Nonlinear Dynamics

, Volume 95, Issue 1, pp 381–390 | Cite as

Hidden and self-excited coexisting attractors in a Lorenz-like system with two equilibrium points

  • Shijian CangEmail author
  • Yue Li
  • Ruiye Zhang
  • Zenghui Wang
Original Paper


This paper reports the finding of unusual hidden and self-excited coexisting dynamical behaviors in an existing Lorenz-like system. For different parameters, the system has different types of equilibrium points, such as saddle-nodes, stable focus-nodes, saddle-foci and nonhyperbolic equilibrium points, which can be used to find different types of hidden and self-excited attractors. The different types of attractors have been vividly demonstrated by several numerical techniques including phase portraits, bifurcation diagrams and basins of attraction. Very interestingly, we find the rare coexistence of chaotic attractor and periodic orbits in the Lorenz-like system with two saddle-foci.


Hidden attractors Coexisting attractors Lorenz-like system Bifurcation Basins of attraction 



We would like to thank the anonymous referees for their valuable suggestions and questions. This work is partly supported by the National Natural Science Foundation of China (Grant Nos. 61873186 and 61773282), the Application Base and Frontier Technology Research Project of Tianjin of China (Grant No. 13JCQNJC03600) and South African National Research Foundation (Grant Nos. 112142 and 112108), South African National Research Foundation Incentive Grant (No. 114911) and Tertiary Education Support Programme (TESP) of South African ESKOM.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflicts of interest.


