Skip to main content

A Three-Dimensional Chaotic System with Square Equilibrium and No-Equilibrium

  • Chapter
  • First Online:
Fractional Order Control and Synchronization of Chaotic Systems

Abstract

Recently, Leonov and Kuznetsov have introduced a new definition “hidden attractor”. Systems with hidden attractors, especially chaotic systems, have attracted significant attention. Some examples of such systems are systems with a line equilibrium, systems without equilibrium or systems with stable equilibria etc. In some interesting new research, systems in which equilibrium points are located on different special curves are reported. This chapter introduces a three-dimensional autonomous system with a square-shaped equilibrium and without equilibrium points. Therefore, such system belongs to a class of systems with hidden attractors. The fundamental dynamics properties of such system are studied through phase portraits, Poincaré map, bifurcation diagram, and Lyapunov exponents. Anti-synchronization scheme for our systems is proposed and confirmed by the Lyapunov stability. Moreover, an electronic circuit is implemented to show the feasibility of the mathematical model. Finally, we introduce the fractional order form of such system.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Aguilar-Lopez, R., Martinez-Guerra, R., & Perez-Pinacho, C. (2014). Nonlinear observer for synchronization of chaotic systems with application to secure data transmission. The European Physical Journal Special Topics, 223, 1541–1548.

    Article  Google Scholar 

  2. Akgul, A., Moroz, I., Pehlivan, I., & Vaidyanathan, S. (2016). A new four-scroll chaotic attractor and its enginearing applications. Optik, 127, 5491–5499.

    Article  Google Scholar 

  3. Akopov, A., Astakhov, V., Vadiasova, T., Shabunin, A., & Kapitaniak, T. (2005). Frequency synchronization in clusters in coupled extended systems. Physics Letters A, 334, 169–172.

    Article  MATH  Google Scholar 

  4. Arneodo, A., Coullet, P., & Tresser, C. (1981). Possible new strange attractors with spiral structure. Communications in Mathematical Physics, 79, 573–579.

    Article  MathSciNet  MATH  Google Scholar 

  5. Azar, A. T., & Vaidyanathan, S. (2015). Chaos modeling and control systems design. Germany: Springer.

    Google Scholar 

  6. Azar, A. T., & Vaidyanathan, S. (2015). Computational intelligence applications in modeling and control. Germany: Springer.

    Google Scholar 

  7. Azar, A. T., & Vaidyanathan, S. (2015). Handbook of research on advanced intelligent control engineering and automation. USA: IGI Global.

    Google Scholar 

  8. Azar, A. T., & Vaidyanathan, S. (2016). Advances in chaos theory and intelligent control. Germany: Springer.

    Book  MATH  Google Scholar 

  9. Bagley, R. L., & Calico, R. A. (1991). Fractional-order state equations for the control of visco-elastically damped structers. Journal of Guidance, Control, and Dynamics, 14, 304–311.

    Article  Google Scholar 

  10. Bao, B., Zou, X., Liu, Z., & Hu, F. (2013). Generalized memory element and chaotic memory system. International Journal of Bifurcation and Chaos, 23, 1350135.

    Article  MathSciNet  MATH  Google Scholar 

  11. Barnerjee, T., Biswas, D., & Sarkar, B. C. (2012). Design and analysis of a first order time-delayed chaotic system. Nonlinear Dynamics, 70, 721–734.

    Article  MathSciNet  Google Scholar 

  12. Behnia, S., Pazhotan, Z., Ezzati, N., & Akhshani, A. (2014). Reconfiguration chaotic logic gates based on novel chaotic circuit. Chaos, Solitons and Fractals, 69, 74–80.

    Article  MathSciNet  MATH  Google Scholar 

  13. Boccaletti, S., Kurths, J., Osipov, G., Valladares, D. L., & Zhou, C. S. (2002). The synchronization of chaotic systems. Physics Reports, 366, 1–101.

