Skip to main content
SpringerLink
Log in

Dynamical behavior and reduced-order combination synchronization of a novel chaotic system

  • Published:
International Journal of Dynamics and Control Aims and scope Submit manuscript

Abstract

This manuscript investigates a novel 3D autonomous chaotic system which generates two strange attractors. The Lyapunov exponent, bifurcation diagram, Poincaré section, Kaplan–Yorke dimension, equilibria and phase portraits are given to justify the chaotic nature of the system. The novel system displays fixed orbit, periodic orbit, chaotic orbit as the parameter value varies. The reduced order combination synchronization is also performed by considering three identical 3D novel chaotic systems in two parts (a) choosing two third order master systems and one second order slave system which is the projection in the 2D plane. (b) choosing one third order master system and two second order slave systems which are the projection in the 2D plane. Numerical simulations justify the validity of the theoretical results discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

References

  1. Chen G, Dong X (1993) From chaos to order—perspectives and methodologies in controlling chaotic nonlinear dynamical systems. Int J Bifurc Chaos 3(06):1363–1409

    Article  MATH  Google Scholar 

  2. Epstein IR, Pojman JA (1998) An introduction to nonlinear chemical dynamics. Oxford University Press, Oxford

    Google Scholar 

  3. Wu X, Wang H (2010) A new chaotic system with fractional order and its projective synchronization. Nonlinear Dyn 61(3):407–417

    Article  MathSciNet  MATH  Google Scholar 

  4. Dadras S, Momeni HR, Qi G, Wang Z-L (2012) Four-wing hyperchaotic attractor generated from a new 4D system with one equilibrium and its fractional-order form. Nonlinear Dyn 67(2):1161–1173

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen G, Ueta T (1999) Yet another chaotic attractor. Int J Bifurc Chaos 9(07):1465–1466

    Article  MathSciNet  MATH  Google Scholar 

  6. Lü J, Chen G (2002) A new chaotic attractor coined. Int J Bifurc Chaos 12(03):659–661

    Article  MathSciNet  MATH  Google Scholar 

  7. Lü J, Chen G, Cheng D, Celikovsky S (2002) Bridge the gap between the Lorenz system and the Chen system. Int J Bifurc Chaos 12(12):2917–2926

    Article  MathSciNet  MATH  Google Scholar 

  8. Yang T (2004) A survey of chaotic secure communication systems. Int J Comput Cognit 2(2):81–130

    Google Scholar 

  9. Rössler OE (1976) An equation for continuous chaos. Phys Lett A 57(5):397–398

    Article  MATH  Google Scholar 

  10. Lorenz E (1963) Deterministic nonperiodic flow. J Atmos Sci 20:130–141

    Article  MATH  Google Scholar 

  11. Azar AT, Ouannas A, Singh S (2017) Control of new type of fractional chaos synchronization. In: International conference on advanced intelligent systems and informatics. Springer, Cham

  12. Zak M (1991) Terminal chaos for information processing in neurodynamics. Biol Cybern 64(4):343–351

    Article  MATH  Google Scholar 

  13. Epstein IR, Showalter K (1996) Nonlinear chemical dynamics: oscillations, patterns, and chaos. J Phys Chem 100(31):13132–13147

    Article  Google Scholar 

  14. Aihara K, Takabe T, Toyoda M (1990) Chaotic neural networks. Phys Lett A 144(6):333–340

    Article  MathSciNet  Google Scholar 

  15. Wu X, Li S (2012) Dynamics analysis and hybrid function projective synchronization of a new chaotic system. Nonlinear Dyn 69(4):1979–1994

    Article  MathSciNet  MATH  Google Scholar 

  16. Kingni ST, Pham V-T, Jafari S, Kol GR, Woafo P (2016) Three-dimensional chaotic autonomous system with a circular equilibrium: analysis, circuit implementation and its fractional-order form. Circuits Syst Signal Process 35(6):1933–1948

    Article  MathSciNet  MATH  Google Scholar 

  17. Tacha OI, Volos CK, Kyprianidis IM, Stouboulos IN, Vaidyanathan S, Pham V-T (2016) Analysis, adaptive control and circuit simulation of a novel nonlinear finance system. Appl Math Comput 276:200–217

    MathSciNet  MATH  Google Scholar 

  18. Tong Y-N (2015) Dynamics of a three-dimensional chaotic system. Opt-Int J Light Electron Opt 126(24):5563–5565

    Article  Google Scholar 

  19. Zhang M, Han Q (2016) Dynamic analysis of an autonomous chaotic system with cubic nonlinearity. Opt-Int J Light Electron Opt 127(10):4315–4319

    Article  Google Scholar 

  20. Çiçek S, Ferikoğlu A, Pehlivan I (2016) A new 3D chaotic system: dynamical analysis, electronic circuit design, active control synchronization and chaotic masking communication application. Opt-Int J Light Electron Opt 127(8):4024–4030

    Article  Google Scholar 

  21. Deng W, Fang J, Zhen-jun W (2015) A dual-parameter hyperchaotic system with constant Lyapunov exponent and its circuit emulation. Opt-Int J Light Electron Opt 126(24):5468–5472

    Article  Google Scholar 

  22. Chen Y, Yang Q (2015) A new Lorenz-type hyperchaotic system with a curve of equilibria. Math Comput Simul 112:40–55

    Article  MathSciNet  Google Scholar 

  23. Wang J, Zhang Q, Chen Z, Li H (2014) Local bifurcation analysis and ultimate bound of a novel 4D hyper-chaotic system. Nonlinear Dyn 78(4):2517–2531