  1. 1.
    Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20(2), 130 (1963)zbMATHCrossRefGoogle Scholar
  2. 2.
    Kapitaniak, M., Lazarek, M., Nielaczny, M., Czolczynski, K., Perlikowski, P., Kapitaniak, T.: Synchronization extends the life time of the desired behavior of globally coupled systems. Sci. Rep. 4, 4391 (2014)CrossRefGoogle Scholar
  3. 3.
    Menck, P.J., Heitzig, J., Kurths, J., Schellnhuber, H.J.: How dead ends undermine power grid stability. Nat. Commun. 5, 3969 (2014)CrossRefGoogle Scholar
  4. 4.
    Zhusubaliyev, Z.T., Mosekilde, E.: Multistability and hidden attractors in a multilevel DC/DC converter. Math. Comput. Simul. 109, 32 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Kuznetsov, N.V.: AETA 2015: Recent Advances in Electrical Engineering and Related Sciences, pp. 13–25. Springer, Berlin (2016)CrossRefGoogle Scholar
  6. 6.
    Vaidyanathan, S., Sambas, A., Mamat, M., Sanjaya, W.M.: A new three-dimensional chaotic system with a hidden attractor, circuit design and application in wireless mobile robot. Arch. Control Sci. 27(4), 541 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Kuznetsov, N.V., Leonov, G.A., Vagaitsev, V.I.: Analytical-numerical method for attractor localization of generalized Chua’s system. IFAC Proc. Vol. 43(11), 29 (2010)CrossRefGoogle Scholar
  8. 8.
    Chen, G., Ueta, T.: Yet another chaotic attractor. Int. J. Bifurc. Chaos 9(07), 1465 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Rössler, O.E.: An equation for continuous chaos. Phys. Lett. A 57(5), 397 (1976)zbMATHCrossRefGoogle Scholar
  10. 10.
    Cang, S., Wu, A., Wang, Z., Xue, W., Chen, Z.: Birth of one-to-four-wing chaotic attractors in a class of simplest three-dimensional continuous memristive systems. Nonlinear Dyn. 83(4), 1987 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Cang, S., Wu, A., Wang, Z., Wang, Z., Chen, Z.: A general method for exploring three-dimensional chaotic attractors with complicated topological structure based on the two-dimensional local vector field around equilibriums. Nonlinear Dyn. 83(1), 1069 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Dudkowski, D., Jafari, S., Kapitaniak, T., Kuznetsov, N.V., Leonov, G.A., Prasad, A.: Hidden attractors in dynamical systems. Phys. Rep. 637, 1 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Pham, V.T., Jafari, S., Kapitaniak, T., Volos, C., Kingni, S.T.: Generating a chaotic system with one stable equilibrium. Int. J. Bifurc. Chaos 27(04), 1750053 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Bao, J., Chen, D.: Coexisting hidden attractors in a 4D segmented disc dynamo with one stable equilibrium or a line equilibrium. Chin. Phys. B 26(8), 080201 (2017)CrossRefGoogle Scholar
  15. 15.
    Molaie, M., Jafari, S., Sprott, J.C., Golpayegani, S.M.R.H.: Simple chaotic flows with one stable equilibrium. Int. J. Bifurc. Chaos 23(11), 1350188 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Wang, X., Chen, G.: A chaotic system with only one stable equilibrium. Commun. Nonlinear Sci. Numer. Simul. 17(3), 1264 (2012)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Bao, B., Li, Q., Wang, N., Xu, Q.: Multistability in Chua’s circuit with two stable node-foci, chaos: an interdisciplinary. J. Nonlinear Sci. 26(4), 043111 (2016)Google Scholar
  18. 18.
    Barati, K., Jafari, S., Sprott, J.C., Pham, V.T.: Simple chaotic flows with a curve of equilibria. Int. J. Bifurc. Chaos 26(12), 1630034 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Jafari, S., Sprott, J.: Simple chaotic flows with a line equilibrium. Chaos Solitons Fractals 57, 79 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Jafari, S., Sprott, J.C., Molaie, M.: A simple chaotic flow with a plane of equilibria. Int. J. Bifurc. Chaos 26(06), 1650098 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Wang, Z., Cang, S., Ochola, E.O., Sun, Y.: A hyperchaotic system without equilibrium. Nonlinear Dyn. 69(1–2), 531 (2012)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Feng, Y., Pan, W.: Hidden attractors without equilibrium and adaptive reduced-order function projective synchronization from hyperchaotic Rikitake system. Pramana 88(4), 62 (2017)CrossRefGoogle Scholar
  23. 23.
    Pham, V.T., Volos, C., Jafari, S., Kapitaniak, T.: Coexistence of hidden chaotic attractors in a novel no-equilibrium system. Nonlinear Dyn. 87(3), 2001 (2017)zbMATHCrossRefGoogle Scholar
  24. 24.
    Jafari, S., Pham, V.T., Kapitaniak, T.: Multiscroll chaotic sea obtained from a simple 3D system without equilibrium. Int. J. Bifurc. Chaos 26(02), 1650031 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Signing, V.F., Kengne, J.: Coexistence of hidden attractors, 2-torus and 3-torus in a new simple 4-D chaotic system with hyperbolic cosine nonlinearity. Int. J. Dyn. Control (2018).
  26. 26.
    Leutcho, G., Kengne, J., Kengne, L.K.: Dynamical analysis of a novel autonomous 4-D hyperjerk circuit with hyperbolic sine nonlinearity: chaos, antimonotonicity and a plethora of coexisting attractors. Chaos Solitons Fractals 107, 67 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Rajagopal, K., Jafari, S., Karthikeyan, A., Srinivasan, A., Ayele, B.: Hyperchaotic memcapacitor oscillator with infinite equilibria and coexisting attractors. Circuits Syst. Signal Process. 37, 1–23 (2018). MathSciNetCrossRefGoogle Scholar
