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Remerging Feigenbaum trees, and multiple coexisting bifurcations in a novel hybrid diode-based hyperjerk circuit with offset boosting

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Abstract

This paper proposes a novel chaotic hyperjerk circuit obtained from the autonomous 4-D hyperjerk circuit (Leutcho et al. in Chaos Solitons Fractals 107:67–87, 2018) via substituting the nonlinear component (pair of anti-parallel diodes) with a first order hybrid diode circuit. The nonlinear component (i.e. hybrid diode) is characterized by a frequency dependent nonlinear I–V characteristic which is responsible for complex behaviours of the whole circuit. The essential dynamic properties of the model are investigated by exploiting numerical tools of nonlinear theory such as bifurcation diagrams, graphs of Lyapunov exponents, as well as phase portraits. Some interesting features are found including period-doubling bifurcation, coexisting bifurcations, symmetry recovering crises, antimonotonicity, and offset-boosting. One of the most interesting features of the new hyperjerk circuit is the presence of various areas of parameter space in which the hyperjerk system exhibits the attractive and intricate property of coexisting bifurcation and several coexisting attractors (e.g. coexistence of two, three, four, five, six, or seven disconnected periodic and chaotic attractors) for the same parameter’ set. Laboratory experimental results show a very good agreement with the theoretical analysis.

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References

  1. Sprott JC (1997) Simplest dissipative chaotic flow. Phys Lett A 228:271–274

    Article  MathSciNet  MATH  Google Scholar 

  2. Sprott JC (2010) Elegant Chaos: algebraically simple flow. World Scientific Publishing, Singapore

    Book  MATH  Google Scholar 

  3. Leutcho GD, Kengne J, Kamdjeu Kengne L (2018) Dynamical analysis of a novel autonomous 4-D hyperjerk circuit with hyperbolic sine nonlinearity. Chaos Solitons Fractals 107:67–87

    Article  MathSciNet  MATH  Google Scholar 

  4. Sprott JC (2011) A new chaotic jerk circuit. IEEE Trans Circuit Syst II Express Briefs 58:240–243

    Article  Google Scholar 

  5. Kengne J, NguomkamNegou A, Tchiotsop D (2017) Antimonotonicity, chaos and multiple attractors in a novel autonomous memristor-based jerk circuit. Nonlinear Dyn 88(4):2589–2608

    Article  Google Scholar 

  6. Kengne J, Folifack VR, Chedjou J, Leutcho GD (2017) Nonlinear behavior of a novel chaotic jerk system: antimonotonicity, crises, and multiple coexisting attractors. Int J Dynam Control. https://doi.org/10.1007/s40435-017-0318-6

    Google Scholar 

  7. Njitacke ZT, Kengne J, Kamdjeu KL (2017) Antimonotonicity, chaos and multiple coexisting attractors in a simple hybrid diode-based jerk circuit. Chaos Solitons Fractals 105:77–91

    Article  MathSciNet  Google Scholar 

  8. Kengne J, Njitacke ZT, Fotsin HB (2016) Coexistence of multiple attractors and crisis route to chaos in autonomous third order Duffing–Holmes type chaotic oscillators. Commun Nonlinear Sci Numer Simul 36:29–44

    Article  MathSciNet  Google Scholar 

  9. Kengne J, Njitacke ZT, Fotsin HB (2016) Dynamical analysis of a simple autonomous jerk system with multiple attractors. Nonlinear Dyn 83:751–765

    Article  MathSciNet  Google Scholar 

  10. Pham VT, Volos CK, Vaidyanathan S, Le TP, Le T, Vu V (2015) A memristor-based hyperchaotic system with hidden attractors. Dynamics, synchronization and circuit emulating. J Eng Sci Technol Rev 2:205–14

    Article  Google Scholar 

  11. Al-Smadi M, Freihat A, Abu Arqub O, Shawagfeh N (2015) A novel multistep generalized differential transform method for solving fractional-order Lü chaotic and hyperchaotic systems. J Comput Anal Appl 19:713–724

    MathSciNet  MATH  Google Scholar 

  12. Abu Arqub O (2016) the reproducing kernel algorithm for handling differential algebraic systems of ordinary differential equations. Math Methods Appl Sci 39:4549–4562

    Article  MathSciNet  MATH  Google Scholar 

  13. El-Ajou A, Abu Arqub O, Momani S (2012) Homotopy analysis method for second-order boundary value problems of integrodifferential equations. Discrete Dyn Nat Soc 365792(2012):18. https://doi.org/10.1155/2012/365792

    MathSciNet  MATH  Google Scholar 

  14. Jafari S, Ahmadi A, Panahi S, Rajagopal K (2018) Extreme multi-stability: when imperfection changes quality. Chaos Solitons Fractals 108:182–186

