Design and control of a multi-wing dissipative chaotic system

  • Amin Zarei
  • Saeed Tavakoli


Considering the dissipative condition, a multi-wing chaotic attractor with a unique perspective is designed, using geometry root locus. By using a nonlinear dissipative term, a new structure is proposed to obtain a dissipative chaotic system with a four-wing attractor. The system properties, namely the phase portraits, bifurcation diagrams, Lyapunov exponents spectrum, Poincare maps and local stability, are investigated by numerical simulations. The phenomenon of multiple attractors is discussed and two double-wing smooth chaotic attractors using two different initial conditions are generated. The nonlinear time-delayed inputs are proposed for chaos control of the new system via bifurcation theory. Based on center manifold and normal form theories, the procedure of determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are investigated.


Dissipative condition Chaotic attractor Multiple attractors Chaos control 


  1. 1.
    Lorenz EN (1963) Deterministic nonperiodic flow. J Atmos Sci 20:130–141CrossRefGoogle Scholar
  2. 2.
    Chua L (1971) Memristor-the missing circuit element. IEEE Trans Circuit Theory 18:507–519CrossRefGoogle Scholar
  3. 3.
    Chua LE, Komuro M, Matsumoto T (1986) The double scroll family. IEEE Trans Circuits Syst 33:1072–1118CrossRefzbMATHGoogle Scholar
  4. 4.
    Chua LO, Wu CW, Huang A, Zhong GQ (1993) A universal circuit for studying and generating chaos. I. Routes to chaos. IEEE Trans Circuits Syst I: Fundam Theory Appl 40:732–744MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Wang X, Chen G (2012) A chaotic system with only one stable equilibrium. Commun Nonlinear Sci Numer Simul 17:1264–1272MathSciNetCrossRefGoogle Scholar
  6. 6.
    Wei Z, Yang Q (2012) Dynamical analysis of the generalized Sprott C system with only two stable equilibria. Nonlinear Dyn 68:543–554MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Wei Z (2012) Delayed feedback on the 3-D chaotic system only with two stable node-foci. Comput Math Appl 63:728–738MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Wang X, Chen G (2013) Constructing a chaotic system with any number of equilibria. Nonlinear Dyn 71:429–436MathSciNetCrossRefGoogle Scholar
  9. 9.
    Li C, Sprott JC (2014) Chaotic flows with a single nonquadratic term. Phys Lett A 378:178–183MathSciNetCrossRefGoogle Scholar
  10. 10.
    Wang Z, Cang S, Ochola EO, Sun Y (2012) A hyperchaotic system without equilibrium. Nonlinear Dyn 69:531–537MathSciNetCrossRefGoogle Scholar
  11. 11.
    Esen O, Choudhury AG, Guha P (2016) Bi-Hamiltonian structures of 3D chaotic dynamical systems. Int J Bifurc Chaos 26:1650215MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lai Q, Chen S (2016) Coexisting attractors generated from a new 4D smooth chaotic system. Int. J. Control Autom Syst 14:1124–1131CrossRefGoogle Scholar
  13. 13.
    Lai Q, Chen S (2016) Research on a new 3D autonomous chaotic system with coexisting attractors. Optik 127:3000–3004CrossRefGoogle Scholar
  14. 14.
    Kais B (2015) Gallery of chaotic attractors generated by fractal network. Int J Bifurc Chaos 25:1530002MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Chen G, Ueta T (1999) Yet another chaotic attractor. Int J Bifurc Chaos 9:1465–1466MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lü J, Chen G (2002) A new chaotic attractor coined. Int J Bifurc Chaos 12:659–661MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Qi G, Chen G, Du S, Chen Z, Yuan Z (2005) Analysis of a new chaotic system. Phys A Stat Mech Appl 352:295–308CrossRefGoogle Scholar
  18. 18.