  28. 28.
    Liu, Y., Chávez, J.P.: Controlling coexisting attractors of an impacting system via linear augmentation. Phys. D Nonlinear Phenom. 348, 1 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Sprott, J.C., Jafari, S., Khalaf, A.J.M., Kapitaniak, T.: Megastability: coexistence of a countable infinity of nested attractors in a periodically-forced oscillator with spatially-periodic damping. Eur. Phys. J. Spec. Top. 226(9), 1979 (2017)CrossRefGoogle Scholar
  30. 30.
    Wang, X., Vaidyanathan, S., Volos, C., Pham, V.T., Kapitaniak, T.: Dynamics, circuit realization, control and synchronization of a hyperchaotic hyperjerk system with coexisting attractors. Nonlinear Dyn. 89(3), 1673 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Sharma, P., Shrimali, M., Prasad, A., Kuznetsov, N., Leonov, G.: Control of multistability in hidden attractors. Eur. Phys. J. Spec. Top. 224(8), 1485 (2015)CrossRefGoogle Scholar
  32. 32.
    Kuznetsov, N., Leonov, G.: Hidden attractors in dynamical systems: systems with no equilibria, multistability and coexisting attractors. IFAC Proc. Vol. 47(3), 5445 (2014)CrossRefGoogle Scholar
  33. 33.
    Lozi, R., Ushiki, S.: Coexisting chaotic attractors in Chua’s circuit. Int. J. Bifurc. Chaos 1(04), 923 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Bao, B., Hu, F., Chen, M., Xu, Q., Yu, Y.: Self-excited and hidden attractors found simultaneously in a modified Chua’s circuit. Int. J. Bifurc. Chaos 25(05), 1550075 (2015)zbMATHCrossRefGoogle Scholar
  35. 35.
    Chen, M., Li, M., Yu, Q., Bao, B., Xu, Q., Wang, J.: Dynamics of self-excited attractors and hidden attractors in generalized memristor-based Chua’s circuit. Nonlinear Dyn. 81(1–2), 215 (2015)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Chen, M., Yu, J., Bao, B.C.: Finding hidden attractors in improved memristor-based Chua‘s circuit. Electron. Lett. 51(6), 462 (2015)CrossRefGoogle Scholar
  37. 37.
    Munmuangsaen, B., Srisuchinwong, B.: A hidden chaotic attractor in the classical Lorenz system. Chaos Solitons Fractals 107, 61 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Yuan, Q., Yang, F.Y., Wang, L.: A note on hidden transient chaos in the \(l\)orenz system. Int. J. Nonlinear Sci. Numer. Simul. 18(5), 427 (2017)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Li, C., Sprott, J.C.: Coexisting hidden attractors in a 4-D simplified Lorenz system. Int. J. Bifurc. Chaos 24(03), 1450034 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Leonov, G., Kuznetsov, N., Mokaev, T.: Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion. Eur. Phys. J. Spec. Top. 224(8), 1421 (2015)CrossRefGoogle Scholar
  41. 41.
    Chen, G., Kuznetsov, N., Leonov, G., Mokaev, T.: Hidden attractors on one path: Glukhovsky–Dolzhansky, Lorenz, and Rabinovich systems. Int. J. Bifurc. Chaos 27(08), 1750115 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Wei, Z., Zhang, W.: Hidden hyperchaotic attractors in a modified Lorenz–Stenflo system with only one stable equilibrium. Int. J. Bifurc. Chaos 24(10), 1450127 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Leonov, G., Kuznetsov, N., Mokaev, T.: Hidden attractor and homoclinic orbit in Lorenz-like system describing convective fluid motion in rotating cavity. Commun. Nonlinear Sci. Numer. Simul. 28(1–3), 166 (2015)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Yang, Q., Chen, G.: A chaotic system with one saddle and two stable node-foci. Int. J. Bifurc. Chaos 18(05), 1393 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Sparrow, C.: The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, vol. 41. Springer, Berlin (2012)zbMATHGoogle Scholar
  46. 46.
    van der Schrier, G., Maas, L.R.: The diffusionless Lorenz equations, Shilnikov bifurcations and reduction to an explicit map. Phys. D Nonlinear Phenom. 141(1–2), 19 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Huang, D.: Periodic orbits and homoclinic orbits of the diffusionless Lorenz equations. Phys. Lett. A 309(3–4), 248 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Yang, Q., Wei, Z., Chen, G.: An unusual 3D autonomous quadratic chaotic system with two stable node-foci. Int. J. Bifurc. Chaos 20(04), 1061 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Wituła, R., Słota, D.: Cardano’s formula, square roots, Chebyshev polynomials and radicals. J. Math. Anal. Appl. 363(2), 639 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Chang, T.S., Chen, C.T.: On the Routh–Hurwitz criterion. IEEE Trans. Autom. Control 19(3), 250 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Kuznetsov, N., Leonov, G., Mokaev, T., Seledzhi, S.: In: AIP Conference Proceedings, vol. 1738, p. 210008. AIP Publishing (2016)Google Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of Product DesignTianjin University of Science and TechnologyTianjinPeople’s Republic of China
  2. 2.School of Electronic Information and AutomationTianjin University of Science and TechnologyTianjinPeople’s Republic of China
  3. 3.School of Electrical and Information EngineeringTianjin UniversityTianjinPeople’s Republic of China
  4. 4.Department of Electrical and Mining EngineeringUniversity of South AfricaFloridaSouth Africa

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