Skip to main content
SpringerLink
Log in

Systems with Stable Equilibria

  • Chapter
  • First Online:
Systems with Hidden Attractors

Part of the book series: SpringerBriefs in Applied Sciences and Technology ((BRIEFSNONLINCIRC))

Abstract

After a recent striking discovery of Wang–Chen system with only one stable equilibrium, various systems with stable equilibria have been investigated. The aim of this chapter is to present such rare systems. In addition, we consider how to find and construct systems having stable equilibrium.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight 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

References

  1. Bagley, R.L., Calico, R.A.: Fractional-order state equations for the control of visco-elastically damped structers. J. Guide Control Dyn. 14, 304–311 (1991)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Gamez-Guzman, L., Cruz-Hernandez, C., Lopez-Gutierrez, R., Garcia-Guerrero, E.E.: Synchronization of Chua’s circuits with multi-scroll attractors: application to communication. Commun. Nonlinear Sci. Numer. Simul. 14, 2765–2775 (2009)

    Article  Google Scholar 

  4. Gejji, D., Jafari, H.: Adomian decomposition: a tool for solving a system of fractional differential equations. J. Math. Anal. Appl. 301, 508–518 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Grigorenko, I., Grigorenko, E.: Chaotic dynamics of the fractional–order lorenz system. Phys. Rev. Lett. 91, 034,101 (2003)

    Google Scholar 

  6. Han, F., Hu, J., Yu, X., Wang, Y.: Fingerprint images encryption via multi-scroll chaotic attractors. Appl. Math. Comput. 185, 931–939 (2007)

    MATH  Google Scholar 

  7. Hartley, T.T., Lorenzo, C.F., Qammer, H.K.: Chaos on a fractional Chua’s system. IEEE Trans. Circ. Syst. I Fundam. Theory Appl. 42, 485–490 (1995)

    Article  Google Scholar 

  8. Heaviside, O.: Electromagnetic Theory. Academic Press, New York (1971)

    MATH  Google Scholar 

  9. Huan, S., Li, Q., Yang, X.: Horseshoes in a chaotic system with only one stable equilibrium. Int. J. Bifurc. Chaos 23, 1350,002 (2013)

    Google Scholar 

  10. Jafari, S., Sprott, J.C.: Simple chaotic flows with a line equilibrium. Chaos, Solitons Fract. 57, 79–84 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Jenson, V.G., Jeffreys, G.V.: Mathematical Methods in Chemical Engineering. Academic Press, New York (1997)

    MATH  Google Scholar 

  12. Kingni, S.T., Jafari, S., Simo, H., Woafo, P.: Three-dimensional chaotic autonomous system with only one stable equilibrium: analysis, circuit design, parameter estimation, control, synchronization and its fractional-order form. Eur. Phys. J. Plus 129, 76 (2014)

    Article  Google Scholar 

  13. Lao, S.K., Shekofteh, Y., Jafari, S., Sprott, J.C.: Cost function based on Gaussian mixture model for parameter estimation of a chaotic circuit with a hidden attractor. Int. J. Bifurc. Chaos 24, 1450,010 (2014)

    Google Scholar 

  14. Li, C.P., Peng, G.J.: Chaos in Chen’s system with a fractional-order. Chaos, Solitons Fract. 20, 443–450 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lü, J.H., Chen, G.R.: Generating multiscroll chaotic attractors: theories, methods and applications. Int. J. Bifurc. Chaos 16, 775–858 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ma, J., Wu, X., Chu, R., Zhang, L.: Selection of multi-scroll attractors in Jerk circuits and their verification using Pspice. Nonlinear Dyn. 76, 1951–1962 (2014)

    Article  Google Scholar 

  17. Molaie, M., Jafari, S., Sprott, J.C., Golpayegani, S.M.R.H.: Simple chaotic flows with one stable equilibrium. Int. J. Bifurc. Chaos 23, 1350,188 (2013)

    Google Scholar 

  18. Orue, A.B., Alvarez, G., Pastor, G., Romera, M., Montoya, F., Li, S.: A new parameter determination method for some double-scroll chaotic systems and its applications to chaotic cryptanalysis. Commun. Nonlinear Sci. Numer. Simul. 15, 3471–3483 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Pham, V.T., Volos, C., Jafari, S., Wei, Z., Wang, X.: Constructing a novel no–equilibrium chaotic system. Int. J. Bifurc. Chaos 24, 1450,073 (2014)

    Google Scholar 

  20. Sprott, J.: Some simple chaotic flows. Phys. Rev. E 50, R647–650 (1994)

    Article  MathSciNet  Google Scholar 

  21. Sprott, J.C., Wang, X., Chen, G.: Coexistence of point, periodic and strange attractors. Int. J. Bifurc. Chaos 23, 1350,093 (2013)

    Google Scholar 

  22. Sun, H.H., Abdelwahad, A.A., Onaral, B.: Linear approximation of transfer function with a pole of fractional-order. IEEE Trans. Autom. Control 29, 441–444 (1894)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  25. Tlelo-Cuautle, E., Pano-Azucena, A.D., Rangel-Magdaleno, J.J., Carbajal-Gomez, V.H., Rodriguez-Gomez, G.: Generating a 50-scroll chaotic attractor at 66 MHz by using FPGAs. Nonlinear Dyn. 85, 2143–2157 (2016)

