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Triple compound combination synchronization of eleven n-dimensional chaotic systems

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Abstract

In this paper, we have constructed a new three-dimensional dynamical system and proposed a novel technique of synchronization to synchronize five drive systems with six response systems together. Motivated by compound combination synchronization and triple compound synchronization, we have extended the idea to triple compound combination synchronization of eleven chaotic systems. Suitable controllers have been constructed to obtain the desired synchronization between drive and response systems. For designing the nonlinear controllers, we used Lyapunov’s stability theory. Numerical simulations are done by using MATLAB, and graphs are presented to show the effectiveness of the proposed method.

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References

  1. Azar AT, Vaidyanathan S (2015) Chaos modeling and control systems design, studies in computational intelligence, vol 581. Springer, New York

    MATH  Google Scholar 

  2. Azar AT, Vaidyanathan S (2015) Computational intelligence applications in modeling and control, vol 575. Studies in computational intelligence. Springer, New York

    Book  Google Scholar 

  3. Chua LO, Itoh M, Kocarev L, Eckert K (1993) Chaos synchronization in Chua’s circuit. J Circuits Syst Comput 3(1):93–108

    Article  MathSciNet  MATH  Google Scholar 

  4. Yau HT, Pu YC, Li SC (2012) Application of a Chaotic synchronization system to secure communication. Inf Technol Control 41:274–282

    Google Scholar 

  5. Yeh JP, Wu KL (2008) A simple method to synchronize chaotic systems and its application to secure communications. Math Comput Model 47:894–902

    Article  MathSciNet  MATH  Google Scholar 

  6. Alsafasfeh QH, Arfoa AA (2011) Image encryption based on the general approach for multiple chaotic systems. J Signal Inf Process 2:238–244

    Google Scholar 

  7. Potapov AB, Ali MK (2001) Nonlinear dynamics and chaos in information processing neural networks. Differ Equ Dyn Syst 9:259–319

    MathSciNet  MATH  Google Scholar 

  8. Upadhyay R, Rai V (2009) Complex dynamics and synchronization in two non-identical chaotic ecological systems. Chaos, Solutions Fractals 40(5):2233–2241

    Article  MathSciNet  MATH  Google Scholar 

  9. Laskin N (2000) Fractional market dynamics. Phys A Stat Mech Appl 287(3–4):482–492

    Article  MathSciNet  Google Scholar 

  10. Turcotte DL (1990) Implications of chaos, scale-invariance, and fractal statistics in geology. Glob Planet Chang 3(3):301–308

    Article  Google Scholar 

  11. Gerig A, Hubler A (2007) Chaos in a one-dimensional compressible flow. Phys Rev E 75(4):045202

    Article  Google Scholar 

  12. Lecar M, Franklin FA, Holman MJ, Murray NW (2001) Chaos in the solar system. Annu Rev Astron Astrophys 39(1):581–631

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Yassen MT (2005) Chaos Synchronization between two different chaotic systems using active control. Chaos Solutions Fractals 23:131–140

    Article  MATH  Google Scholar 

  15. Odibat ZM (2010) Adaptive feedback control and synchronization of non-identical chaotic fractional order systems. Nonlinear Dyn 60:479–487

    Article  MathSciNet  MATH  Google Scholar 

  16. Rakkiyappan R, Sivasamy R, Li X (2015) Synchronization of identical and nonidentical memristor-based chaotic systems via active backstepping control technique. Circuits Syst Signal Process 34:763–778

    Article  Google Scholar 

  17. Hosseinnia SH, Ghaderi R, Ranjbar A, Mahmoudian M, Momani S (2010) Sliding mode synchronization of an uncertain fractional order chaotic system. Comput Math Appl 59:1637–1643

    Article  MathSciNet  MATH  Google Scholar 

  18. Guo Q, Wan F (2017) Complete synchronization of the global coupled dynamical network induced by Poisson noises. PLoS ONE 12:e0188632. https://doi.org/10.1371/journal.pone.0188632

    Article  Google Scholar 

  19. Kim C, Rim S, Kye W, Ryu J, Park Y (2003) Anti-synchronization of chaotic oscillators. Phys Lett A 320(1):39–46. https://doi.org/10.1016/j.physleta.2003.10.051

    Article  MathSciNet  MATH  Google Scholar 

  20. Sudheer K, Sabir M (2009) Hybrid synchronization of hyperchaotic Lu system. Pramana 73(4):781–786. https://doi.org/10.1007/s12043-009-0145-1

