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Mathematical Problems in Engineering
Volume 2017, Article ID 3927184, 11 pages
https://doi.org/10.1155/2017/3927184
Research Article

A No-Equilibrium Hyperchaotic System and Its Fractional-Order Form

1Modeling Evolutionary Algorithms Simulation and Artificial Intelligence, Faculty of Electrical & Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam
2Department of Mechanical and Electrical Engineering, Institute of Mines and Petroleum Industries, University of Maroua, P.O. Box 46, Maroua, Cameroon
3School of Electronics and Telecommunications, Hanoi University of Science and Technology, 01 Dai Co Viet, Hanoi, Vietnam

Correspondence should be addressed to Viet-Thanh Pham; moc.liamg@0103tvp

Received 25 March 2017; Accepted 1 June 2017; Published 29 June 2017

Academic Editor: Jonathan N. Blakely

Copyright © 2017 Duy Vo Hoang 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. O. E. Rössler, “An equation for hyperchaos,” Physics Letters. A, vol. 71, no. 2-3, pp. 155–157, 1979. View at Publisher · View at Google Scholar · View at MathSciNet
  2. G. Grassi and S. Mascolo, “A system theory approach for designing cryptosystems based on hyperchaos,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 46, no. 9, pp. 1135–1138, 1999. View at Publisher · View at Google Scholar · View at Scopus
  3. Y. Huang and X. Yang, “Hyperchaos and bifurcation in a new class of four-dimensional Hopfield neural networks,” Neurocomputing, vol. 69, no. 13–15, pp. 1787–1795, 2006. View at Publisher · View at Google Scholar · View at Scopus
  4. V. S. Udaltsov, J. P. Goedgebuer, L. Larger, J. B. Cuenot, P. Levy, and W. T. Rhodes, “Communicating with hyperchaos: the dynamics of a dnlf emitter and recovery of transmitted information,” Optics and Spectroscopy, vol. 95, no. 1, pp. 114–118, 2003. View at Publisher · View at Google Scholar · View at Scopus
  5. S. Sadoudi, C. Tanougast, M. S. Azzaz, and A. Dandache, “Design and FPGA implementation of a wireless hyperchaotic communication system for secure real-time image transmission,” EURASIP Journal on Image and Video Processing, vol. 2013, article 43, pp. 1–18, 2013. View at Publisher · View at Google Scholar
  6. R. Vicente, J. Daudén, P. Colet, and R. Toral, “Analysis and characterization of the hyperchaos generated by a semiconductor laser subject to a delayed feedback loop,” IEEE Journal of Quantum Electronics, vol. 41, no. 4, pp. 541–548, 2005. View at Publisher · View at Google Scholar · View at Scopus
  7. T. Matsumoto, L. O. Chua, and K. Kobayashi, “Hyperchaos: laboratory experiment and numerical confirmation,” Institute of Electrical and Electronics Engineers. Transactions on Circuits and Systems, vol. 33, no. 11, pp. 1143–1149, 1986. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. Y. Li, W. K. S. Tang, and G. Chen, “Hyperchaos evolved from the generalized Lorenz equation,” International Journal of Circuit Theory and Applications, vol. 33, no. 4, pp. 235–251, 2005. View at Publisher · View at Google Scholar · View at Scopus
  9. A. Qi, C. Zhang, and H. Wang, “A switched hyperchaotic system and its FPGA circuitry implementation,” Journal of Electronics, vol. 28, no. 3, pp. 383–388, 2011. View at Publisher · View at Google Scholar · View at Scopus
  10. S. Dadras, H. R. Momeni, G. Qi, and Z.-l. Wang, “Four-wing hyperchaotic attractor generated from a new 4D system with one equilibrium and its fractional-order form,” Nonlinear Dynamics. An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems, vol. 67, no. 2, pp. 1161–1173, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. D. Cafagna and G. Grassi, “New 3D-scroll attractors in hyperchaotic Chua's circuits forming a ring,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 13, no. 10, pp. 2889–2903, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. A. Chen, J. Lu, J. Lü, and S. Yu, “Generating hyperchaotic Lü attractor via state feedback control,” Physica A: Statistical Mechanics and its Applications, vol. 376, pp. 102–108, 2011. View at Publisher · View at Google Scholar · View at Scopus
  13. 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
  14. V.-T. Pham, F. Rahma, M. Frasca, and L. Fortuna, “Dynamics and synchronization of a novel hyperchaotic system without equilibrium,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 24, no. 6, Article ID 1450087, 1450087, 11 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. 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
  16. V.-T. Pham, C. Volos, and L. V. Gambuzza, “A memristive hyperchaotic system without equilibrium,” Scientific World Journal, vol. 2014, Article ID 368986, 2014. View at Publisher · View at Google Scholar · View at Scopus
  17. V. T. Pham, S. Vaidyanathan, C. K. Volos, and S. Jafari, “Hidden attractors in a chaotic system with an exponential nonlinear term,” European Physical Journal: Special Topics, vol. 224, no. 8, Article ID A1507, pp. 1507–1517, 2015. View at Publisher · View at Google Scholar · View at Scopus
  18. 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
