholar
  43. 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. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 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)CrossRefGoogle Scholar
  48. 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. 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. 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)CrossRefGoogle Scholar
  51. 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. 52.
    K. Oldham, J. Spanier, The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order, vol. 111 (Elsevier, Amsterdam, 1974)zbMATHGoogle Scholar
  53. 53.
    S.S. Ray, Fractional Calculus with Applications for Nuclear Reactor Dynamics (CRC Press, Boca Raton, 2015)CrossRefGoogle Scholar
  54. 54.
    O.M. Duarte, Fractional Calculus for Scientists and Engineers (Springer, Berlin, 2011), 114 pp.zbMATHGoogle Scholar
  55. 55.
    F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models (World Scientific, Singapore, 2010)zbMATHCrossRefGoogle Scholar
  56. 56.
    V.E. Tarasov, Fractional Dynamics; Applications of the Fractional Calculus to Dynamics of Particles, Fields and Media (Springer, Berlin, 2010), 522 pp.zbMATHGoogle Scholar
  57. 57.
    D. Baleanu, Z.B. Günvec, M.J.A. Tenreiro, New Trends in Nanotechnology and Fractional Calculus Applications (Springer, Berlin, 2010), 544 pp.CrossRefGoogle Scholar
  58. 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)CrossRefGoogle Scholar
  59. 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)CrossRefGoogle Scholar
  60. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  61. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations (Wiley, Hoboken, 1993)zbMATHGoogle Scholar
  63. 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. 64.
    M. Caputo, Linear models of dissipation whose Q is almost frequency independent-II. Geophys. J. Int. 13(5), 529–539 (1967)CrossRefGoogle Scholar
  65. 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. 19–20Google Scholar
  66. 66.
    I. Pan, S. Das, Intelligent Fractional Order Systems and Control: An Introduction, vol. 438 (Springer, Berlin, 2012)zbMATHGoogle Scholar
  67. 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)zbMATHCrossRefGoogle Scholar
  68. 68.
    N.J. Ford, A.C. Simpson, The numerical solution of fractional differential equations: speed versus accuracy. Numer. Algorithms 26(4), 333–346 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  69. 69.
    Y. Chen, I. Petras, D. Xue, Fractional order control - a tutorial, in 2009 American Control Conference (2009), pp. 1397–1411Google Scholar
  70. 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)zbMATHGoogle Scholar
  71. 71.
    D. Cafagna, G. Grassi, On the simplest fractional-order memristor-based chaotic system. Nonlinear Dynam. 70(2), 1185–1197 (2012)MathSciNetCrossRefGoogle Scholar
  72. 72.
    R. Garrappa, Short tutorial: solving fractional differential equations by Matlab codes. Department of Mathematics, University of Bari (2014)Google Scholar
  73. 73.
    M.-F. Danca, N. Kuznetsov, Matlab code for Lyapunov exponents of fractional-order systems. Int. J. Bifurcation Chaos 28(5), 1850067 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  74. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  75. 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)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Esteban Tlelo-Cuautle
    • 1
  • Ana Dalia Pano-Azucena
    • 1
  • Omar Guillén-Fernández
    • 1
  • Alejandro Silva-Juárez
    • 1
  1. 1.INAOETonantzintlaMexico

Personalised recommendations