Abstract
A novel autonomous chaotic Jerk system with a nonlinearity in the form \( \phi_{k} \left( x \right) = 0.5\left( {\exp \left( {kx} \right) - \exp \left( { - x} \right)} \right) \) which for \( k = 1 \) reduces to the hyperbolic sine is proposed. Two general purpose semiconductor diodes connected in antiparallel are utilized to synthesize the tuneable nonlinearity. The new system presents three rest points among which two unstable ones. Thus, the dynamics is organized around the zero equilibrium point. Correspondingly, the model develops only mono-scroll strange attractors. The numerical analysis of the system reveals some interesting phenomena such as period doubling cascades to chaos, period-doubling reversals, periodic windows, hysteresis, and coexisting bifurcations as well. The presence of parallel bifurcation branches justifies the occurrence of multiple coexisting attractors in some ranges of parameters. Several basins of attraction with extremely complex structures are provided to illustrate the magnetization of the state space due to the presence of numerous competing attractors. The practical implication of this phenomenon is that it might be very difficult to predict the dynamical state of the system. Multistability in the symmetry boundary is studied by using \( k \) as control parameter. A very good agreement is observed between practical experiments and the theoretical predictions.
Similar content being viewed by others
References
Upadhyay RK (2003) Multiple attractors and crisis route to chaos in a model of food-chain. Chaos, Solitons Fractals 16:737–747
Cushing JM, Henson SM, Blackburn CC (2007) Multiple mixed attractors in a competition model. J Biol Dyn 1:347–362
Masoller C (1994) Coexistence of attractors in a laser diode with optical feedback from a large external cavity. Phys Rev 50:2569–2578
Njitacke ZT, Kengne J, Fotsin HB, Nguomkam NA, Tchiotsop D (2016) Coexistence of multiple attractors and crisis route to chaos in a novel memristive diode bidge-based Jerk circuit. Chaos, Solitons Fractals 91:180–197
Pham VT, Vaidyanathan S, Volos CK, Jafari S, Kuznetsov NV, Hoang TM (2016) A novel memristive timedelay chaotic system without equilibrium points. Eur Phys J Spec Top 225(1):127–136
Hens C, Dana SK, Feudel U (2015) Extreme multistability: attractors manipulation and robustness. Chaos 25:053112
Bao BC, Xu B, Bao H, Chen M (2016) Extreme multistability in a memristive circuit. Electron Lett 52(12):1008–1010
Bao B, Jiang T, Xu Q, Chen M, Wu H, Hu Y (2016) Coexisting infinitely many attractors in active bandpass filter-based memristive circuit. Nonlinear Dyn 86(3):1711–1723
Sprott JC (1997) Some simple Jerk functions. Am. J. Phys A. 65:537–543
Sprott JC (1997) Simplest dissipative chaotic flow. Phys Lett A 228:271–274
Sprott JC (2000) Simple chaotic systems and circuits. Am J Phys 68:758–763
Sprott JC (2010) Elegant chaos: algebraically simple flow. World Scientific Publishing, Singapore
Sprott JC (2011) A new chaotic Jerk circuit. IEEE Trans Circuits Syst II Expr Br 58:240–243
Njitacke ZT, Kengne J, Nguomkam NA, Fouodji TM, Fotsin HB (2015) Coexistence of multiple attractors and crisis route to chaos in a novel chaotic Jerk circuit. Int J Bifurcat Chaos 25(4):1550052
Kengne J, Negou AN, Tchiotsop D (2017) Antimonotonicity chaos and multiple attractors in a novel autonomous memristor-based Jerk circuit. Nonlinear Dyn 88:2589–2608
Parlitz U, Lauterborn W (1985) Superstructure in the bifurcation set of the Duffing equation ẍ + dẋ+ x+ = f cos (ωt). Phys Lett A 107:351–355
Nguomkam Negou A, Kengne J, Tchiotsop D (2018) Periodicity, chaos and multiple coexisting attractors in a generalized Moore–Spiegel system. Chaos, Solitons Fractals 107:275–289
Kengne J, Njitacke ZT, Fotsin HB (2016) Dynamical analysis of a simple autonomous Jerk system with multiple attractors. Nonlinear Dyn 83:751–765
Kengne J, Nguomkam Negou A, Njitacke ZT (2017) Antimonotonicity, chaos and multiple attractors in a novel autonomous Jerk circuit. Int J Bifurcat Chaos 27(7):1750100
Wolf A, Swift JB, Swinney HL, Wastano JA (1985) Determining Lyapunov exponents from time series. Physica D 16:285–317
Strogatz SH (1994) Nonlinear dynamics and chaos. Addison-Wesley, Reading
Sprott JC (2011) A proposed standard for the publication of new chaotic systems. Int J Bifurcat Chaos 21(9):2391–2394
Kuznetsov AP, Kuznetsov SP, Mosekilde E, Stankevich NV (2015) Co-existing hidden attractors in a radio-physical oscillator. J Phys A: Math Theor 48:125101
Vaidyanathan S, Volos CK, Kyprianidis IM, Stouboulos IN, Pham VT (2015) Analysis, adaptive control and antisynchronization of a six-term novel Jerk chaotic system with two exponential nonlinearities and its circuit simulation. J Eng Sci Technol Rev 8:24–36
Kengne J, Njitacke ZT, Fotsin HB (2016) Dynamical analysis of a simple autonomous Jerk system with multiple attractors. Nonlinear Dyn 83:751
Vaidyanathan S, Azar AT (2016) Adaptive backstepping control and synchronization of a novel 3-D Jerk system with an exponential nonlinearity. In: Azar A, Vaidyanathan S (eds) Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing., vol 337. Springer, Cham
Upadhyay RK (2003) Multiple attractors and crisis route to chaos in a model of food-chain. Chaos Solit Fract 16:737–747
Vaithianathan V, Veijun J (1999) Coexistence of four different attractors in a fundamental power system model. IEEE Trans Circutes Syst I 46:405–409
Nguomkam NA, Kengne J (2018) 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
Pivka L, Wu CW, Huang A (1994) Chua’s oscillator: a com- pendium of chaotic phenomena. J Frankl Inst 331B(6):705–741
Kuznetsov AP, Kuznetsov SP, Mosekilde E, Stankevich NV (2015) Co-existing hidden attractors in a radio-physical oscillator. J Phys A: Math Theor 48:125101
Li C, Sprott JC (2013) Amplitude control approach for chaoticsignals. Nonlinear Dyn 73:1335–1341
Swathy PS, Thamilmaran K (2013) An experimental study on SC-CNN based canonical Chua’s circuit. Nonlinear Dyn 71:505–514
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Tagne, R.L.M., Kengne, J. & Negou, A.N. Multistability and chaotic dynamics of a simple Jerk system with a smoothly tuneable symmetry and nonlinearity. Int. J. Dynam. Control 7, 476–495 (2019). https://doi.org/10.1007/s40435-018-0458-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40435-018-0458-3