Reversal of period-doubling and extreme multistability in a novel 4D chaotic system with hyperbolic cosine nonlinearity

  • V. R. Folifack SigningEmail author
  • J. Kengne


Recently, the study of nonlinear systems with an infinite number of coexisting attractors has become one of the most followed topics owing to their fundamental and technological importance. This contribution is focused on a new 4D autonomous system (whose nonlinearity is a hyperbolic function) inspired by the quadratic system introduced by Jay and Roy (Optik,, 2017). Basic properties of the new system are discussed and its complex behaviors are characterized using classical nonlinear diagnostic tools. This system exhibits a rich repertoire of dynamic behaviors including chaos, chaos 2-torus, and quasi-periodicity. Interesting and striking phenomena such as antimonotonicity and extreme multistability are reported. Moreover, the hyperbolic cosine nonlinearity is easily implemented by using only two semiconductor diodes (no analog multiplier is involved) connected in parallel. We confirm the feasibility of the proposed theoretical model using PSpice simulations based on an analog computer of the model.


4D chaotic system Antimonotonicity Extreme multistability Hyperbolic cosine nonlinearity PSpice simulations 



The authors would like to thank the editor and the anonymous reviewers for their valuable comments and suggestions that helped to greatly improve this work.


  1. 1.
    Saucedo-Solorio JM, Pisarchik AN, Kir’yanov AV (2003) Generalized multistability in a fiber laser with modulated losses. J Opt Soc Am B 20:490CrossRefGoogle Scholar
  2. 2.
    Arecchi FT, Meucci R, Puccioni G, Tredicce J (1982) Experimental evidence of subharmonic bifurcations, multistability, and turbulence in a -switched gas laser. Phys Rev Lett 49:1217CrossRefGoogle Scholar
  3. 3.
    Arecchi FT (1991) Rate processes in nonlinear optical dynamics with many attractors. Chaos 1:357MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Power SB, Kleeman R (1994) Multiple equilibria in a global ocean general circulation model. J Phys Oceanogr 23:1670CrossRefGoogle Scholar
  5. 5.
    Huisman J, Weissing F (2001) Fundamental unpredictability in multispecies competition. Am Nat 157:488CrossRefGoogle Scholar
  6. 6.
    Schwarz G, Lehmann C, Scholl E (2000) Self-organized symmetry-breaking current filamentation and multistability in Corbino disks. Phys Rev B 61:10194CrossRefGoogle Scholar
  7. 7.
    Borresen J, Lynch S (2002) Further investigation of hysteresis in Chua’s circuit. Int J Bifurc Chaos 12:129CrossRefGoogle Scholar
  8. 8.
    Hongyan S, Stephen SK, Kenneth S (1999) Uncertain destination dynamics. Phys Rev E 60:3876CrossRefGoogle Scholar
  9. 9.
    Njitacke ZT, Kengne J, Guomkam AN (2017) Dynamical analysis and electronic circuit realization of an equilibrium free 3D chaotic system with a large number of coexisting attractors. Optik 130:356–364CrossRefGoogle Scholar
  10. 10.
    Sen Z, Yicheng Z, Zhijun L, Mengjiao W, Le X (2018) Generating one to four-wing hidden attractors in a novel 4D no-equilibrium chaotic system with extreme multistability. Chaos 28:013113MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Leonov GA, Kuznetsov NV, Vagaitsev VI (2011) Localization of hidden Chua’s attractors. Phys Lett A 375(23):2230–2233MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Leonov GA, Kuznetsov NV, Vagaitsev VI (2012) Hidden attractor in smooth Chua systems. Phys D 241(18):1482–1486MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    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):1330002MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Leonov G, Kuznetsov N, Kiseleva M, Solovyeva E, Zaretskiy A (2014) Hidden oscillations in mathematical model of drilling system actuated by induction motor with a wound rotor. Nonlinear Dyn 77(1–2):277–288CrossRefGoogle Scholar
  15. 15.
