International Journal of Dynamics and Control

, Volume 7, Issue 4, pp 1225–1241 | Cite as

Scenario to chaos and multistability in a modified Coullet system: effects of broken symmetry

  • Jacques KengneEmail author
  • Leandre Kamdjeu Kengne


Recently, the study of nonlinear systems with multiple attractors has drained the attention of researchers worldwide owing to its fundamental and technological relevance. In this work, we consider a modified Coullet system with a cubic polynomial nonlinearity in the form \( \varphi_{m} \left( x \right) = x - mx^{2} - x^{3} \) where \( m \) is a control parameter. The new system bridges the gap between chaotic systems with symmetry and those without symmetry. In fact for \( m = 0 \), the system displays a perfect symmetry which is reflected on the location of equilibria, the type of attractors, and the shape of basins of attraction as well. In this special case, multistability involves a pair of mutually symmetric periodic or chaotic attractors, a pair of chaotic attractors with a pair of limit cycles, two pairs of limit cycles with a pair of chaotic attractors, and so on. For \( m \ne 0 \), the system is non symmetric and more complex nonlinear phenomena are observed such as parallel bifurcation branches, coexisting multiple asymmetric attractors, and hysteresis. For instance, the coexistence of a stable fixed point with a limit cycle, two different non symmetric limit cycles or chaotic attractors, three or four different non symmetric periodic and chaotic attractors are reported when monitoring the system parameters and initial conditions as well. Basins of attraction with non-trivial basin boundaries structures are computed to visualize the magnetization of the state space in the presence of multiple attractors. Experimental results based on a suitable electronic implementation of the proposed jerk system are included.


Modified Coullet system Cubic polynomial nonlinearity Multiple attractors Basins of attraction Experimental study 