    Qi G, Chen G, Li S, Zhang Y (2006) Four-wing attractors: from pseudo to real. Int J Bifurc Chaos 16:859–885MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Chen Z, Yang Y, Yuan Z (2008) A single three-wing or four-wing chaotic attractor generated from a three-dimensional smooth quadratic autonomous system. Chaos Solitons Fractals 38:1187–1196MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Grassi G (2008) Novel four-wing and eight-wing attractors using coupled chaotic Lorenz systems. Chin Phys B 17:3247CrossRefGoogle Scholar
  21. 21.
    Dadras S, Momeni HR (2009) A novel three-dimensional autonomous chaotic system generating two, three and four-scroll attractors. Phys Lett A 373:3637–3642MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lai Q, Guan ZH, Wu Y, Liu F, Zhang DX (2013) Generation of multi-wing chaotic attractors from a Lorenz-like system. Int J Bifurc Chaos 23:1350152MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Zhou L, Chen Z, Wang Z, Wang J (2016) On the analysis of local bifurcation and topological horseshoe of a new 4D hyper-chaotic system. Chaos Solitons Fractals 91:148–156MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Zarei A (2015) Complex dynamics in a 5-D hyper-chaotic attractor with four-wing, one equilibrium and multiple chaotic attractors. Nonlinear Dyn 81:585–605MathSciNetCrossRefGoogle Scholar
  25. 25.
    Zhou T, Chen G, Yang Q (2004) Constructing a new chaotic system based on the Silnikov criterion. Chaos Solitons Fractals 19:985–993MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Wei Z, Sprott JC, Chen H (2015) Elementary quadratic chaotic flows with a single non-hyperbolic equilibrium. Phys Lett A 379:2184–2187MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Tian X, Xu R (2016) Stability and Hopf bifurcation of a delayed CohenGrossberg neural network with diffusion. Math Meth Appl Sci 40:293–305Google Scholar
  28. 28.
    Dong E, Liang Z, Du S, Chen Z (2016) Topological horseshoe analysis on a four-wing chaotic attractor and its FPGA implement. Nonlinear Dyn 83:623–630MathSciNetCrossRefGoogle Scholar
  29. 29.
    Qi G, Wang Z, Guo Y (2012) Generation of an eight-wing chaotic attractor from Qi 3-D four-wing chaotic system. Int J Bifurc Chaos 22:1250287MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Guan ZH, Lai Q, Chi M, Cheng XM, Liu F (2014) Analysis of a new three-dimensional system with multiple chaotic attractors. Nonlinear Dyn 75:331–343MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Ott E, Grebogi C, Yorke JA (1990) Controlling chaos. Phys Rev Lett 64:1196MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Pyragas K (1992) Continuous control of chaos by self-controlling feedback. Phys Lett A 170:421–428CrossRefGoogle Scholar
  33. 33.
    Pyragas K (1995) Control of chaos via extended delay feedback. Phys Lett A 206:323–330MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Zarei A, Tavakoli S (2016) Hopf bifurcation analysis and ultimate bound estimation of a new 4-D quadratic autonomous hyper-chaotic system. Appl Math Comput 291:323–339MathSciNetGoogle Scholar
  35. 35.
    Tian X, Xu R, Gan Q (2015) Hopf bifurcation analysis of a BAM neural network with multiple time delays and diffusion. Appl Math Comput 266:909–926MathSciNetGoogle Scholar
  36. 36.
    Xiao M, Jiang G, Zhao L, Xu W, Wan Y, Fan C, Wang Z (2015) Stability switches and Hopf bifurcations of an isolated population model with delay-dependent parameters. Appl Math Comput 264:99–115MathSciNetGoogle Scholar
  37. 37.
    Zang H, Zhang T, Zhang Y (2015) Bifurcation analysis of a mathematical model for genetic regulatory network with time delays. Appl Math Comput 260:204–226MathSciNetGoogle Scholar
  38. 38.
    Hassard BD, Kazarinoff ND, Wan YH (1981) Theory and applications of Hopf bifurcation. CUP Arch 129–138Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Faculty of Electrical and Computer EngineeringUniversity of Sistan and BaluchestanZahedanIran

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