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Symmetry-breaking, amplitude control and constant Lyapunov exponent based on single parameter snap flows

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Abstract

In this paper, we introduce a novel snap system with a unique parameterized piecewise quadratic nonlinearity in the form \(\psi _{n} \left( x \right) =x-n\left| x \right| -\mathrm{d}x\left| x \right| \)where n controls the symmetry of the system and serves as total amplitude control of the signals. The model is described by a continuous time 4D autonomous system (ODE) with smooth conditional nonlinearity. We study the chaos mechanism with respect to system parameters both in the symmetric and asymmetric modes of oscillations by exploiting bifurcation diagrams, basin of attractions and phase portraits as main tools. In particular, for\(n=0\), the system displays a perfect symmetry and develops rich dynamics including period doubling sequences, merging crisis, hysteresis, and coexisting multiple symmetric attractors. For\(n\ne 0\), the system is non-symmetric and the space magnetization induced more complex and striking effects including asymmetric double scroll strange attractors, parallel branches, and asymmetric basin boundary leads to many coexisting asymmetric stable states and so on. Apart from all these complex and rich phenomena, many others including offset-boosting with total amplitude control, and antimonotonicity are also presented. Finally, Pspice based simulations of the proposed system are included.

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References

  1. C. Li, J.C. Sprott, Variable-boostable chaotic flows. Optik 127, 10389–10398 (2016)

    ADS  Google Scholar 

  2. C. Li, J.C. Sprott, H. Xing, Constructing chaotic systems with conditional symmetry. Nonlinear Dyn. 87, 1351–1358 (2017)

    MATH  Google Scholar 

  3. C. Li, J.C. Sprott, H. Xing, Hypogenetic chaotic jerk flows. Phys. Lett. A 380, 1172–1177 (2016)

    ADS  MathSciNet  Google Scholar 

  4. C. Li, J.C. Sprott, Z. Yuan, H. Li, Constructing chaotic systems with total amplitude control. Int. J. Bifurc. Chaos 25, 1530025 (2015)

    MathSciNet  MATH  Google Scholar 

  5. C. Li, J.C. Sprott, Amplitude control approach for chaotic signals. Nonlinear Dyn. 73, 1335–1341 (2013)

    MathSciNet  MATH  Google Scholar 

  6. C. Li, W.J.-C. Thio, J.C. Sprott, R.X. Zhang, T.A. Lu, Linear synchronization and circuit implementation of chaotic system with complete amplitude control. Chin. Phys. B 26, 120501 (2017)

    ADS  Google Scholar 

  7. G.D. Leutcho, J. Kengne, A unique chaotic snap system with a smoothly adjustable symmetry and nonlinearity: chaos, offset-boosting, antimonotonicity, and coexisting multiple attractors. Chaos Solitons Fractals 113, 275–293 (2018)

    ADS  MathSciNet  Google Scholar 

  8. A. Nguomkam Negou, J. Kengne, Dynamic analysis of a unique jerk system with a smoothly adjustable symmetry and nonlinearity: reversals of period doubling, offset boosting and coexisting bifurcations. Int. J. Electron. Commun. (AEÜ) 90, 1–19 (2018)

    Google Scholar 

  9. C. Li, J.C. Sprott, T. Kapitaniak, T. Lu, Infinite lattice of hyperchaotic strange attractors. Chaos Solitons Fractals 109, 76–82 (2018)

    ADS  MathSciNet  MATH  Google Scholar 

  10. C. Masoller, Coexistence of attractors in a laser diode with optical feedback from a large external cavity. Phys. Rev. A 50, 2569–2578 (1994)

    ADS  Google Scholar 

  11. J.M. Cushing, S.M. Henson, Blackburn, multiple mixed attractors in a competition model. J. Biol. Dyn. 1, 347–362 (2007)

    MathSciNet  MATH  Google Scholar 

  12. A. Massoudi, M.G. Mahjani, M. Jafarian, Multiple attractors in Koper–Gaspard model of electrochemical. J. Electroanal. Chem. 647, 74–86 (2010)

    Google Scholar 

  13. J. Kengne, A. NguomkamNegou, D. Tchiotsop, Antimonotonicity, chaos and multiple attractors in a novel autonomous memristor-based jerk circuit. Nonlinear Dyn. (2017). https://doi.org/10.1007/s11071-017-3397-1

  14. J. Kengne, Z.T. Njitacke, A. NguomkamNegou, M. FouodjiTsotsop, H.B. Fotsin, Coexistence of multiple attractors and crisis route to chaos in a novel chaotic jerk circuit. Int. J. Bifurc. Chaos 25, 1550052 (2015)

    Google Scholar 

  15. S. Jafari, J.C. Sprott, F. Nazarimehr, Recent new examples of hidden attractors. Eur. Phys. J. Spec. Topics 224, 1469–1476 (2015)