    Article  Google Scholar 

  26. Tlelo-Cuautle, E., Rangel-Magdaleno, J.J., Pano-Azucena, A.D., Obeso-Rodelo, P.J., Nunez-Perez, J.C.: FPGA realization of multi-scroll chaotic oscillators. Commun. Nonlinear Sci. Numer. Simul. 27, 66–80 (2015)

    Article  MathSciNet  Google Scholar 

  27. Wang, X., Chen, G.: A chaotic system with only one stable equilibrium. Commun. Nonlinear Sci. Numer. Simul. 17, 1264–1272 (2012)

    Article  MathSciNet  Google Scholar 

  28. Wei, Z.: Delayed feedback on the 3-D chaotic system only with two stable node-foci. Comput. Math. Appl. 63, 728–738 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wei, Z., Moroz, I., Liu, A.: Degenerate Hopf bifurcation, hidden attractors, and control in the extented Sprott E system with only one stable equilibrium. Turk. J. Math. 38, 672–687 (2014)

    Article  MATH  Google Scholar 

  30. Wei, Z., Moroz, I., Wang, Z., Sprott, J.C., Kapitaniak, T.: Dynamics at infinity, degenerate Hopf and zero–Hopf bifurcation for Kingni–Jafari system with hidden attractors. Int. J. Bifurc. Chaos 26, 1650,125 (2016)

    Google Scholar 

  31. Wei, Z., Pham, V.T., Kapitaniak, T., Wang, Z.: Bifurcation analysis and circuit realization for multiple-delayed Wang-Chen with hidden chaotic attractors. Nonlinear Dyn. 85, 1635–1650 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  32. Wei, Z., Wang, Z.: Chaotic behavior and modified function projective synchronization of a simple system with one stable equilibrium. Kybernetika 49, 359–374 (2013)

    MathSciNet  MATH  Google Scholar 

  33. Wei, Z., Yang, Q.: Dynamical analysis of a new autonomous 3–D chaotic system only with stable equilibria. Nonlinear Anal. Real World Appl. 12, 106–118 (2011)

    Google Scholar 

  34. Wei, Z., Yang, Q.: Dynamical analysis of the generalized Sprott C system with only two stable equilibria. Nonlinear Dyn. 68, 543–554 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  35. Wei, Z.C., Pehlivan, I.: Chaos, coexisting attractors, and circuit design of the generalized Sprott C system with only two stable equilibria. Optoelectron. Adv. Mater. Rapid Commun. 6, 742–745 (2012)

    Google Scholar 

  36. Westerlund, S., Ekstam, L.: Capacitor theory. IEEE Trans. Dielectr. Electr. Insul. 1, 826–839 (1994)

    Article  Google Scholar 

  37. Yalcin, M.E.: Increasing the entropy of a random number generator using n-scroll chaotic attractors. Int. J. Bifurc. Chaos 17, 4471–4479 (2007)

    Article  Google Scholar 

  38. Yang, Q., Chen, G.: A chaotic system with one saddle and two stable node-foci. Int. J. Bifurc. Chaos 18, 1393–1414 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  39. Yang, Q., Wei, Z., Chen, G.: An unusual 3D chaotic system with two stable node-foci. Int. J. Bifurc. Chaos 20, 1061–1083 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  40. Yang, Q.G., Zeng, C.B.: Chaos in fractional conjugate lorenz system and its scaling attractor. Commun. Nonlinear Sci. Numer. Simul. 15, 4041–4051 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  41. Yeniceri, R., Yalcin, M.E.: Multi-scroll chaotic attractors from a generalized time-delayed sampled-data system. Int. J. Circ. Theory Appl. 44, 1263–1276 (2016)

    Article  Google Scholar 

  42. Zidan, M.A., Radwan, A.G., Salama, K.N.: Controllable V–shape multiscroll butterfly attractors system and circuit implementation. Int. J. Bifurc. Chaos 22, 1250,142 (2012)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Electronics and Telecommunications, Hanoi University of Science and Technology, Hanoi, Vietnam

    Viet-Thanh Pham

  2. Department of Physics, Aristotle University of Thessaloniki, Thessaloniki, Greece

    Christos Volos

  3. Faculty of Mechanical Engineering, Lodz University of Technology, Łódź, Poland

    Tomasz Kapitaniak

Authors
  1. Viet-Thanh Pham
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christos Volos
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tomasz Kapitaniak
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Viet-Thanh Pham .

Rights and permissions

Reprints and permissions

Copyright information

© 2017 The Author(s)

About this chapter

Cite this chapter

Pham, VT., Volos, C., Kapitaniak, T. (2017). Systems with Stable Equilibria. In: Systems with Hidden Attractors. SpringerBriefs in Applied Sciences and Technology(). Springer, Cham. https://doi.org/10.1007/978-3-319-53721-4_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-53721-4_2

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-53720-7

  • Online ISBN: 978-3-319-53721-4

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation