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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Stability Analysis and Combination-Combination Synchronization of the Chaotic System

Journal of Applied Nonlinear Dynamics 14(4) (2025) 819--834 | DOI:10.5890/JAND.2025.12.005

Vijay K. Shukla, Mahesh C. Joshi

Department of Mathematics, D.S.B. Campus, Kumaun University, Nainital-263001, Uttarakhand, India

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Abstract

This work examines the analysis of Jacobi stability of the Sprott-C system by using the KCC-theory. The five KCC invariants are obtained to study the system's characteristics. We also calculate the spherical and ellipsoidal ultimate bound of the ecological model. Further, we analyze combination-combination synchronization among chaotic systems with delay and without delay terms. To see the effect of time-delay on combination-combination synchronization we are considering the time-delay ecological chaotic system. By designing suitable controllers, the synchronization among these systems is achieved. Finally, numerical results validate the feasibility of the designed control method.

References

  1. [1]  Kosambi, D.D. (1933), Parallelism and path-space, Mathematische Zeitschrift, 37, 608.
  2. [2]  Cartan, E. (1933), Observations sur le mémoire précédent, Mathematische Zeitschrift, 37, 619.
  3. [3]  Chern, S.S. (1939), Sur la géométrie d'un système d'équations différentielles du second ordre, Bulletin of Mathematical Sciences, 63, 206-212.
  4. [4]  Antonelli, P.L., Ingarden, R., and Matsumoto, M. (1993), The Theory of Sprays and Finsler Spaces with Applications, Kluwer Academic Publishers, Dordrecht.
  5. [5]  Sabau, S.V. (2005), Some remarks on Jacobi stability, Nonlinear Analysis: Theory, Methods and Applications, 63(5), 143-153.
  6. [6]  Harko, T., Ho, C.Y., Leung, C.S., and Yip, S. (2015), Jacobi stability analysis of the Lorenz system, International Journal of Geometric Methods in Modern Physics, 12(7), 1550081.
  7. [7]  Boehmer, C.G., Harko, T., and Sabau, S.V. (2012), Jacobi stability analysis of dynamical systems - applications in gravitation and cosmology, Advances in Theoretical and Mathematical Physics, 16(4), 1145-1196.
  8. [8]  Gupta, M.K., and Yadav, C.K. (2017), Jacobi stability analysis of modified Chua circuit system, International Journal of Geometric Methods in Modern Physics, 14(06), 1750089.
  9. [9]  Gupta, M.K., and Yadav, C.K. (2016), Jacobi stability analysis of Rikitake system, International Journal of Geometric Methods in Modern Physics, 13(07), 1650098.
  10. [10]  Huang, Q., Liu, A., and Liu, Y. (2019), Jacobi stability analysis of the Chen system, International Journal of Bifurcation and Chaos, 29(10), 1950139.
  11. [11]  Leonov, G.A. (2001), Bounds for attractors and the existence of homoclinic orbits in the Lorenz system, Journal of Applied Mathematics and Mechanics, 65(1), 19-32.
  12. [12]  Li, D., Lu, J.A., Wu, X., and Chen, G. (2005), Estimating the bounds for the Lorenz family of chaotic systems, Chaos, Solitons $\&$ Fractals, 23(2), 529-534.
  13. [13]  Liao, X. (2004), On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization, Science in China Series E, 34(12), 1404-1419.
  14. [14]  Leonov, G.A., Bunin, A.I., and Koksch, N. (1987), Attraktorlokalisierung des Lorenz-Systems, ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 67(12), 649-656.
  15. [15]  Zhang, F., and Zhang, G. (2014), Boundedness solutions of the complex Lorenz chaotic system, Applied Mathematics and Computation, 243, 12-23.
  16. [16]  Lin, M., Hou, Y., Al-Towailb, M.A., and Saberi-Nik, H. (2023), The global attractive sets and synchronization of a fractional-order complex dynamical system, AIMS Mathematics, 8(2), 3523-3541.
  17. [17]  Li, D., Lu, J.A., Wu, X., and Chen, G. (2006), Estimating the ultimate bound and positively invariant set for the Lorenz system and a unified chaotic system, Journal of Mathematical Analysis and Applications, 323(2), 844-853.
  18. [18]  Pan, Y., Er, M.J., and Sun, T. (2012), Composite adaptive fuzzy control for synchronizing generalized Lorenz systems, Chaos: An Interdisciplinary Journal of Nonlinear Science, 22(2), 1-7.
  19. [19]  Park, C.W., Lee, C.H., and Park, M. (2002), Design of an adaptive fuzzy model-based controller for chaotic dynamics in Lorenz systems with uncertainty, Information Sciences, 147(4), 245-266.