  1. 1.
    Argyris J, Faust G, Haase M (1994) An exploration of chaos. North-Holland, AmsterdamzbMATHGoogle Scholar
  2. 2.
    Bao B, Jiang T, Xu Q, Chen M, Wu H, Hu Y (2016) Coexisting infinitely many attractors in active band-pass filter-based memristive circuit. Nonlinear Dyn 86(3):1711–1723Google Scholar
  3. 3.
    Bier M, Bountis TC (1994) Remerging Feigenbaum trees in dynamical systems. Phys Lett A 104:239–244MathSciNetGoogle Scholar
  4. 4.
    Coman K, McCormick WD, Swinney HL (1986) Multiplicity in a chemical reaction with one-dimensional dynamics. Phys Rev Lett 56:999Google Scholar
  5. 5.
    Cushing JM, Henson SM, Blackburn CC (2007) Multiple mixed attractors in a competition model. J Biol Dyn 1:347–362MathSciNetzbMATHGoogle Scholar
  6. 6.
    Dawson SP, Grebogi C, Yorke JA, Kan I, Koçak H (1992) Antimonotonicity: inevitable reversals of period-doubling cascades. Phys Lett A 162:249–254MathSciNetGoogle Scholar
  7. 7.
    Eichhorn R, Linz SJ, Hanggi P (2002) Simple polynomial classes of chaotic jerky dynamics. Chaos Solitons Fractals 13:1–15MathSciNetzbMATHGoogle Scholar
  8. 8.
    Elsonbaty AR, El-Sayed AMA (2017) Further nonlinear dynamical analysis of simple jerk system with multiple attractors. Nonlinear Dyn 83(2):1169–1186zbMATHGoogle Scholar
  9. 9.
    Kengne J, Njitacke ZT, Kamdoum Tamba V, Nguomkam Negou A (2015) Periodicity, chaos and multiple attractors in a memristor-based Shinriki’s circuit. Chaos Interdiscip. J Nonlinear Sci 25:103126zbMATHGoogle Scholar
  10. 10.
    Kengne J (2017) On the dynamics of Chua’s oscillator with a smooth cubic nonlinearity: occurrence of multiple attractors. Nonlinear Dyn 87(1):363–375MathSciNetGoogle Scholar
  11. 11.
    Kengne J (2015) Coexistence of chaos with hyperchaos, period-3 doubling bifurcation, and transient chaos in the hyperchaotic oscillator with gyrators. Int J Bifurc Chaos 25(4):1550052MathSciNetGoogle Scholar
  12. 12.
    Kengne J, Chedjou JC, Fonzin Fozin T, Kyamakya K, Kenne G (2014) On the analysis of semiconductor diode based chaotic and hyperchaotic chaotic generators-a case study. Nonlinear Dyn 77:373–386MathSciNetGoogle Scholar
  13. 13.
    Kengne J, Mogue RLT (2018) Dynamic analysis of a novel jerk system with composite tanh-cubic nonlinearity: chaos, multi-scroll, and multiple coexisting attractors. Int J Dyn Control. CrossRefGoogle Scholar
  14. 14.
    Kengne J, Njitacke ZT, Nguomkam Negou A, Fouodji Tsotsop M, Fotsin HB (2015) Coexistence of multiple attractors and crisis route to chaos in a novel chaotic jerk circuit. Int J Bifurc Chaos 25(4):1550052zbMATHGoogle Scholar
  15. 15.
    Kengne J, Folifack Signing VR, Chedjou JC, Leutcho GD (2017) Nonlinear behavior of a novel chaotic jerk system: antimonotonicity, crises, and multiple coexisting attractors. Int J Dyn Control. CrossRefGoogle Scholar
  16. 16.
    Kengne J, Njitacke ZT, Fotsin HB (2016) Dynamical analysis of a simple autonomous jerk system with multiple attractors. Nonlinear Dyn 83:751–765MathSciNetGoogle Scholar
  17. 17.
    Li C, Sprott JC (2014) Coexisting hidden attractors in a 4-D simplified Lorenz system. Int J Bifurc Chaos 24:1450034MathSciNetzbMATHGoogle Scholar
  18. 18.
    Li C, Hu W, Sprott JC, Wang X (2015) Multistability in symmetric chaotic systems. Eur Phys J Spec Top 224:1493–1506Google Scholar
  19. 19.
    Li C, Sprott JC (2013) Multistability in a butterfly flow. Int J Bifurc Chaos 23(12):1350199MathSciNetzbMATHGoogle Scholar
  20. 20.
    Pisarchik AN, Feudel U (2014) Control of multistability. Phys Rep 540(4):167–218MathSciNetzbMATHGoogle Scholar
  21. 21.
    Luo X, Small M (2007) On a dynamical system with multiple chaotic attractors. Int J Bifurc Chaos 17(9):3235–3251MathSciNetzbMATHGoogle Scholar
  22. 22.
    Masoller C (1994) Coexistence of attractors in a laser diode with optical feedback from a large external cavity. Phys Rev A 50:2569–2578Google Scholar
  23. 23.
    Massoudi A, Mahjani MG, Jafarian M (2010) Multiple attractors in Koper-Gaspard model of electrochemical. J Electroanal Chem 647:74–86Google Scholar
  24. 24.