  19. B. C. Bao, H. Bao, N. Wang, M. Chen, and Q. Xu, “Hidden extreme multistability in memristive hyperchaotic system,” Chaos, Solitons Fractals, vol. 94, pp. 102–111, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  20. D. Cafagna and G. Grassi, “Fractional-order systems without equilibria: the first example of hyperchaos and its application to synchronization,” Chinese Physics B, vol. 24, no. 8, Article ID 080502, 2015. View at Publisher · View at Google Scholar · View at Scopus
  21. K. Rajagopal, A. Karthikeyan, and A. K. Srinivasan, “FPGA implementation of novel fractional-order chaotic systems with two equilibriums and no equilibrium and its adaptive sliding mode synchronization,” Nonlinear Dynamics. An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems, vol. 87, no. 4, pp. 2281–2304, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  22. G. A. Leonov and N. V. Kuznetsov, “Hidden attractors in dynamical systems. From hidden oscillations 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, 1330002, 69 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. D. Dudkowski, S. Jafari, T. Kapitaniak, N. V. Kuznetsov, G. A. Leonov, and A. Prasad, “Hidden attractors in dynamical systems,” Physics Reports. A Review Section of Physics Letters, vol. 637, pp. 1–50, 2016. 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. An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems, vol. 71, no. 3, pp. 429–436, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. S. Brezetskyi, D. Dudkowski, and T. Kapitaniak, “Rare and hidden attractors in Van der Pol-Duffing oscillators,” European Physical Journal: Special Topics, vol. 224, no. 8, pp. 1459–1467, 2015. View at Publisher · View at Google Scholar · View at Scopus
  26. 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
  27. 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
  28. Z. Wei and Q. Yang, “Dynamical analysis of a new autonomous 3-D chaotic system only with stable equilibria,” Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, vol. 12, no. 1, pp. 106–118, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  29. N. Levinson, “Transformation theory of non-linear differential equations of the second order,” Annals of Mathematics. Second Series, vol. 45, pp. 723–737, 1944. View at Publisher · View at Google Scholar · View at MathSciNet
  30. G. A. Leonov, N. V. Kuznetsov, and T. N. Mokaev, “Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion,” European Physical Journal: Special Topics, vol. 224, no. 8, pp. 1421–1458, 2015. View at Publisher · View at Google Scholar · View at Scopus
  31. 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
  32. P. Frederickson, J. L. Kaplan, E. D. Yorke, and J. A. Yorke, “The Liapunov dimension of strange attractors,” Journal of Differential Equations, vol. 49, no. 2, pp. 185–207, 1983. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  33. 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 MathSciNet
  34. 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
  35. 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
  36. A. Buscarino, L. Fortuna, M. Frasca, and L. V. Gambuzza, “A chaotic circuit based on Hewlett-Packard memristor,” Chaos. An Interdisciplinary Journal of Nonlinear Science, vol. 22, no. 2, Article ID 023136, 023136, 9 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  37. C. Li, J. C. Sprott, Z. Yuan, and H. Li, “Constructing chaotic systems with total amplitude control,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 25, no. 10, Article ID 1530025, 1530025, 14 pages, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  38. C. Li and J. C. Sprott, “Variable-boostable chaotic flows,” Optik, vol. 127, no. 22, pp. 10389–10398, 2016. View at Publisher · View at Google Scholar · View at Scopus
  39. C. Li and J. C. Sprott, “Amplitude control approach for chaotic signals,” Nonlinear Dynamics. An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems, vol. 73, no. 3, pp. 1335–1341, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  40. C. Li and J. C. Sprott, “Finding coexisting attractors using amplitude control,” Nonlinear Dynamics. An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems, vol. 78, no. 3, pp. 2059–2064, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  41. W. Deng, “Short memory principle and a predictor-corrector approach for fractional differential equations,” Journal of Computational and Applied Mathematics, vol. 206, no. 1, pp. 174–188, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  42. K. Diethelm, N. J. Ford, and A. D. Freed, “A predictor-corrector approach for the numerical solution of fractional differential equations,” Nonlinear Dynamics. An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems, vol. 29, no. 1-4, pp. 3–22, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  43. R. Caponetto, G. Dongola, L. Fortuna, and I. Petras, “Fractional Order Systems: Modeling and Control Applicationss, World Scientific,” 2010, Singapore. View at Google Scholar
  44. P. Perlikowski, S. Yanchuk, M. Wolfrum, A. Stefanski, P. Mosiolek, and T. Kapitaniak, “Routes to complex dynamics in a ring of unidirectionally coupled systems,” Chaos. An Interdisciplinary Journal of Nonlinear Science, vol. 20, no. 1, 013111, 10 pages, 2010. View at Publisher · View at Google Scholar · View at MathSciNet