Table of Contents Author Guidelines Submit a Manuscript
Complexity
Volume 2017, Article ID 7871467, 11 pages
https://doi.org/10.1155/2017/7871467
Research Article

A Novel Chaotic System without Equilibrium: Dynamics, Synchronization, and Circuit Realization

1Faculty of Computers and Information, Benha University, Benha, Egypt
2Nanoelectronics Integrated Systems Center (NISC), Nile University, Giza, Egypt
3Department of Physics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
4Faculty of Physics, Department of Electronics, Computers, Telecommunications and Control, National and Kapodistrian University of Athens, 15784 Athens, Greece
5School of Electronics and Telecommunications, Hanoi University of Science and Technology, 01 Dai Co Viet, Hanoi, Vietnam
6Engineering Mathematics and Physics, Cairo University, Giza, Egypt
7Research and Development Center, Vel Tech University, Avadi, Chennai, Tamil Nadu 600062, India
8Laboratory of Mathematics, Informatics and Systems (LAMIS), University of Larbi Tebessi, 12002 Tebessa, Algeria
9University of Puebla, Puebla, PUE, Mexico

Correspondence should be addressed to Christos Volos; moc.liamg@solovhc

Received 21 October 2016; Revised 12 December 2016; Accepted 22 December 2016; Published 2 February 2017

Academic Editor: Carlos Gershenson

Copyright © 2017 Ahmad Taher Azar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. E. Lorenz, “Deterministic nonperiodic flow,” Journal of the Atmospheric Sciences, vol. 20, pp. 130–141, 1963. View at Google Scholar
  2. O. Rössler, “An equation for continuous chaos,” Physics Letters A, vol. 57, pp. 397–398, 1976. View at Google Scholar
  3. G. Chen and T. Ueta, “Yet another chaotic attractor,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 9, no. 7, pp. 1465–1466, 1999. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. J. C. Sprott, “Some simple chaotic flows,” Physical Review E, vol. 50, no. 2, pp. R647–R650, 1994. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. J. C. Sprott, Elegant Chaos Algebraically Simple Chaotic Flows, World Scientific, Singapore, 2010.
  6. J. Wu, L. Wang, G. Chen, and S. Duan, “A memristive chaotic system with heart-shaped attractors and its implementation,” Chaos, Solitons & Fractals, vol. 92, pp. 20–29, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  7. R. Wu and C. Wang, “A new simple chaotic circuit based on memristor,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 26, no. 9, Article ID 1650145, 11 pages, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  8. A. G. Radwan, K. Moaddy, and I. Hashim, “Amplitude modulation and synchronization of fractional-order memristor-based Chua's circuit,” Abstract and Applied Analysis, vol. 2013, Article ID 758676, 10 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  9. A. G. Radwan, A. M. Soliman, and A.-L. El-Sedeek, “An inductorless CMOS realization of Chua's circuit,” Chaos, Solitons and Fractals, vol. 18, no. 1, pp. 149–158, 2003. View at Publisher · View at Google Scholar · View at Scopus
  10. A. G. Radwan, A. M. Soliman, and A.-L. El-Sedeek, “MOS realization of the conjectured simplest chaotic equation,” Circuits, Systems, and Signal Processing, vol. 22, no. 3, pp. 277–285, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. A. Radwan, A. Soliman, and A. S. Elwakil, “1-D digitally-controlled multiscroll chaos generator,” International Journal of Bifurcation and Chaos, vol. 17, no. 1, pp. 227–242, 2007. View at Publisher · View at Google Scholar · View at Scopus
  12. M. A. Zidan, A. G. Radwan, and K. N. Salama, “Controllable V-shape multiscroll butterfly attractor: system and circuit implementation,” International Journal of Bifurcation and Chaos, vol. 22, no. 6, Article ID 1250143, 2012. View at Publisher · View at Google Scholar · View at Scopus