    Upadhyay RK (2003) Multiple attractors and crisis route to chaos in a model of food-chain. Chaos Solitons Fractals 16:737–747MathSciNetzbMATHGoogle Scholar
  25. 25.
    Vaithianathan V, Veijun J (1999) Coexistence of four different attractors in a fundamental power system model. IEEE Trans Circuits Syst I 46:405–409Google Scholar
  26. 26.
    Lai Q, Chen S (2016) Generating multiple chaotic attractors from sprott B system. Int J Bifurc Chaos 26(11):1650177MathSciNetzbMATHGoogle Scholar
  27. 27.
    Lai Q, Chen S (2016) Coexisting attractors generated from a new 4D smooth chaotic system. Int J Control Autom Syst 14(4):1124–1131Google Scholar
  28. 28.
    Hens C, Dana SK, Feudel U (2015) Extreme multistability: attractors manipulation and robustness. Chaos 25:053112MathSciNetzbMATHGoogle Scholar
  29. 29.
    Yuan F, Wang GY, Wang XW (2016) Extreme multistability in a memristor-based multi-scroll hyper-chaotic system. Chaos 26(7):073107MathSciNetGoogle Scholar
  30. 30.
    Letellier C, Gilmore R (2007) Symmetry groups for 3D dynamical systems. J Phys A Math Theor 40:5597–5620MathSciNetzbMATHGoogle Scholar
  31. 31.
    Sprott JC, Wang X, Chen G (2013) Coexistence of point, periodic and strange attractors. Int J Bifurc Chaos 23(5):1350093MathSciNetGoogle Scholar
  32. 32.
    Sprott JC (2010) Elegant chaos: algebraically simple flow. World Scientific Publishing, SingaporezbMATHGoogle Scholar
  33. 33.
    Sprott JC (2000) Simple chaotic systems and circuits. Am J Phys 68:758–763Google Scholar
  34. 34.
    Sprott JC (2011) A new chaotic jerk circuit. IEEE Trans Circuits Syst II Express Br 58:240–243Google Scholar
  35. 35.
    Louodop P, Kountchou M, Fotsin H, Bowong S (2014) Practical finite-time synchronization of jerk systems: theory and experiment. Nonlinear Dyn 78:597–607MathSciNetzbMATHGoogle Scholar
  36. 36.
    Njitacke ZT, Kengne J, Fotsin HB, Nguomkam Negou A, Tchiotsop D (2016) Coexistence of multiple attractors and crisis route to chaos in a novel memristive diode bridge-based Jerk circuit. Chaos Solitons Fractals 91:180–197zbMATHGoogle Scholar
  37. 37.
    Pivka L, Wu CW, Huang A (1994) Chua’s oscillator: a compendium of chaotic phenomena. J Franklin Inst 331B(6):705–741MathSciNetzbMATHGoogle Scholar
  38. 38.
    Yang J, Zhao L (2015) Bifurcation analysis and chaos control of the modified Chua’s circuit system. Chaos Solitons Fractals 77:332–339MathSciNetzbMATHGoogle Scholar
  39. 39.
    Zhao H, Lin Y, Dai Y (2017) Hopf bifurcation and hidden attractor of a modified Chua’s equation. Nonlinear Dyn 90(3):2013–2021MathSciNetzbMATHGoogle Scholar
  40. 40.
    Zhong GQ (1994) Implementation of Chua’s circuit with a cubic nonlinearity. IEEEE Trans Circuits Syst I Fund Theor Appl 41:934–941Google Scholar
  41. 41.
    Li C, Sprott JC (2013) Amplitude control approach for chaotic signals. Nonlinear Dyn 73:1335–1341MathSciNetzbMATHGoogle Scholar
  42. 42.
    Strogatz SH (1994) Nonlinear dynamics and chaos. Addison-Wesley, ReadingGoogle Scholar
  43. 43.
    Malasoma JM (2000) What is the simplest dissipative chaotic jerk equation which is parity invariant? Phys Lett A 264:383–389MathSciNetzbMATHGoogle Scholar
  44. 44.
    Jafari A, Mliki E, Akgul A, Pham VT, Kingni ST, Wang X, Jafari S (2017) Chameleon: the most hidden chaotic flow. Nonlinear Dyn. MathSciNetCrossRefGoogle Scholar
  45. 45.
    Jafari S, Sportt JC, Nazarimehr F (2015) Recent new examples of hidden attractors. Eur Phys J Spec Top 224:1469–1476Google Scholar
  46. 46.
    Jafari S, Sprott JC, Golpayegani S (2013) Elementary chaotic flows with no equilibria. Phys Lett A 377:699–702MathSciNetGoogle Scholar
  47. 47.
    Nayfeh AH, Balachandran B (1995) Applied nonlinear dynamics: analytical, computational and experimental methods. Wiley, New YorkzbMATHGoogle Scholar
  48. 48.
    Pham V-T, Jafari S, Volos C, Giakoumis A, Vaidyanathan S, Kapitaniak T (2016) A chaotic system with equilibria located on the rounded square loop and its circuit implementation. IEEE Trans Circuits Syst II Express Briefs 6(9):878–882Google Scholar
  49. 49.
    Kuznetsov YA (1995) Elements of applied bifurcation theory. Springer, New YorkzbMATHGoogle Scholar