    Jafari S, Sprott JC, Nazarimehr F (2015) Recent new examples of hidden attractors. Eur Phys J Spec Top 224:1469–1476CrossRefGoogle Scholar
  16. 16.
    Ngonghala Calistus N, Ulrike F, Kenneth S (2011) Extreme multistability in a chemical model system. Phys Rev E 83:056206CrossRefGoogle Scholar
  17. 17.
    Patel Mitesh S, Unnati P, Abhijit S, Gautam CS, Chittaranjan H, Syamal KD, Ulrike F, Kenneth S, Calistus NN, Ravindra EA (2014) Experimental observation of extreme multistability in an electronic system of two coupled Rössler oscillators. Phys Rev E 89:022918CrossRefGoogle Scholar
  18. 18.
    Bao B, Tao J, Quan X, Chen M, Wu H, Hu Y (2016) Coexisting infinitely many attractors in active band-pass filter-based memristive circuit. Nonlinear Dyn 86:1711–1723CrossRefGoogle Scholar
  19. 19.
    Bao BC, Bao H, Wang N, Chen M, Xu Q (2017) Hidden extreme multistability in memristive hyperchaotic system. Chaos Solitons Fractals 94:102–111MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Njitacke ZT, Kengne J, Wafo TR, Pelap FB (2018) Uncertain destination dynamics of a novel memristive 4D autonomous system. Chaos, Solitons Fractals 107:177–185MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Singh JP, Roy BK (2017) Coexistence of asymmetric hidden chaotic attractors in a new simple 4-D chaotic system with curve of equilibria. Optik–Int J Light Electron Opt. Google Scholar
  22. 22.
    Rössler OE (1976) An equation for continuous chaos. Phys Lett A 57:397–398CrossRefzbMATHGoogle Scholar
  23. 23.
    Constantin P, Foias C (1988) Navier-stokes equations. The University of Chicago Press, Chicago and LondonzbMATHGoogle Scholar
  24. 24.
    Henon M, Heiles C (1964) The applicability of the third integral of motion some numerical experiments. Astron J 69:73MathSciNetCrossRefGoogle Scholar
  25. 25.
    Qi G, Chen G, Zhang Y (2008) On a new asymmetric chaotic system. Chaos, Solitons Fractals 37(2):409–423CrossRefGoogle Scholar
  26. 26.
    Sprott JC (2011) A proposed standard for the publication of new chaotic systems. Int J Bifurc Chaos 21(9):2391–2394CrossRefGoogle Scholar
  27. 27.
    Willems JC (1972) Dissipative dynamical systems part II: linear systems with quadratic supply rates. Archive for rational mechanics and analysis 45:352–393MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Hilborn RC (1994) Chaos and nonlinear dynamics-an introduction for scientists and engineers. Oxford University Press, OxfordzbMATHGoogle Scholar
  29. 29.
    Signing VRF, Kengne J (2018) 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. MathSciNetGoogle Scholar
  30. 30.
    Wolf A, Swift JB, Swinney HL, Wastano JA (1985) Determining Lyapunov exponents from time series. Phys D 16:285–317MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Dawson SP, Grebogi C, Yorke JA, Kan I, Ko CH (1992) Antimonotonicity: inevitable reversals of period-doubling cascades. Phys Lett A 162:249–254MathSciNetCrossRefGoogle Scholar
  32. 32.
    Srinivasan K, Chandrasekar VK, Venkatesan A, Raja MI (2016) Duffing– van der Pol oscillator type dynamics in Murali–Lakshmanan–Chua (MLC) circuit. Chaos, Solitons Fractals 82:60–71MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Kengne J, Njikam SM, Signing VRF (2018) A plethora of coexisting strange attractors in a simple jerk system with hyperbolic tangent nonlinearity. Chaos, Solitons Fractals 106:201–213MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Kengne J, Signing VRF, Chedjou JC, Leutcho GD (2017) Nonlinear behavior of a novel chaotic jerk system: antimonotonicity, crises, and multiple coexisting attractors. J Dyn Control, Int. Google Scholar
  35. 35.
    Kengne J, Negou NA, Tchiotsop D (2017) Antimonotonicity, chaos and multiple attractors in a novel autonomous memristor-based jerk circuit. Nonlinear Dyn. zbMATHGoogle Scholar
  36. 36.
    Njitacke ZT, Kengne J, Kamdjeu LK (2017) Antimonotonicity, chaos and multiple coexisting attractors in a simple hybrid diode-based jerk circuit. Chaos, Solitons Fractals 105:77–91MathSciNetCrossRefGoogle Scholar