  13. A. S. Mansingka, M. Affan Zidan, M. L. Barakat, A. G. Radwan, and K. N. Salama, “Fully digital jerk-based chaotic oscillators for high throughput pseudo-random number generators up to 8.77 Gbits/s,” Microelectronics Journal, vol. 44, no. 9, pp. 744–752, 2013. View at Publisher · View at Google Scholar · View at Scopus
  14. A. Buscarino, C. Famoso, L. Fortuna, and M. Frasca, “A new chaotic electro-mechanical oscillator,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 26, no. 10, Article ID 1650161, 7 pages, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  15. H. Liu, A. Kadir, and Y. Li, “Asymmetric color pathological image encryption scheme based on complex hyper chaotic system,” Optik, vol. 127, pp. 5812–5819, 2016. View at Google Scholar
  16. A. G. Radwan, S. H. AbdElHaleem, and S. K. Abd-El-Hafiz, “Symmetric encryption algorithms using chaotic and non-chaotic generators: a review,” Journal of Advanced Research, vol. 7, no. 2, pp. 193–208, 2016. View at Publisher · View at Google Scholar · View at Scopus
  17. K. Moaddy, A. G. Radwan, K. N. Salama, S. Momani, and I. Hashim, “The fractional-order modeling and synchronization of electrically coupled neuron systems,” Computers & Mathematics with Applications, vol. 64, no. 10, pp. 3329–3339, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. A. G. Radwan, K. Moaddy, K. N. Salama, S. Momani, and I. Hashim, “Control and switching synchronization of fractional order chaotic systems using active control technique,” Journal of Advanced Research, vol. 5, no. 1, pp. 125–132, 2014. View at Publisher · View at Google Scholar · View at Scopus
  19. Z. Lin, S. Yu, C. Li, J. Lü, and Q. Wang, “Design and smartphone-based implementation of a chaotic video communication scheme via WAN remote transmission,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 26, no. 9, Article ID 1650158, 8 pages, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  20. L. Min, X. Yang, G. Chen, and D. Wang, “Some polynomial chaotic maps without equilibria and an application to image encryption with avalanche effects,” International Journal of Bifurcation and Chaos, vol. 25, no. 9, Article ID 1550124, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. H. Liu, A. Kadir, and Y. Li, “Audio encryption scheme by confusion and diffusion based on multi-scroll chaotic system and one-time keys,” Optik, vol. 127, no. 19, pp. 7431–7438, 2016. View at Publisher · View at Google Scholar · View at Scopus
  22. J. Lü and G. Chen, “Generating multiscroll chaotic attractors: theories, methods and applications,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 16, no. 4, pp. 775–858, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. X. Wang and G. Chen, “A chaotic system with only one stable equilibrium,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 3, pp. 1264–1272, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. X. Wang and G. Chen, “Constructing a chaotic system with any number of equilibria,” Nonlinear Dynamics, vol. 71, no. 3, pp. 429–436, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. L. P. Shilnikov, “A case of the existence of a countable number of periodic motions,” Soviet Mathematics. Doklady, vol. 6, pp. 163–166, 1965. View at Google Scholar
  26. L. Shilnikov, A. Shilnikov, D. Turaev, and L. Chua, Methods of Qualitative Theory in Nonlinear Dynamics, World Scientific, Singapore, 1998.
  27. S. Jafari, J. C. Sprott, and S. M. R. H. Golpayegani, “Elementary quadratic chaotic flows with no equilibria,” Physics Letters A, vol. 377, no. 9, pp. 699–702, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  28. A. Sommerfeld, “Beitrage zum dynamischen ausbau der festigkeitslehre,” Zeitschrift des Vereins Deutscher Ingenieure, vol. 46, pp. 391–394, 1902. View at Google Scholar