  50. 50.
    Kuznetsov N, Leonov G, Vagaitsev V (2010) Analytical-numerical method for attractor localization of generalized Chua’s system. IFAC Proc 4(1):29–33Google Scholar
  51. 51.
    Kuznetsov NV, Leonov GA, Yuldashev MV, Yuldashev RV (2017) Hidden attractors in dynamical models of phase-locked loop circuits: limitations of simulation in MATLAB and SPICE. Commun Nonlinear Sci Numer Simul 51:39–49Google Scholar
  52. 52.
    Leonov G, Kuznetsov N, Vagaitsev V (2011) Localization of hidden Chuaʼs attractors. Phys Lett A 375(23):2230–2233MathSciNetzbMATHGoogle Scholar
  53. 53.
    Leonov G, Kuznetsov N, Vagaitsev V (2012) Hidden attractor in smooth Chua systems. Physica D 241(18):1482–1486MathSciNetzbMATHGoogle Scholar
  54. 54.
    Leonov GA, Kuznetsov NV (2013) Hidden attractors in dynamical systems. From hidden oscillations in Hilbert Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits. Int J Bifurc Chaos 23(01):1330002MathSciNetzbMATHGoogle Scholar
  55. 55.
    Leonov GA, Kuznetsov NV, Mokaev TN (2015) Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion. Eur Phys J Spec Top 224:1421–1458Google Scholar
  56. 56.
    Wolf A, Swift JB, Swinney HL, Wastano JA (1985) Determining Lyapunov exponents from time series. Physica D 16:285–317MathSciNetzbMATHGoogle Scholar
  57. 57.
    Kuznetsov AP, Kuznetsov SP, Mosekilde E, Stankevich NV (2015) Co-existing hidden attractors in a radio-physical oscillator. J Phys A Math Theor 48:125101MathSciNetzbMATHGoogle Scholar
  58. 58.
    Leipnik RB, Newton TA (1981) Double strange attractors in rigid body motion with linear feedback control. Phys Lett A 86:63–87MathSciNetGoogle Scholar
  59. 59.
    Manimehan I, Philominathan P (2012) Composite dynamical behaviors in a simple series–parallel LC circuit. Chaos Solitons Fractals 45:1501–1509Google Scholar
  60. 60.
    Ogawa T (1988) Quasiperiodic instability and chaos in the bad-cavity laser with modulated inversion: numerical analysis of a Toda oscillator system. Phys Rev A 37:4286MathSciNetGoogle Scholar
  61. 61.
    Parlitz U, Lauterborn W (1985) Superstructure in the bifurcation set of the Duffing equation ẍ + dẋ + x + x3 = f cos (ωt). Phys Lett A 107:351–355MathSciNetGoogle Scholar
  62. 62.
    Parlitz U, Lauterborn W (1987) Period-doubling cascades and devil’s staircases of the driven van der Pol oscillator. Phys Rev A 36:1428Google Scholar
  63. 63.
    Kocarev L, Halle K, Eckert K, Chua LO (1993) Experimental observation of antimonotonicity in Chua’s circuit. Int J Bifurc Chaos 3:1051–1055zbMATHGoogle Scholar
  64. 64.
    Kyprianidis I, Stouboulos I, Haralabidis P, Bountis T (2000) Antimonotonicity and chaotic dynamics in a fourth-order autonomous nonlinear electric circuit. Int J Bifurc Chaos 10:1903–1915Google Scholar
  65. 65.
    Kingni ST, 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. Eur Phys J Plus 129:76Google Scholar
  66. 66.
    Kingni ST, Keuninckx L, Woafo P, van der Sande G, Danckaert J (2013) Dissipative chaos, Shilnikov chaos and bursting oscillations in a three-dimensional autonomous system: theory and electronic implementation. Nonlinear Dyn 73:1111–1123MathSciNetzbMATHGoogle Scholar
  67. 67.
    Kiers K, Schmidt D (2004) Precision measurement of a simple chaotic circuit. Am J Phys 76(4):503–509Google Scholar
  68. 68.
    Hamill DC (1993) Learning about chaotic circuits with SPICE. IEEE Trans Edu. 36:28–35Google Scholar
  69. 69.
    Roberts W, Sedra S (1997) SPICE. Oxford university Press, NYGoogle Scholar
  70. 70.
    Kamdoum Tamba V, Fotsin HB, Kengne J, Megam Ngouonkadi EB, Talla PK (2017) Emergence of complex dynamical behaviors in improved Colpitts oscillators: antimonotonicity, coexisting attractors, and metastable chaos. Int J Dyn Control 5:395–406MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratoire d’Automatique et Informatique Appliquée (LAIA), Department of Electrical Engineering, IUT-FV BandjounUniversity of DschangDschangCameroon
  2. 2.Laboratory of Electronics and Signal Processing, Department of PhysicsUniversity of DschangDschangCameroon

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