    Article  Google Scholar 

  15. Jafari S, Sprott JC, Pham VT, Volos K, Li C (2016) Simple chaotic 3D flows with surfaces of equilibria. Nonlinear Dyn 86:1349–1358

    Article  Google Scholar 

  16. Akif A, Calgan H, Koyuncu I, Pehlivan I, Istanbullu A (2016) Chaos-based engineering applications with a 3D chaotic system without equilibrium points. Nonlinear Dyn 84:481–495

    Article  MathSciNet  Google Scholar 

  17. Akif A, Shafqat H, Ihsan P (2016) A new three-dimensional chaotic system, its dynamical analysis and electronic circuit applications. Optik Int J Light Electron Opt 127:7062–7071

    Article  Google Scholar 

  18. Vaidyanathan S, Sampath S (2012) Anti-synchronization of four-wing chaotic systems via sliding mode control. Int J Autom Comput 9:274–279

    Article  Google Scholar 

  19. Vaidyanathan S, Azar AT (2014) Analysis, control and synchronization of a nine-term 3-D novel chaotic system. Chaos Model Control Syst Des 581:19–38

    MathSciNet  Google Scholar 

  20. Akif A, Li C, Pehlivan I (2017) Amplitude control analysis of a four-wing chaotic attractor, its electronic circuit designs and microcontroller-based random number generator. J Circuit Syst Comput 26:1750190

    Article  Google Scholar 

  21. Akif A, Pehlivan I (2016) A new three-dimensional chaotic system without equilibrium points, its dynamical analyses and electronic circuit application. Tehnički vjesnik. https://doi.org/10.17559/TV-20141212125942

    Google Scholar 

  22. Vaidyanathan S, Akgul A, Kaçar S, Çavusoglu U (2018) A new 4-D chaotic hyperjerk system, its synchronization, circuit design and applications in RNG, image encryption and chaos-based steganography. Eur Phys J Plus 133:46

    Article  Google Scholar 

  23. Klouverakis KE, Sprott JC (2006) Chaotic hyperjerk systems. Chaos Solitons Fractals 28:739–746

    Article  MathSciNet  MATH  Google Scholar 

  24. Linz SJ (2008) On hyperjerk systems. Chaos Solitons Fractals 37:741–747

    Article  MathSciNet  MATH  Google Scholar 

  25. Elhadj Z, Sprott JC (2008) Transformation of 4-D dynamical systems to hyperjerk form. Palest J Math 2013:38–45

    MathSciNet  MATH  Google Scholar 

  26. Munmuangsaen B, Srisuchinwong B (2011) Elemetary chaotic snap flows. Chaos Solitons Fractals 44:995–1003

    Article  Google Scholar 

  27. Fatma YD, Sprott JC (2016) Simple chaotic hyperjerk system. Int J Bifurc Chaos 26:1650189

    Article  MathSciNet  Google Scholar 

  28. Vaidyanathan S, Volos C, Pham V-T, Madhavan K (2015) Analysis, adaptive control and synchronization of a novel 4-D hyperchaotic hyperjerk system and its SPICE implementation. Arch Control Sci 25:135–158

    Article  MathSciNet  Google Scholar 

  29. Wang X, Vaidyanathan S, Volos C, Pham V-T, Kapitaniak T (2017) Dynamics, circuit realization, control and synchronization of a hyperchaotic hyperjerk system with coexisting attractors. Nonlinear Dyn. https://doi.org/10.1007/s11071-017-3542-x

    MathSciNet  MATH  Google Scholar 

  30. Peter D, Sundarapandian V, Viet-Thanh P, Volos CK, Nistazakis E, Tombras G (2017) Hyperchaotic atractor in a novel hyperjerk system with two nonlinearities. Circuits Syst Signal Process 37:613

    Google Scholar 

  31. Pham VT, Vaidyanathan S, Volos CK, Jafari S, Wang X (2016) A chaotic hyperjerk system based on memristive device. In: Vaidyanathan S, Volos C (eds) Advances and applications in chaotic systems. Studies in computational intelligence, vol 636. Springer, Berlin

    Google Scholar 

  32. Bao B, Zou X, Liu Z, Hu F (2013) Generalized memory element and chaotic memory system. Int J Bifurc Chaos 23:1350135–1350412

    Article  MathSciNet  MATH  Google Scholar 

  33. Bao B, Hu F, Chen M, Xu Q (2015) Self-excited and hidden attractors found simultaneously in a modified Chua’s circuit. Int J Bifurc Chaos 5:1550075

    Article  MATH  Google Scholar 

  34. Buscarino A, Fortuna L, Frasca M, Gambuzza LV (2012) A chaotic circuit based on Hewlett–Packard memristor. Chaos 22:023136