    Article  MathSciNet  MATH  Google Scholar 

  14. Boulkroune, A., Bouzeriba, A., Bouden, T., & Azar, A. T. (2016). Fuzzy adaptive synchronization of uncertain fractional-order chaotic systems. In A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control (Vol. 337, pp. 681–697). Studies in fuzziness and soft computing. Springer: Germany.

    Google Scholar 

  15. Boulkroune, A., Hamel, S., & Azar, A. T. (2016). Fuzzy control-based function synchronization of unknown chaotic systems with dead-zone input. In A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control (Vol. 337, pp. 699–718). Studies in fuzziness and soft computing. Springer: Germany.

    Google Scholar 

  16. Brezetskyi, S., Dudkowski, D., & Kapitaniak, T. (2015). Rare and hidden attractors in van der pol-duffing oscillators. The European Physical Journal Special Topics, 224, 1459–1467.

    Article  Google Scholar 

  17. Buscarino, A., Fortuna, L., & Frasca, M. (2009). Experimental robust synchronization of hyperchaotic circuits. Physica D, 238, 1917–1922.

    Article  MATH  Google Scholar 

  18. Chen, G., & Ueta, T. (1999). Yet another chaotic attractor. International Journal of Bifurcation and Chaos, 9, 1465–1466.

    Article  MathSciNet  MATH  Google Scholar 

  19. Chen, G., & Yu, X. (2003). Chaos control: Theory and applications. Berlin: Springer.

    Book  MATH  Google Scholar 

  20. Chenaghlu, M. A., & Khasmakhi, S. J. N. N. (2016). A novel keyed parallel hashing scheme based on a new chaotic system. Chaos, Solitons and Fractals, 87, 216–225.

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. Fortuna, L., & Frasca, M. (2007). Experimental synchronization of single-transistor-based chaotic circuits. Chaos, 17, 043118-1–5.

    Article  MATH  Google Scholar 

  23. Frederickson, P., Kaplan, J. L., Yorke, E. D., & York, J. (1983). The Lyapunov dimension of strange attractors. Journal of Differential Equations, 49, 185–207.

    Article  MathSciNet  MATH  Google Scholar 

  24. Gamez-Guzman, L., Cruz-Hernandez, C., Lopez-Gutierrez, R., & Garcia-Guerrero, E. E. (2009). Synchronization of Chua’s circuits with multi-scroll attractors: Application to communication. Communications in Nonlinear Science and Numerical Simulation, 14, 2765–2775.

    Google Scholar 

  25. Gejji, D., & Jafari, H. (2005). A domian decomposition: A tool for solving a system of fractional differential equations. Journal of Mathematical Analysis and Applications, 301, 508–518.

    Google Scholar 

  26. Gotthans, T., & Petržela, J. (2015). New class of chaotic systems with circular equilibrium. Nonlinear Dynamics, 73, 429–436.

    Google Scholar 

  27. Gotthans, T., Sportt, J. C., & Petržela, J. (2016). Simple chaotic flow with circle and square equilibrium. International Journal of Bifurcation and Chaos, 26, 1650137.

    Article  MathSciNet  MATH  Google Scholar 

  28. Grigorenko, I., & Grigorenko, E. (2003). Chaotic dynamics of the fractional-order lorenz system. Physical Review Letters, 91, 034101.

    Article  Google Scholar 

  29. Hartley, T. T., Lorenzo, C. F., & Qammer, H. K. (1995). Chaos on a fractional Chua’s system. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 42, 485–490.

    Article  Google Scholar 

  30. Heaviside, O. (1971). Electromagnetic theory. New York, USA: Academic Press.

    MATH  Google Scholar 

  31. Hoang, T. M., & Nakagawa, M. (2007). Anticipating and projective–anticipating synchronization of coupled multidelay feedback systems. Physics Letters A, 365, 407–411.

    Google Scholar 

  32. Hu, J., Chen, S., & Chen, L. (2005). Adaptive control for anti-synchronization of Chua’s chaotic system. Physics Letters A, 339, 455–460.