    Article  MathSciNet  MATH  Google Scholar 

  24. Leipnik RB, Newton TA (1981) Double strange attractors in rigid body motion with linear feedback control. Phys Lett A 86(2):63–67

    Article  MathSciNet  Google Scholar 

  25. Pecora LM, Carroll TL (1990) Synchronization of chaotic systems. Phys Rev Lett 64:821–824

    Article  MathSciNet  MATH  Google Scholar 

  26. Ott E, Grebogi C, Yorke JA (1990) Controlling chaos. Phys Rev Lett 64:1196–1199

    Article  MathSciNet  MATH  Google Scholar 

  27. Ma M, Zhou J, Cai J (2012) Practical synchronization of nonautonomous systems with uncertain parameter mismatch via a single state feedback control. Int J Mod Phys C 23:12500731

    Article  Google Scholar 

  28. Zhu H (2010) Anti-synchronization of two different chaotic systems via optimal control with fully unknown parameters. J Inf Comput Sci 5:011–018

    Google Scholar 

  29. Li Y, Tong S, Li T (2013) Adaptive fuzzy feedback control for a single-link flexible robot manipulator driven DC via backstepping. Nonlinear Anal 14:483–494

    Article  MathSciNet  MATH  Google Scholar 

  30. Kareem SO, Ojo KS, Njah AN (2012) Function projective synchronization of identical and non-identical modified finance and Shimizu–Morioka systems. Pramana 79:71–79

    Article  Google Scholar 

  31. Khan A, Shikha S (2016) Mixed tracking and projective synchronization of 6D hyperchaotic system using active control. Int J Nonlinear Sci 22(1):44–53

    MathSciNet  Google Scholar 

  32. Yang CC (2012) Robust synchronization and antisynchronization of identical 6 oscillators via adaptive sliding mode control. J Sound Vib 331:501–509

    Article  Google Scholar 

  33. Li S-Y, Yang C-H, Lin C-T, Ko L-W, Chin T-T (2012) Adaptive synchronization of chaotic systems with unknown parameters via new backstepping strategy. Nonlinear Dyn 70:2129–2142

    Article  MathSciNet  Google Scholar 

  34. Hong-Yan Z, Le-Quan M, Geng Z, Guan-Rong C (2013) Generalized chaos synchronization of bidirectional arrays of discrete systems. Chin Phys Lett 30(4):040502

    Article  Google Scholar 

  35. Chen J, Jiao L, Wu J, Wang X (2010) Projective synchronization with different scale factors in driven-response complex network and its application to image encryption. Nonlinear Anal 11:3045–3058

    Article  MATH  Google Scholar 

  36. Li GH (2007) Modified projective synchronization of chaotic system. Chaos Solitons Fractals 32(5):1786–1790

    Article  MathSciNet  MATH  Google Scholar 

  37. Runiz L, Zhengmin W (2009) Adaptive function projective synchronization of unified chaotic systems with uncertain parameters. Chaos Solitons Fractals 42(2):1266–1272

    Article  MATH  Google Scholar 

  38. Kareem SO, Ojo KS, Njah AN (2012) Function projective synchronization of identical and non-identical modified finance and Shimizu-Morioka systems. Pramana 79:71–79

    Article  Google Scholar 

  39. Sudheer KS, Sabir M (2009) Hybrid synchronization of hyperchaotic Lu system. Pramana 73(4):781–786

    Article  MATH  Google Scholar 

  40. Khan A, Shikha, (2017) Hybrid function projective synchronization of chaotic systems via adaptive control . Int J Dyn Control 5(4):1114–1121

  41. Ogunjo S (2013) Increased and reduced order synchronization of 2D and 3D dynamical systems. Int J Nonlinear Sci 16(2):105–112

    MathSciNet  Google Scholar 

  42. Khan A (2017) Increased and reduced order synchronisations between 5D and 6D hyperchaotic systems. Indian J Ind Appl Math 8(1):118–131

    Article  Google Scholar 

  43. Runzi L, Yinglan W, Shucheng D (2011) Combination synchronization of three classic chaotic systems using active backstepping design. Chaos: Interdiscip J Nonlinear Sci 21(4):043114

    Article  MATH  Google Scholar 

  44. Khan A, Shikha (2017) Combination synchronization of Genesio time delay chaotic system via robust adaptive sliding mode control. Int J Dyn Control. https://doi.org/10.1007/s40435-017-0339-1

  45. Khan A (2017) Combination synchronization of time-delay chaotic system via robust adaptive sliding mode control. Pramana 88(6):91

    Article  Google Scholar 

  46. Ojo KS, Njah AN, Olusola OI, Omeike MO (2014) Generalized reduced-order hybrid combination synchronization of three Josephson junctions via backstepping technique. Nonlinear Dyn 77(3):583–595

    Article  MathSciNet  MATH  Google Scholar 

  47. Chen G, Hill DJ, Yu X (eds) (2003) Bifurcation control: theory and applications, vol 293. Springer, Berlin

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Jamia Millia Islamia, New Delhi, 110 025, India

    Khan Ayub &  Shikha

Authors
  1. Khan Ayub
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shikha
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shikha.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ayub, K., Shikha Dynamical behavior and reduced-order combination synchronization of a novel chaotic system. Int. J. Dynam. Control 6, 1160–1174 (2018). https://doi.org/10.1007/s40435-017-0382-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40435-017-0382-y

Keywords

Search

Navigation