  37. 37.
    Ogawa T (1988) Quasi-periodic instability and chaos in the badcavity laser with modulated inversion: numerical analysis of a Toda oscillator system. Phys Rev A 37:4286–4302MathSciNetCrossRefGoogle Scholar
  38. 38.
    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
  39. 39.
    Knobloch E, Weiss NO (1983) Bifurcations in a model of magnetoconvection. Physica D 9:379–407MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Bier M, Bountis TC (1984) Remerging Feigenbaum trees in dynamical systems. Phys Lett A 104:239–244MathSciNetCrossRefGoogle Scholar
  41. 41.
    Kyprianidis IM, Stouboulos IN, Haralabidis P (2000) Antimonotonicity and chaotic dynamics in a fourth-order autonomous nonlinear electric circuit Int. J Bifurc Chaos 10(8):1903–1915CrossRefGoogle Scholar
  42. 42.
    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):1550052MathSciNetCrossRefGoogle Scholar
  43. 43.
    Kengne J, Njitacke ZT, Nguomkam NA, Tsotsop FM, Fotsin HB (2015) Coexistence of multiple attractors and crisis route to chaos in a novel chaotic jerk circuit. Int J Bifurc Chaos 25(4):1550052CrossRefzbMATHGoogle Scholar
  44. 44.
    Kengne J (2016) On the dynamics of chua’s oscillator with a smooth cubic nonlinearity: occurrence of multiple attractors. Nonlinear Dyn 87(1):363–375MathSciNetCrossRefGoogle Scholar
  45. 45.
    Yujun N, Xingyuan W, Mingjun W, Huaguang Z (2010) A new hyperchaotic system and its circuit implementation. Commun Nonlinear Sci Numer Simul 15:3518–3524CrossRefGoogle Scholar
  46. 46.
    Jafari A, Mliki E, Akgul A, Pham VT, Kingni ST, Wang X, Jafari S (2017) Chameleon: the most hidden chaotic flow. Nonlinear Dyn. MathSciNetGoogle Scholar
  47. 47.
    Pham VT, Volos C, Jafari S, Sundarapandian V, Kapitaniak T, Wang X (2016) A chaotic system with different families of hidden attractors. Int J Bifurc Chaos 26(8):1650139MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Bianchi G, Kuznetsov NV, Leonov GA, Seledzhi SM, Yuldashev MV, Yuldashev RV (2016) Hidden oscillations in SPICE simulation of two-phase Costas loop with non-linear. VCO, IFAC Pap Online 49(14):45–50CrossRefGoogle Scholar
  49. 49.
    Kamdoum TV, Fotsin HB, Kengne J, Megam NEB, Talla PK (2016) Emergence of complex dynamical behaviors in improved Colpitts oscillators: antimonotonicity, coexisting attractors, and metastable chaos. Int J Dyn Control. Google Scholar
  50. 50.
    Singh JP, Roy BK (2018) A more chaotic and easily hardware implementable new 3-D chaotic system in comparison with 50 reported systems. Nonlinear Dyn. zbMATHGoogle Scholar
  51. 51.
    Singh JP, Roy BK, Wei Z (2018) A new four-dimensional chaotic system with first Lyapunov exponent of about 22, hyperbolic curve and circular paraboloid types of equilibria and its switching synchronization by an adaptive global integral sliding mode control. Chin Phys B 27(4):040500–040514Google Scholar
  52. 52.
    Singh JP, Roy BK (2018) Hidden attractors in a new complex generalised Lorenz hyper- chaotic system, its synchronisation using adaptive contraction theory, circuit validation and application. Nonlinear Dyn 92(2):373–394CrossRefzbMATHGoogle Scholar
  53. 53.
    Singh JP, Roy BK, Jafari S (2017) New family of 4-D hyperchaotic and chaotic systems with quadric surfaces of equilibria. Chaos, Solitons Fractals 106:243–257MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Singh JP, Roy BK (2017) The simplest 4-D chaotic system with line of equilibria, chaotic 2-torus and 3-torus behavior. Nonlinear Dyn 89(3):1845–1862CrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Unité de Recherche de Laboratoire d’Automatique et Informatique Appliquée (LAIA), Department of Electrical EngineeringIUT-FV Bandjoun, University of DschangDschangCameroon
  2. 2.Unité de Recherche de Matière Condensée, d’Electronique et de Traitement du Signal, Department of PhysicsUniversity of DschangDschangCameroon

Personalised recommendations