  29. R. Evan-Iwanowski, Resonance Oscillations in Mechanical Systems, Elsevier, Amsterdam, The Netherlands, 1976.
  30. M. Eckert, Arnold Sommerfeld: Science, Life and Turbulent Times 1868–1951, Springer, New York, NY, USA, 2013.
  31. S. Nosé, “A molecular dynamics method for simulations in the canonical ensemble,” Molecular Physics, vol. 52, no. 2, pp. 255–268, 1984. View at Publisher · View at Google Scholar · View at Scopus
  32. W. G. Hoover, “Canonical dynamics: equilibrium phase-space distributions,” Physical Review A, vol. 31, no. 3, pp. 1695–1697, 1985. View at Publisher · View at Google Scholar · View at Scopus
  33. H. A. Posch, W. G. Hoover, and F. J. Vesely, “Canonical dynamics of the Nosé oscillator: stability, order, and chaos,” Physical Review. A. Third Series, vol. 33, no. 6, pp. 4253–4265, 1986. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  34. J. C. Sprott, W. G. Hoover, and C. G. Hoover, “Heat conduction, and the lack thereof, in time-reversible dynamical systems: generalized Nosé-Hoover oscillators with a temperature gradient,” Physical Review E, vol. 89, no. 4, Article ID 042914, 2014. View at Publisher · View at Google Scholar · View at Scopus
  35. J. C. Sprott, “Strange attractors with various equilibrium types,” European Physical Journal: Special Topics, vol. 224, no. 8, pp. 1409–1419, 2015. View at Publisher · View at Google Scholar · View at Scopus
  36. L. Wang and X.-S. Yang, “The invariant tori of knot type and the interlinked invariant tori in the Nosè-Hoover oscillator,” The European Physical Journal B, vol. 88, article 78, 5 pages, 2015. View at Publisher · View at Google Scholar
  37. Z. Wei, “Dynamical behaviors of a chaotic system with no equilibria,” Physics Letters A, vol. 376, no. 2, pp. 102–108, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  38. J.-L. Zuo and C.-L. Li, “Multiple attractors and dynamic analysis of a no-equilibrium chaotic system,” Optik, vol. 127, no. 19, pp. 7952–7957, 2016. View at Publisher · View at Google Scholar · View at Scopus
  39. A. Akgul, H. Calgan, I. Koyuncu, I. Pehlivan, and A. Istanbullu, “Chaos-based engineering applications with a 3D chaotic system without equilibrium points,” Nonlinear Dynamics, vol. 84, no. 2, pp. 481–495, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  40. Z. Wang, S. Cang, E. O. Ochola, and Y. Sun, “A hyperchaotic system without equilibrium,” Nonlinear Dynamics. An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems, vol. 69, no. 1-2, pp. 531–537, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  41. Z. Wei, R. Wang, and A. Liu, “A new finding of the existence of hidden hyperchaotic attractors with no equilibria,” Mathematics and Computers in Simulation, vol. 100, pp. 13–23, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  42. Z. Wang, J. Ma, S. Cang, Z. Wang, and Z. Chen, “Simplified hyper-chaotic systems generating multi-wing non-equilibrium attractors,” Optik, vol. 127, no. 5, pp. 2424–2431, 2016. View at Publisher · View at Google Scholar · View at Scopus
  43. G. A. Leonov, N. V. Kuznetsov, O. A. Kuznetsova, S. M. Seledzhi, and V. I. Vagaitsev, “Hidden oscillations in dynamical systems,” WSEAS Transactions on Systems and Control, vol. 6, no. 2, pp. 54–67, 2011. View at Google Scholar · View at Scopus
  44. G. A. Leonov, N. V. Kuznetsov, and V. I. Vagaitsev, “Localization of hidden Chua's attractors,” Physics Letters. A, vol. 375, no. 23, pp. 2230–2233, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  45. T. Kapitaniak and G. A. Leonov, “Multistability: uncovering hidden attractors,” European Physical Journal: Special Topics, vol. 224, no. 8, pp. 1405–1408, 2015. View at Publisher · View at Google Scholar · View at Scopus