    ADS  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  18. V.T. Pham, C. Volos, S.T. Kingni, S. Jafari, T. Kapitaniak, Coexistence of hidden chaotic attractors in a novel no-equilibrium system. Nonlinear Dyn. (2016) (3170-x)

  19. W. Zhouchao, Y. Pei, W. Zhang, Y. Minghui, Study of hidden attractors, multiple limit cycles from Hopf bifurcation and boundedness of motion in the generalized hyperchaotic Rabinovich system. Nonlinear Dyn. 82, 131–141 (2015)

    MathSciNet  MATH  Google Scholar 

  20. W. Zhouchao, L. Yingying, B. Sang, L. Yongjian, Z. Wei, Complex dynamical behaviors in a 3D simple chaotic flow with 3D stable or 3D unstable manifolds of a single equilibrium. Int. J. Bifurc. Chaos 29(7), 1950095 (2019)

    MathSciNet  MATH  Google Scholar 

  21. Q. Lai, S. Chen, Coexisting attractors generated from a new 4D smooth chaotic system. Int. J. Control Autom. 14, 1124–1131 (2016)

    Google Scholar 

  22. B. Bao, Q. Li, N. Wang, Q. Xu, Multistability in Chua’s circuit with two stable node-foci. Chaos 26, 043111 (2016)

    ADS  MathSciNet  Google Scholar 

  23. V.T. Pham, A. Akgul, C. Volos, S. Jafari, T. Kapitaniake, Dynamics and circuit realization of a no-equilibrium chaotic system with a boostable variable. Int. Jo. Electron. Commun. (2017). https://doi.org/10.1016/j.aeue.2017.05.034

  24. A. Akif, I. Moroz, I. Pehlivan, S. Vaidyanathan, A new four-scroll chaotic attractor and its engineering applications. Optik 127(13), 5491–5499 (2016)

    ADS  Google Scholar 

  25. G.D. Leutcho, J. Kengne, L. Kamdjeu Kengne, 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, 67–87 (2018)

    ADS  MathSciNet  MATH  Google Scholar 

  26. G.D. Leutcho, J. Kengne, R. Kengne, 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 (2018)

    Google Scholar 

  27. G.D. Leutcho, J. Kengne, T. Fonzin Fozin, K. Srinivasan, Z. Njitacke Tabekoueng, S. Jafari, M. Borda, Multistability control of space magnetization in hyperjerk oscillator: a case study. J. Comput. Nonlinear Dyn. 15(5) (2020)

  28. G.D. Leutcho, S. Jafari, I.I. Hamarash, J. Kengne, Z.T. Njitacke, I. Hussain, A new megastable nonlinear oscillator with infinite attractors. Chaos Solitons Fractals 134, 109703 (2020)

    MathSciNet  Google Scholar 

  29. G.D. Leutcho, A.J.M. Khalaf, Z. Njitacke Tabekoueng, T.F. Fozin, J. Kengne, S. Jafari, I. Hussain, A new oscillator with mega-stability and its Hamilton energy: Infinite coexisting hidden and self-excited attractors. Chaos Interdiscip. J. Nonlinear Sci. 30(3), 033112 (2020)

    MathSciNet  MATH  Google Scholar 

  30. Z. Wei, Y. Li, B. Sang, Y. Liu, W. Zhang, Complex dynamical behaviors in a 3D simple chaotic flow with 3D stable or 3D unstable manifolds of a single equilibrium. Int. J. Bifurc. Chaos 29(07), 1950095 (2019)

    MathSciNet  MATH  Google Scholar 

  31. C. Kahllert, The effects of symmetry breaking in Chua’s circuit and related piecewise-linear dynamical system. Int. J. Bifurc. Chaos 3, 963–979 (1993)

    MathSciNet  Google Scholar 

  32. H. Cao, Z. Jing, Chaotic dynamics of Josephson equation driven by constant and ac forcings. Chaos Solitons Fractals 12, 1887–1895 (2001)

    ADS  MathSciNet  MATH  Google Scholar 

  33. R. Rynio, A. Okninski, Symmetry breaking and fractal dependence on initial conditions in dynamical systems: ordinary differential equations of thermal convection. Chaos Solitons Fractals 9, 1723–1732 (1198)

  34. K.E. Chlouverakis, J.C. Sprott, Chaotic hyperjerk systems. Chaos Solitons Fractals 28, 739–746 (2006)

    ADS  MathSciNet  MATH  Google Scholar 

  35. S.J. Linz, On hyperjerk systems. Chaos Solitons Fractals 37, 741–747 (2008)

    ADS  MathSciNet  MATH  Google Scholar 

  36. B. Munmuangsaen, Srisuchinwong, Elementary chaotic snap flows. Chaos Solitons Fractals 44, 995–1003 (2011)

    ADS  Google Scholar 

  37. S. Vaidyanathan, A. Akgul, S. Kaçar, U. Çavusoglu, 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 (2018)

    Google Scholar 

  38. V.T. Pham, S. Vaidyanathan, C. Volos, S. Jafari, X. Wang, A chaotic hyperjerk system based on memristive device. In Advances and Applications in Chaotic Systems. Studies in Computational Intelligence, vol. 636, ed. by S. Vaidyanathan, C. Volos (Springer, 2016)