  20. [20]  Shukla, V.K., Kumar, A., and Mishra, P.K. (2023), Chaos synchronization of complex chaotic systems via nonlinear control method, AIP Conference Proceedings, 2819(1), 1-13.
  21. [21]  Bouzeriba, A., Boulkroune, A., and Bouden, T. (2016), Projective synchronization of two different fractional-order chaotic systems via adaptive fuzzy control, Neural Computing and Applications, 27, 1349-1360.
  22. [22]  Sun, J., Shen, Y., Zhang, G., Xu, C., and Cui, G. (2013), Combination–combination synchronization among four identical or different chaotic systems, Nonlinear Dynamics, 73, 1211-1222.
  23. [23]  Kharabian, B., and Mirinejad, H. (2023), Switched combination synchronization of nonidentical fractional-order chaotic systems using neuro-fuzzy sliding mode control, American Control Conference, 2203-2209.
  24. [24]  Sun, J., Shen, Y., Wang, X., and Chen, J. (2014), Finite-time combination-combination synchronization of four different chaotic systems with unknown parameters via sliding mode control, Nonlinear Dynamics, 76, 383-397.
  25. [25]  Ojo, K.S., Njah, A.N., Olusola, O.I., and Omeike, M.O. (2014), Reduced order projective and hybrid projective combination-combination synchronization of four chaotic Josephson junctions, Journal of Chaos, 2014(282407), 1-9.
  26. [26]  Dong, W., Wang, C., Zhang, H., and Ma, P. (2023), Finite-time combination-combination synchronization of hyperchaotic systems with different structures and its application, Journal of System Simulation, 35(7), 1590-1601.
  27. [27]  Ibraheem, A., and Kumar, N. (2021), Dual combination–combination synchronization of time-delayed dynamical systems via adaptive sliding mode control under uncertainties and external disturbances, International Journal of Dynamics and Control, 9, 737-754.
  28. [28]  Zhang, Y., Yang, Z., and Fath, B.D. (2010), Ecological network analysis of an urban water metabolic system: model development, and a case study for Beijing, Science of the Total Environment, 408(20), 4702-4711.
  29. [29]  Bhardwaj, R., and Das, S. (2019), Chaos control dynamics in competitive herbivore species network, Jñānābha, 49(2), 22-28.
  30. [30]  Bagchi, D., Arumugam, R., Chandrasekar, V.K., and Senthilkumar, D.V. (2024), Generalized synchronization in a tritropic food web metacommunity, Journal of Theoretical Biology, 567, 111759.
  31. [31]  Arellano-Véliz, N.A., Jeronimus, B.F., Kunnen, E.S., and Cox, R.F. (2024), The interacting partner as the immediate environment: Personality, interpersonal dynamics, and bodily synchronization, Journal of Personality, 92(1), 180-201.
  32. [32]  Takyi, E.M., Parshad, R.D., Upadhyay, R.K., and Rai, V. (2023), Blow-Up Dynamics and Synchronization in Tri-Trophic Food Chain Models, Algorithms, 16(4), 180.
  33. [33]  Dwivedi, S., and Kumari, N. (2023), Effectiveness of phase synchronization in chaotic food chain model with refugia and Allee effects during seasonal fluctuations, Chaos: An Interdisciplinary Journal of Nonlinear Science, 33(6), 1-24.
  34. [34]  Jafari, S., Sprott, J.C., and Golpayegani, S.M.R.H. (2013), Elementary quadratic chaotic flows with no equilibria, Physics Letters A, 377(9), 699-702.
  35. [35]  Rahman, Q.I., and Schmeisser, G. (2002), Analytic Theory of Polynomials, Oxford University Press, Oxford.
  36. [36]  Mahmoud, E.E., Trikha, P., Jahanzaib, L.S., and Almaghrabi, O.A. (2020), Dynamical analysis and chaos control of the fractional chaotic ecological model, Chaos, Solitons $\&$ Fractals, 141, 110348.
  37. [37]  Bateman, P.W., Fleming, P.A., and Wolfe, A.K. (2017), A different kind of ecological modelling: the use of clay model organisms to explore predator–prey interactions in vertebrates, Journal of Zoology, 301(4), 251-262.
  38. [38]  Park, J.H., and Kwon, O.M. (2005), A novel criterion for delayed feedback control of time-delay chaotic systems, Chaos, Solitons $\&$ Fractals, 23(2), 495-501.
  39. [39]  Zhu, Z.Y., Zhao, Z.S., Zhang, J., Wang, R.K., and Li, Z. (2020), Adaptive fuzzy control design for synchronization of chaotic time-delay system, Information Sciences, 535, 225-241.
  40. [40]  Hale, J.K. (2006), Functional Differential Equations, Springer, Berlin, Heidelberg.
  41. [41]  Dănilă, B., Harko, T., Mak, M.K., Pantaragphong, P., and Sabau, S.V. (2016), Jacobi stability analysis of scalar field models with minimal coupling to gravity in a cosmological background, Advances in High Energy Physics, 2016, 1-26.

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