    Article  MathSciNet  MATH  Google Scholar 

  35. Li C, Sprott JC (2014) Coexisting hidden attractors in a 4-D simplified Lorenz system. Int J Bifurc Chaos 24:1450034

    Article  MathSciNet  MATH  Google Scholar 

  36. Qiang L, Tsafack N, Kengne J, Xiao-Wen Z (2018) Coexisting attractors and circuit implementation of a new 4D chaotic system with two equilibria. Chaos Solitons Fractals 107:92–102

    Article  MathSciNet  MATH  Google Scholar 

  37. Vaithianathan V, Veijun J (1999) Coexistence of four different attractors in a fundamental power system model. IEEE Trans Circuit Syst I Fundam Theory Appl 6:405–409

    Google Scholar 

  38. Xu Q, Lin Y, Bao B, Chen M (2016) Multiple attractors in a non-ideal active voltage-controlled memristor based Chua’s circuit. Chaos Solitons Fractals 83:186–200

    Article  MathSciNet  MATH  Google Scholar 

  39. 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 23:133000239-1–69

    MathSciNet  MATH  Google Scholar 

  40. 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 224:1421–1458

    Article  Google Scholar 

  41. Jafari S, Sprott JC, Nazarimehr F (2015) Recent new examples of hidden attractors. Eur Phys J Spec Top 224:1469–1476

    Article  Google Scholar 

  42. Jafari S, Pham VT, Kapitaniak T (2016) Multiscroll chaotic sea obtained from a simple 3D system without equilibrium. Int J Bifurc Chaos 26:1650031

    Article  MathSciNet  MATH  Google Scholar 

  43. Jafari S, Sprott JC, Molaie M (2016) A simple chaotic flow with a plane of equilibria. Int J Bifurc Chaos 26:1650098

    Article  MathSciNet  MATH  Google Scholar 

  44. Pham VT, Volos C, Kingni ST, Jafari S, Kapitaniak T (2016) Coexistence of hidden chaotic attractors in a novel no-equilibrium system. Nonlinear Dyn 87:2001

    Article  Google Scholar 

  45. Li C, Wang X, Chen G (2017) Diagnosing multistability by offset boosting. Nonlinear Dyn 90:1335

    Article  MathSciNet  Google Scholar 

  46. Li C, Sprott JC, Akgul A, Lu Herbert HC, Zhao Y (2017) A new chaotic oscillator with free control. Chaos 27:083101

    Article  MathSciNet  MATH  Google Scholar 

  47. Li C, Sprott JC, Mei Y (2017) An infinite 2-D lattice of strange attractors. Nonlinear Dyn 89:2629

    Article  MathSciNet  Google Scholar 

  48. Li C, Sprott JC (2018) An infinite 3-D quasiperiodic lattice of chaotic attractors. Phys Lett A 382:581–587

    Article  MathSciNet  MATH  Google Scholar 

  49. Dawson SP, Grebogi C, Yorke JA, Kan I, Koçak H (1992) Antimonotonicity: inevitable reversals of period-doubling cascades. Phys Lett A 162:249–254

    Article  MathSciNet  Google Scholar 

  50. Ren S, Panahi S, Rajagopal K, Akgul A, Pham V-T, Jafari S (2018) A new chaotic flow with hidden attractor: the first hyperjerk system with no equilibrium. Z Naturforsch. https://doi.org/10.1515/zna-2017-0409

    Google Scholar 

  51. Kocarev L, Halle K, Eckert K, Chua L (1993) Experimental observation of antimonotonicity in Chua’s circuit. Int J Bifurc Chaos 3:1051–1055

    Article  MATH  Google Scholar 

  52. Volos C, Akgul A, Pham VT, Stouboulos I, Kyprianidis I (2017) A simple chaotic circuit with a hyperbolic sine function and its use in a sound encryption scheme. Nonlinear Dyn 84:481–495

    Google Scholar 

  53. Bier M, Bountis TC (1984) Remerging Feigenbaum trees in dynamical systems. Phys Lett A 104:239–244

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Unité de Recherche d’Automatique et Informatique Appliquée (LAIA), Department of Electrical Engineering, IUT-FV Bandjoun, University of Dschang, Dschang, Cameroon

    Gervais Dolvis Leutcho & Jacques Kengne

  2. Unité de Recherche de Matière Condensée, d’Electronique et de Traitement du Signal (URMACETS), Department of Physics, University of Dschang, P.O. Box 67, Dschang, Cameroon

    Gervais Dolvis Leutcho & Romanic Kengne

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  1. Gervais Dolvis Leutcho
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  2. Jacques Kengne
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Correspondence to Gervais Dolvis Leutcho.

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Leutcho, G.D., Kengne, J. & Kengne, R. Remerging Feigenbaum trees, and multiple coexisting bifurcations in a novel hybrid diode-based hyperjerk circuit with offset boosting. Int. J. Dynam. Control 7, 61–82 (2019). https://doi.org/10.1007/s40435-018-0438-7

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