  46. D. Dudkowski, S. Jafari, T. Kapitaniak, N. V. Kuznetsov, G. A. Leonov, and A. Prasad, “Hidden attractors in dynamical systems,” Physics Reports, vol. 637, pp. 1–50, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  47. G. A. Leonov, N. V. Kuznetsov, and V. I. Vagaitsev, “Hidden attractor in smooth Chua systems,” Physica D. Nonlinear Phenomena, vol. 241, no. 18, pp. 1482–1486, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  48. G. A. Leonov and N. V. Kuznetsov, “Hidden attractors in dynamical systems: From hidden oscillation in Hilbert-Kolmogorov, Aizerman and Kalman problems to hidden chaotic attractor in Chua circuits,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 23, no. 1, Article ID 1330002, 69 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  49. G. A. Leonov, N. V. Kuznetsov, M. A. Kiseleva, E. P. Solovyeva, and A. M. Zaretskiy, “Hidden oscillations in mathematical model of drilling system actuated by induction motor with a wound rotor,” Nonlinear Dynamics, vol. 77, no. 1-2, pp. 277–288, 2014. View at Publisher · View at Google Scholar · View at Scopus
  50. G. A. Leonov, N. V. Kuznetsov, and T. N. Mokaev, “Hidden attractor and homoclinic orbit in Lorenz-like system describing convective fluid motion in rotating cavity,” Communications in Nonlinear Science and Numerical Simulation, vol. 28, no. 1-3, pp. 166–174, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  51. Q. Li, H. Zeng, and J. Li, “Hyperchaos in a 4D memristive circuit with infinitely many stable equilibria,” Nonlinear Dynamics. An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems, vol. 79, no. 4, pp. 2295–2308, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  52. S. Brezetskyi, D. Dudkowski, and T. Kapitaniak, “Rare and hidden attractors in Van der Pol-Duffing oscillators,” The European Physical Journal Special Topics, vol. 224, no. 8, pp. 1459–1467, 2015. View at Publisher · View at Google Scholar · View at Scopus
  53. B. Munmuangsaen, J. C. Sprott, W. J. Thio, A. Buscarino, and L. Fortuna, “A simple chaotic flow with a continuously adjustable attractor dimension,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 25, no. 12, Article ID 1530036, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  54. C. Li, J. C. Sprott, Z. Yuan, and H. Li, “Constructing chaotic systems with total amplitude control,” International Journal of Bifurcation and Chaos, vol. 25, no. 10, Article ID 1530025, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  55. A. Chudzik, P. Perlikowski, A. Stefanski, and T. Kapitaniak, “Multistability and rare attractors in van der pol-duffing oscillator,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 21, no. 7, pp. 1907–1912, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  56. P. R. Sharma, M. D. Shrimali, A. Prasad, N. V. Kuznetsov, and G. A. Leonov, “Control of multistability in hidden attractors,” European Physical Journal: Special Topics, vol. 224, no. 8, pp. 1485–1491, 2015. View at Publisher · View at Google Scholar · View at Scopus
  57. Z. T. Zhusubaliyev, E. Mosekilde, A. N. Churilov, and A. Medvedev, “Multistability and hidden attractors in an impulsive Goodwin oscillator with time delay,” European Physical Journal: Special Topics, vol. 224, no. 8, pp. 1519–1539, 2015. View at Publisher · View at Google Scholar · View at Scopus
  58. Z. T. Zhusubaliyev and E. Mosekilde, “Multistability and hidden attractors in a multilevel DC/DC converter,” Mathematics and Computers in Simulation, vol. 109, pp. 32–45, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  59. A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, “Determining Lyapunov exponents from a time series,” Physica D: Nonlinear Phenomena, vol. 16, no. 3, pp. 285–317, 1985. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  60. N. V. Kuznetsov, “The Lyapunov dimension and its estimation via the Leonov method,” Physics Letters A, vol. 380, no. 25-26, pp. 2142–2149, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  61. N. V. Kuznetsov, T. A. Alexeeva, and G. A. Leonov, “Invariance of LYApunov exponents and LYApunov dimension for regular and irregular linearizations,” Nonlinear Dynamics. An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems, vol. 85, no. 1, pp. 195–201, 2016. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  62. G. A. Leonov, N. V. Kuznetsov, N. A. Korzhemanova, and D. V. Kusakin, “Lyapunov dimension formula for the global attractor of the Lorenz system,” Communications in Nonlinear Science and Numerical Simulation, vol. 41, pp. 84–103, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  63. S. Banerjee, Chaos Synchronization and Cryptography for Secure Communication, IGI Global, Hershey, Pa, USA, 2010.