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

    Article  Google Scholar 

  40. J.C. Sprott, Elegant Chaos: Algebraically Simple Flow (World Scientific Publishing, Singapore, 2010)

    MATH  Google Scholar 

  41. S.H. Strogatz, Nonlinear Dynamics and Chaos (Addison-Wesley, Reading, 1994)

    Google Scholar 

  42. C. Li, J. Wang, W. Hu, Absolute term introduced to rebuild the chaotic attractor with constant Lyapunov exponent spectrum. Nonlinear Dyn. 68, 575–587 (2012)

    MathSciNet  MATH  Google Scholar 

  43. S. Zhang, Y. Zeng, Z. Li, M. Wang, X. Zhang, D. Chang, A novel simple no-equilibrium chaotic system with complex hidden dynamics. Int. J. Dyn. Control 6, 1465–1476 (2018)

    MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

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

    ADS  MathSciNet  Google Scholar 

  46. H. Bao, N. Wang, B. Bao, M. Chen, P. Jin, G. Wang, Initial condition-dependent dynamics and transient period in memristor-based hypogenetic jerk system with four line equilibria. Commun. Nonlinear Sci. Numer. Simul. 57, 264–275 (2018). https://doi.org/10.1016/j.cnsns.2017.10.001

    Article  ADS  MathSciNet  MATH  Google Scholar 

  47. A. Bayani, K. Rajagopal, A.J.M. Khalaf, S. Jafari, G.D. Leutcho, J. Kengne, Dynamical analysis of a new multistable chaotic system with hidden attractor: antimonotonicity, coexisting multiple attractors, and offset boosting. Phys. Lett. A 383(13), 1450–1456 (2019). https://doi.org/10.1016/j.physleta.2019.02.005

    Article  ADS  MathSciNet  Google Scholar 

  48. J.C. Sprott, Do we need more chaos examples? Chaos Theory Appl. 2, 1–3 (2020)

    Google Scholar 

  49. H. Öztürk, A novel chaos application to observe performance of asynchronous machine under chaotic load. Chaos Theory Appl. 2(2), 90–97 (2020)

  50. M.E. Cimen, Z.B. Garip, M.A. Pala, A.F. Boz, A. Akgul, Modelling of a chaotic system motion in video with artiıficial neural networks. Chaos Theory Appl. 1(1), 38–50 (2019)

    Google Scholar 

  51. Y. Adiyaman, S. Emiroglu, M.K. Uçar, M. Yildiz, Dynamical analysis, electronic circuit design and control application of a different chaotic system. Chaos Theory Appl. 2(1), 8–14 (2019)

    Google Scholar 

  52. Z. Njitacke, T. Fozin, L.K. Kengne, G. Leutcho, E.M. Kengne, J. Kengne, Multistability and its Annihilation in the Chua’s Oscillator with Piecewise-Linear Nonlinearity. Chaos Theory Appl. 2, 77–89 (2020)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Research Unit of Laboratory of Condensed Matter, Electronics and Signal Processing (UR-MACETS) Department of Physics, Faculty of Sciences, University of Dschang, P.O. Box 67, Dschang, Cameroon

    Gervais Dolvis Leutcho, Romanic Kengne & Léandre Kamdjeu Kengne

  2. Department of Physics, University of Mons, Place du Parc 20, 7000, Mons, Belgium

    Gervais Dolvis Leutcho

  3. School of Physics and Electronics, Central South University Changsha, Hunan, 410083, China

    Huihai Wang

  4. Research Unit of Laboratory of Automation and Applied Computer (LAIA), Electrical Engineering Department of IUT-FV, University of Dschang, P.O. Box 134, Bandjoun, Cameroon

    Léandre Kamdjeu Kengne & Zeric Tabekoueng Njitacke

  5. Department of Electrical and Electronic Engineering, College of Technology (COT), University of Buea, P.O. Box 63, Buea, Cameroon

    Zeric Tabekoueng Njitacke

  6. Department of Electrical and Electronic Engineering, Faculty of Engineering and Technology (FET), University of Buea, P.O. Box 63, Buea, Cameroon

    Theophile Fonzin Fozin

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  1. Gervais Dolvis Leutcho
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Correspondence to Gervais Dolvis Leutcho.

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Leutcho, G.D., Wang, H., Kengne, R. et al. Symmetry-breaking, amplitude control and constant Lyapunov exponent based on single parameter snap flows. Eur. Phys. J. Spec. Top. 230, 1887–1903 (2021). https://doi.org/10.1140/epjs/s11734-021-00136-7

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