    Article  MATH  Google Scholar 

  33. Huang, Y., Wang, Y., Chen, H., & Zhang, S. (2016). Shape synchronization control for three-dimensional chaotic systems. Chaos, Solitons and Fractals, 87, 136–145.

    Article  MathSciNet  MATH  Google Scholar 

  34. Jafari, S., & Sprott, J. C. (2013). Simple chaotic flows with a line equilibrium. Chaos, Solitons and Fractals, 57, 79–84.

    Article  MathSciNet  MATH  Google Scholar 

  35. Jafari, S., Sprott, J. C., & Golpayegani, S. M. R. H. (2013). Elementary quadratic chaotic flows with no equilibria. Physics Letters A, 377, 699–702.

    Article  MathSciNet  Google Scholar 

  36. Jafari, S., Sprott, J. C., & Nazarimehr, F. (2015). Recent new examples of hidden attractors. The European Physical Journal Special Topics, 224, 1469–1476.

    Article  Google Scholar 

  37. Jenson, V. G., & Jeffreys, G. V. (1997). Mathematical methods in chemical engineering. New York, USA: Academic Press.

    MATH  Google Scholar 

  38. Kajbaf, A., Akhaee, M. A., & Sheikhan, M. (2016). Fast synchronization of non-identical chaotic modulation-based secure systems using a modified sliding mode controller. Chaos, Solitons and Fractals, 84, 49–57.

    Article  MATH  Google Scholar 

  39. Kapitaniak, T. (1994). Synchronization of chaos using continuous control. Physical Review E, 50, 1642–1644.

    Article  Google Scholar 

  40. Karthikeyan, R., & Vaidyanathan, S. (2014). Hybrid chaos synchronization of four-scroll systems via active control. Journal of Electrical Engineering, 65, 97–103.

    Article  Google Scholar 

  41. Khalil, H. (2002). Nonlinear systems. New Jersey, USA: Prentice Hall.

    MATH  Google Scholar 

  42. Kim, C. M., Rim, S., Kye, W. H., Ryu, J. W., & Park, Y. J. (2003). Anti-synchronization of chaotic oscillators. Physics Letters A, 320, 39–46.

    Article  MathSciNet  MATH  Google Scholar 

  43. Kingni, S. T., Jafari, S., Simo, H., & Woafo, P. (2014). Three-dimensional chaotic autonomous system with only one stable equilibrium: Analysis, circuit design, parameter estimation, control, synchronization and its fractional-order form. The European Physical Journal Plus, 129, 76.

    Google Scholar 

  44. Koupaei, J. A., & Hosseini, S. M. M. (2015). A new hybrid algorithm based on chaotic maps for solving systems of nonlinear equations. Chaos, Solitons and Fractals, 81, 233–245.

    Google Scholar 

  45. Kuznetsov, N. V., Leonov, G. A., & Seledzhi, S. M. (2011). Hidden oscillations in nonlinear control systems. IFAC Proceedings, 18, 2506–2510.

    Google Scholar 

  46. Leonov, G. A., & Kuznetsov, N. V. (2011). Algorithms for searching for hidden oscillations in the Aizerman and Kalman problems. Doklady Mathematics, 84, 475–481.

    Google Scholar 

  47. Leonov, G. A., & Kuznetsov, N. V. (2011). Analytical–numerical methods for investigation of hidden oscillations in nonlinear control systems. IFAC Proceedings, 18, 2494–2505.

    Google Scholar 

  48. Leonov, G. A., & Kuznetsov, N. V. (2013). 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, 23, 1330002.

    Article  MathSciNet  MATH  Google Scholar 

  49. Leonov, G. A., Kuznetsov, N. V., Kiseleva, M. A., Solovyeva, E. P., & Zaretskiy, A. M. (2014). Hidden oscillations in mathematical model of drilling system actuated by induction motor with a wound rotor. Nonlinear Dynamics, 77, 277–288.

    Article  Google Scholar 

  50. Leonov, G. A., Kuznetsov, N. V., Kuznetsova, O. A., Seldedzhi, S. M., & Vagaitsev, V. I. (2011). Hidden oscillations in dynamical systems. Transmission Systems Control, 6, 54–67.