  64. J. L. Mata-Machucaa, R. Martínez-Guerraa, R. Aguilar-Lópezb, and C. Aguilar-Ibañez, “A chaotic system in synchronization and secure communications,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 4, pp. 1706–1713, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  65. C. K. Volos, I. M. Kyprianidis, and I. N. Stouboulos, “Image encryption process based on chaotic synchronization phenomena,” Signal Processing, vol. 93, no. 5, pp. 1328–1340, 2013. View at Publisher · View at Google Scholar · View at Scopus
  66. R. Aguilar-López, R. Martínez-Guerra, and C. A. Perez-Pinacho, “Nonlinear observer for synchronization of chaotic systems with application to secure data transmission,” The European Physical Journal Special Topics, vol. 223, no. 8, pp. 1541–1548, 2014. View at Publisher · View at Google Scholar · View at Scopus
  67. S. Çiçek, A. Ferikoğlu, and I. Pehlivan, “A new 3D chaotic system: dynamical analysis, electronic circuit design, active control synchronization and chaotic masking communication application,” Optik, vol. 127, no. 8, pp. 4024–4030, 2016. View at Publisher · View at Google Scholar · View at Scopus
  68. C. Hua and X. Guan, “Adaptive control for chaotic systems,” Chaos, Solitons & Fractals, vol. 22, no. 1, pp. 55–60, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  69. G. Feng and G. Chen, “Adaptive control of discrete–time chaotic systems: a fuzzy control approach,” Chaos, Solitons & Fractals, vol. 23, no. 2, pp. 459–467, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  70. S.-Y. Li, C.-H. Yang, C.-T. Lin, L.-W. Ko, and T.-T. Chiu, “Adaptive synchronization of chaotic systems with unknown parameters via new backstepping strategy,” Nonlinear Dynamics, vol. 70, no. 3, pp. 2129–2143, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  71. H. K. Khalil, Nonlinear Systems, Prentice Hall, Upper Saddle River, NJ, USA, 3rd edition, 2002.
  72. M. s. Yalcin, J. A. Suykens, and J. Vandewalle, “True random bit generation from a double-scroll attractor,” IEEE Transactions on Circuits and Systems. I. Regular Papers, vol. 51, no. 7, pp. 1395–1404, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  73. G. Y. Wang, X. L. Bao, and Z. L. Wang, “Design and implementation of a new hyperchaotic system,” Chinese Physics B, vol. 17, pp. 3596–3602, 2008. View at Google Scholar
  74. C. K. Volos, I. M. Kyprianidis, and I. N. Stouboulos, “A chaotic path planning generator for autonomous mobile robots,” Robotics and Autonomous Systems, vol. 60, no. 4, pp. 651–656, 2012. View at Publisher · View at Google Scholar · View at Scopus
  75. D. Valli, B. Muthuswamy, S. Banerjee et al., “Synchronization in coupled Ikeda delay systems experimental observations using Field Programmable Gate Arrays,” The European Physical Journal Special Topics, vol. 223, pp. 1465–1479, 2014. View at Google Scholar
  76. A. Akgul, I. Moroz, I. Pehlivan, and S. Vaidyanathan, “A new four–scroll chaotic attractor and its engineering applications,” Optik, vol. 127, no. 13, pp. 5491–5499, 2016. View at Publisher · View at Google Scholar · View at Scopus