    Article  MATH  Google Scholar 

  21. Khan A, Khattar D, Agrawal N (2018) Hybrid projective synchronization between the fractional order systems. J Math Comput Sci 8(2):253–269

    Google Scholar 

  22. Ho M, Hung YC, Chou C (2002) Phase and anti-phase synchronization of two chaotic systems by using active control. Phys Lett A 296(1):43–48. https://doi.org/10.1016/S0375-9601(02)00074-9

    Article  MATH  Google Scholar 

  23. Du H, Zeng Q, Wang C, Ling M (2010) Function projective synchronization in coupled chaotic systems. Nonlinear Anal Real World Appl 11(2):705–712. https://doi.org/10.1016/j.nonrwa.2009.01.016

    Article  MathSciNet  MATH  Google Scholar 

  24. Shahverdiev E, Sivaprakasam S, Shore K (2002) Lag synchronization in time-delayed systems. Phys Lett A 292(6):320–324. https://doi.org/10.1016/S0375-9601(01)00824-6

    Article  MATH  Google Scholar 

  25. Liu Y, Davids P (2000) Dual synchronization of chaos. Phys Rev E 61(3):R2176. https://doi.org/10.1103/PhysRevE.61.R2176

    Article  Google Scholar 

  26. Runzi L, Yinglan W, Shucheng D (2011) Combination synchronization of three classic chaotic systems using active backstepping design. Chaos Intersiscip J Nonlinear Sci 21(4):043114. https://doi.org/10.1063/1.3655366

    Article  MATH  Google Scholar 

  27. Zhou X, Xiong L, Cai X (2014) Combination-combination synchronization of four nonlinear complex chaotic systems. Abstr Appl Anal. https://doi.org/10.1155/2014/953265

    Article  MathSciNet  MATH  Google Scholar 

  28. Sun J, Jiang S, Cui G, Wang Y (2016) Dual combination synchronization of six chaotic systems. J Comput Nonlinear Dyn 11(3):034501. https://doi.org/10.1115/1.4031676

    Article  Google Scholar 

  29. Jahanzaib SJ, Trikha P, Chaudhary H, Haider SM (2020) Compound synchronization using disturbance observer based adaptive sliding mode control technique. J Math Comput Sci 10:1463–1480. https://doi.org/10.28919/jmcs/4637

    Article  Google Scholar 

  30. Ojo KS, Njah AN, Olusola OI (2015) Compound-combination synchronization of chaos in identical and different orders chaotic systems. Arch Control Sci 25(4):463–490

    Article  MathSciNet  MATH  Google Scholar 

  31. Zhang B, Deng F (2014) Double-compound synchronization of six memristor-based Lorenz systems. Nonlinear Dyn 77(4):1519–1530. https://doi.org/10.1007/s11071-014-1396-z

    Article  Google Scholar 

  32. Mahmoud G, Abed-Elhameed T, Farghaly A (2018) Double compound combination synchronization among eight n-dimensional chaotic systems. Chin Phys B 27(8):080502. https://doi.org/10.1088/1674-1056/27/8/080502

    Article  Google Scholar 

  33. Yadav V, Prasad G, Srivastava M, Das S (2019) Triple compound synchronization among eight chaotic systems with external disturbances via nonlinear approach. Differ Equ Dyn Syst. https://doi.org/10.1007/s12591-019-00477-9

    Article  MATH  Google Scholar 

  34. Wolf A, Swift JB, Swinney HL, Vastano JA (1985) Determining Lyapunov exponents from a time series. Phys D Nonlinear Phenom 16:285–317

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Kirori Mal College, University of Delhi, Delhi, India

    Dinesh Khattar & Neha Agrawal

  2. Department of Mathematics, Indian Institute of Technology Delhi, Delhi, India

    Govind Singh

Authors
  1. Dinesh Khattar
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  2. Neha Agrawal
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  3. Govind Singh
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Contributions

A new chaotic system was introduced. A novel scheme of synchronization was proposed and named as triple compound combination synchronization.

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Correspondence to Govind Singh.

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Khattar, D., Agrawal, N. & Singh, G. Triple compound combination synchronization of eleven n-dimensional chaotic systems. Int. J. Dynam. Control 11, 2499–2513 (2023). https://doi.org/10.1007/s40435-023-01140-6

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  • DOI: https://doi.org/10.1007/s40435-023-01140-6

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