    Google Scholar 

  51. Leonov, G. A., Kuznetsov, N. V., & Vagaitsev, V. I. (2011). Localization of hidden Chua’s attractors. Physics Letters A, 375, 2230–2233.

    Google Scholar 

  52. Leonov, G. A., Kuznetsov, N. V., and Vagaitsev, V. I. (2012). Hidden attractor in smooth Chua system. Physica D, 241, 1482–1486.

    Google Scholar 

  53. Li, C. P., & Peng, G. J. (2004). Chaos in Chen’s system with a fractional-order. Chaos, Solitons and Fractals, 20, 443–450.

    Article  MathSciNet  MATH  Google Scholar 

  54. Lorenz, E. N. (1963). Deterministic non-periodic flow. Journal of Atmospheric Science, 20, 130–141.

    Article  Google Scholar 

  55. Lü, J., & Chen, G. (2002). A new chaotic attractor coined. International Journal of Bifurcation and Chaos, 12, 659–661.

    Article  MathSciNet  MATH  Google Scholar 

  56. Molaei, M., Jafari, S., Sprott, J. C., & Golpayegani, S. (2013). Simple chaotic flows with one stable equilibrium. International Journal of Bifurcation and Chaos, 23, 1350188.

    Article  MathSciNet  MATH  Google Scholar 

  57. Ojoniyi, O. S., & Njah, A. N. (2016). A 5D hyperchaotic Sprott B system with coexisting hidden attractor. Chaos, Solitons and Fractals, 87, 172–181.

    Article  MathSciNet  MATH  Google Scholar 

  58. Orlando, G. (2016). A discrete mathematical model for chaotic dynamics in economics: Kaldor’s model on business cycle. Mathematics and Computers in Simulation, 125, 83–98.

    Article  MathSciNet  Google Scholar 

  59. Pecora, L. M., & Carroll, T. L. (1990). Synchronization in chaotic signals. Physical Review A, 64, 821–824.

    MATH  Google Scholar 

  60. Pham, V.-T., Jafari, S., Volos, C., Wang, X., & Golpayegani, S. M. R. H. (2014). Is that really hidden? The presence of complex fixed-points in chaotic flows with no equilibria. International Journal of Bifurcation and Chaos, 24, 1450146.

    Google Scholar 

  61. Pham, V.-T., Vaidyanathan, S., Volos, C. K., Hoang, T. M., & Yem, V. V. (2016). Dynamics, synchronization and SPICE implementation of a memristive system with hidden hyperchaotic attractor. In A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing (Vol. 337, pp. 35–52). Germany: Springer.

    Google Scholar 

  62. Pham, V. T., Vaidyanathan, S., Volos, C. K., & Jafari, S. (2015). Hidden attractors in a chaotic system with an exponential nonlinear term. The European Physical Journal Special Topics, 224, 1507–1517.

    Google Scholar 

  63. Pham, V.-T., Volos, C. K., Jafari, S., Wei, Z., & Wang, X. (2014). Constructing a novel no-equilibrium chaotic system. International Journal of Bifurcation and Chaos, 24, 1450073.

    Google Scholar 

  64. Pham, V. T., Volos, C. K., Vaidyanathan, S., Le, T. P., & Vu, V. Y. (2015). A memristor-based hyperchaotic system with hidden attractors: Dynamics, sychronization and circuital emulating. Journal of Engineering Science and Technology Review, 8, 205–214.

    Google Scholar 

  65. Rosenblum, M. G., Pikovsky, A. S., & Kurths, J. (1997). From phase to lag synchronization in coupled chaotic oscillators. Physical Review Letters, 78, 4193–4196.

    Article  MATH  Google Scholar 

  66. Rössler, O. E. (1976). An equation for continuous chaos. Physics Letters A, 57, 397–398.

    Article  Google Scholar 

  67. Sastry, S. (1999). Nonlinear systems: Analysis, stability, and control. USA: Springer.

    Book  MATH  Google Scholar 

  68. Shahzad, M., Pham, V. T., Ahmad, M. A., Jafari, S., & Hadaeghi, F. (2015). Synchronization and circuit design of a chaotic system with coexisting hidden attractors. The European Physical Journal Special Topics, 224, 1637–1652.

    Article  Google Scholar 

  69. Sharma, P. R., Shrimali, M. D., Prasad, A., Kuznetsov, N. V., & Leonov, G. A. (2015). Control of multistability in hidden attractors. The European Physical Journal Special Topics, 224, 1485–1491.

    Article  Google Scholar 

  70. Soriano-Sanchez, A. G., Posadas-Castillo, C., Platas-Garza, M. A., & Diaz-Romero, D. A. (2015). Performance improvement of chaotic encryption via energy and frequency location criteria. Mathematics and Computers in Simulation, 112, 14–27.

    Article  MathSciNet  Google Scholar 

  71. Sprott, J. C. (2003). Chaos and times-series analysis. Oxford: Oxford University Press.

    MATH  Google Scholar 

  72. Sprott, J. C. (2010). Elegant chaos: Algebraically simple chaotic flows. Singapore: World Scientific.

    Book  MATH  Google Scholar 

  73. Sprott, J. C. (2015). Strange attractors with various equilibrium types. The European Physical Journal Special Topics, 224, 1409–1419.

    Article  Google Scholar 

  74. Srinivasan, K., Senthilkumar, D. V., Murali, K., Lakshmanan, M., & Kurths, J. (2011). Synchronization transitions in coupled time-delay electronic circuits with a threshold nonlinearity. Chaos, 21, 023119.

    Google Scholar 

  75. Stefanski, A., Perlikowski, P., & Kapitaniak, T. (2007). Ragged synchronizability of coupled oscillators. Physical Review E, 75, 016210.

    Article  MathSciNet  Google Scholar 

  76. Strogatz, S. H. (1994). Nonlinear dynamics and chaos: With applications to physics, biology, chemistry, and engineering. Massachusetts: Perseus Books.

    MATH  Google Scholar 

  77. Sun, H. H., Abdelwahad, A. A., & Onaral, B. (1894). Linear approximation of transfer function with a pole of fractional-order. IEEE Transactions on Automatic Control, 29, 441–444.

    Article  Google Scholar 

  78. Tacha, O. I., Volos, C. K., Kyprianidis, I. M., Stouboulos, I. N., Vaidyanathan, S., & Pham, V. T. (2016). Analysis, adaptive control and circuit simulation of a novel nonlineaar finance system. Applied Mathematics and Computation, 276, 200–217.

    Article  MathSciNet  MATH  Google Scholar 

  79. Tang, Y., Wang, Z., & Fang, J. A. (2010). Image encryption using chaotic coupled map lattices with time-varying delays. Communications in Nonlinear Science and Numerical Simulation, 15, 2456–2468.

    Article  MathSciNet  MATH  Google Scholar 

  80. Tavazoei, M. S., & Haeri, M. (2008). Limitations of frequency domain approximation for detecting chaos in fractional-order systems. Nonlinear Analysis, 69, 1299–1320.

    Article  MathSciNet  MATH  Google Scholar 

  81. Tavazoei, M. S., & Haeri, M. (2009). A proof for non existence of periodic solutions in time invariant fractional-order systems. Automatica, 45, 1886–1890.

    Article  MathSciNet  MATH  Google Scholar 

  82. Vaidyanathan, S. (2012). Anti-synchronization of four-wing chaotic systems via sliding mode control. International Journal of Automation and Computing, 9, 274–279.

    Article  Google Scholar 

  83. Vaidyanathan, S. (2013). A new six-term 3-D chaotic system with an exponential nonlineariry. Far East Journal of Mathematical Sciences, 79, 135–143.

    MATH  Google Scholar 

  84. Vaidyanathan, S. (2014). Analysis and adaptive synchronization of eight-term novel 3-D chaotic system with three quadratic nonlinearities. The European Physical Journal Special Topics, 223, 1519–1529.

    Article  Google Scholar 

  85. Vaidyanathan, S. (2016). Analysis, control and synchronization of a novel 4-D highly hyperchaotic system with hidden attractors. In A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control (Vol. 337, pp. 529–552). Studies in fuzziness and soft computing. Springer: Germany.

    Google Scholar 

  86. Vaidyanathan, S., & Azar, A. T. (2015). Anti-synchronization of identical chaotic systems using sliding mode control and an application to Vaidhyanathan-Madhavan chaotic systems. Studies in Computational Intelligence, 576, 527–547.

    Google Scholar 

  87. Vaidyanathan, S., & Azar, A. T. (2015). Hybrid synchronization of identical chaotic systems using sliding mode control and an application to Vaidhyanathan chaotic systems. Studies in Computational Intelligence, 576, 549–569.

    Google Scholar 

  88. Vaidyanathan, S., & Azar, A. T. (2016). A novel 4-D four-wing chaotic system with four quadratic nonlinearities and its synchronization via adaptive control method. In A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing (Vol. 337, pp. 203–224). Germany: Springer.

    Google Scholar 

  89. Vaidyanathan, S., & Azar, A. T. (2016). Adaptive backstepping control and synchronization of a novel 3-D jerk system with an exponential nonlinearity. In A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing (Vol. 337, pp. 249–274). Germany: Springer.

    Google Scholar 

  90. Vaidyanathan, S., & Azar, A. T. (2016). Adaptive control and synchronization of Halvorsen circulant chaotic systems. In A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing (Vol. 337, pp. 225–247). Germany: Springer.

    Google Scholar 

  91. Vaidyanathan, S., & Azar, A. T. (2016). Dynamic analysis, adaptive feedback control and synchronization of an eight-term 3-D novel chaotic system with three quadratic nonlinearities. In A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing (Vol. 337, pp. 155–178). Germany: Springer.

    Google Scholar 

  92. Vaidyanathan, S., & Azar, A. T. (2016). Generalized projective synchronization of a novel hyperchaotic four-wing system via adaptive control method. In A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing (Vol. 337, pp. 275–296). Germany: Springer.

    Google Scholar 

  93. Vaidyanathan, S., & Azar, A. T. (2016). Qualitative study and adaptive control of a novel 4-D hyperchaotic system with three quadratic nonlinearities. In A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing (Vol. 337, pp. 179–202). Germany: Springer.

    Google Scholar 

  94. Vaidyanathan, S., Idowu, B. A., & Azar, A. T. (2015). Backstepping controller design for the global chaos synchronization of Sprott’s jerk systems. Studies in Computational Intelligence, 581, 39–58.

    Google Scholar 

  95. Vaidyanathan, S., Pham, V. T., & Volos, C. K. (2015). A 5-d hyperchaotic rikitake dynamo system with hidden attractors. The European Physical Journal Special Topics, 224, 1575–1592.

    Google Scholar 

  96. Vaidyanathan, S., Volos, C., Pham, V. T., Madhavan, K., & Idowo, B. A. (2014). Adaptive backstepping control, synchronization and circuit simualtion of a 3-D novel jerk chaotic system with two hyperbolic sinusoidal nonlinearities. Archives of Control Sciences, 33, 257–285.

    MATH  Google Scholar 

  97. Vaidyanathan, S., Volos, C. K., & Pham, V. T. (2015). Analysis, control, synchronization and spice implementation of a novel 4-d hyperchaotic rikitake dynamo system without equilibrium. Journal of Engineering Science and Technology Review, 8, 232–244.

    Google Scholar 

  98. Volos, C. K., Kyprianidis, I. M., & Stouboulos, I. N. (2011). Various synchronization phenomena in bidirectionally coupled double scroll circuits. Communications in Nonlinear Science and Numerical Simulation, 71, 3356–3366.

    Article  MathSciNet  MATH  Google Scholar 

  99. Wang, X., & Chen, G. (2012). A chaotic system with only one stable equilibrium. Communications in Nonlinear Science and Numerical Simulation, 17, 1264–1272.

    Article  MathSciNet  Google Scholar 

  100. Wang, X., & Chen, G. (2013). Constructing a chaotic system with any number of equilibria. Nonlinear Dynamics, 71, 429–436.

    Article  MathSciNet  Google Scholar 

  101. Wei, Z. (2011). Dynamical behaviors of a chaotic system with no equilibria. Physics Letters A, 376, 102–108.

    Article  MathSciNet  MATH  Google Scholar 

  102. Westerlund, S., & Ekstam, L. (1994). Capacitor theory. IEEE Transactions on Dielectrics and Electrical Insulation, 1, 826–839.

    Article  Google Scholar 

  103. Woafo, P., & Kadji, H. G. E. (2004). Synchronized states in a ring of mutually coupled self-sustained electrical oscillators. Physical Review E, 69, 046206.

    Article  Google Scholar 

  104. Wolf, A., Swift, J. B., Swinney, H. L., & Vastano, J. A. (1985). Determining Lyapunov exponents from a time series. Physica D, 16, 285–317.

    Google Scholar 

  105. Xiao-Yu, D., Chun-Biao, L., Bo-Cheng, B., & Hua-Gan, W. (2015). Complex transient dynamics of hidden attractors in a simple 4d system. Chinese Physics B, 24, 050503.

    Article  Google Scholar 

  106. Yalcin, M. E., Suykens, J. A. K., & Vandewalle, J. (2005). Cellular neural networks: Multi-scroll chaos and synchronization. Singapore: World Scientific.

    Book  MATH  Google Scholar 

  107. Yang, Q. G., & Zeng, C. B. (2010). Chaos in fractional conjugate lorenz system and its scaling attractor. Communications in Nonlinear Science and Numerical Simulation, 15, 4041–4051.

    Article  MathSciNet  MATH  Google Scholar 

  108. Zhang, Y., & Sun, J. (2004). Chaotic synchronization and anti-synchronization based on suitable separation. Physics Letters A, 330, 442–447.

    Article  MathSciNet  MATH  Google Scholar 

  109. Zhu, Q., & Azar, A. T. (2015). Complex system modelling and control through intelligent soft computations. Germany: Springer.

    Book  MATH  Google Scholar 

  110. Zhusubaliyev, Z. T., & Mosekilde, E. (2015). Multistability and hidden attractors in a multilevel DC/DC converter. Mathematics and Computers in Simulation, 109, 32–45.

    Article  MathSciNet  Google Scholar 

  111. Zhusubaliyev, Z. T., Mosekilde, E., Churilov, A., & Medvedev, A. (2015). Multistability and hidden attractors in an impulsive Goodwin oscillator with time delay. The European Physical Journal Special Topics, 224, 1519–1539.

    Google Scholar 

  112. Zhusubaliyev, Z. T., Mosekilde, E., Rubanov, V. G., & Nabokov, R. A. (2015). Multistability and hidden attractors in a relay system with hysteresis. Physica D, 306, 6–15.

    Google Scholar 

Download references

Acknowledgements

Research described in this paper was supported by Czech Ministry of Education in frame of National Sustainability Program under grant GA15-22712S. V.-T. Pham is grateful to Le Thi Van Thu, Philips Electronics—Vietnam, for her help.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Viet-Thanh Pham .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Pham, VT., Vaidyanathan, S., Volos, C.K., Jafari, S., Gotthans, T. (2017). A Three-Dimensional Chaotic System with Square Equilibrium and No-Equilibrium. In: Azar, A., Vaidyanathan, S., Ouannas, A. (eds) Fractional Order Control and Synchronization of Chaotic Systems. Studies in Computational Intelligence, vol 688. Springer, Cham. https://doi.org/10.1007/978-3-319-50249-6_21

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-50249-6_21

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-50248-9

  • Online ISBN: 978-3-319-50249-6

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics