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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

Jacobi and Linear Stability Analysis of T Chaotic System

Journal of Applied Nonlinear Dynamics 12(3) (2023) 523--536 | DOI:10.5890/JAND.2023.09.007

Vijay K. Shukla$^{1}$, Vijay K. Yadav$^{2}$, Abhishek Kumar$^{3}$, Prashant K. Mishra$^{4}$

$^{1}$ Department of Mathematics, Shiv Harsh Kisan P.G. College, Basti-272001, India

$^{2}$ Department of Mathematics, Nirma University, Ahmedabad-382481, India

$^{3}$ Department of Mathematics, Jai Prakash University, Chapra-841301, India

$^{4}$ Department of Mathematics, P. C. Vigyan Mahavidyalaya, Jai Prakash University, Chapra-841301, India

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Abstract

In this paper, the stability analysis of T chaotic system has been discussed by two different methods viz., Jacobi stability and linear stability methods. Jacobi stability analysis of T chaotic system has been studied by using geometro-dynamical approach with Kosambi-Cartan-Chern (KCC) theory. The deviation curvature tensor and five KCC invariants are obtained which express the intrinsic properties of nonlinear dynamical system. The dynamical behaviors of deviation vector components near the equilibrium points of T system are also discussed. The phase portrait of deviation vector components reflects the dynamical behavior of T system, which shows the instability and chaotic behaviour near the equilibrium points for a set of parameters.

References

  1. [1] Lorenz, E.N. (1963), Deterministic non-periods flows, Journal of the Atmospheric Sciences, 20, 130-141.
  2. [2] Chen, G. and Ueta, T. (1999), Yet another chaotic attractor, International Journal of Bifurcation and Chaos, 9(7), 1465-1466.
  3. [3] Lu, J. and Chen, G. (2002), A new chaotic attractor coined, International Journal of Bifurcation and Chaos, 12(3), 659-661.
  4. [4] Alvarez, G., Li, S., Montoya, F., Pastor, G., and Romera, M. (2005), Breaking projective chaos synchronization secure communication using filtering and generalized synchronization, Chaos, Solitons $\&$ Fractals, 24, 775-83.
  5. [5] Pecora, L.M. and Carroll, T.L. (1990), Synchronization in chaotic systems, Physics Review Letter, 64(8), 821-824.
  6. [6] Pecora, L.M. and Carroll, T.L. (1991), Driving systems with chaotic signals, Physics Review Letter, 44(4), 2374-2383.
  7. [7] Kosambi, D.D. (1933), Parallelism and path-space, Mathematische Zeitschrift, 37, 608-618.
  8. [8] Cartan, E. (1933), Observations sur le memoirprecedent, Mathematische Zeitschrift, 37, 619-622.
  9. [9] Chern, S.S. (1939), Sur la geometrie dun systemed equations differentialles du second ordre, Bulletin des Sciences Mathematiques, 63, 206-212.
  10. [10] Gupta, M.K. and Yadav, C.K. (2017), Jacobi stability analysis of Rossler system, International Journal of Bifurcation and Chaos, 27, 63-76.
  11. [11] Abolghasem, H. (2013), Jacobi stability of Hamiltonian systems, International Journal Pure and Applied Mathematics, 87(1), 181-194.
  12. [12] 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, 121-142.
  13. [13] Gupta, M.K. and Yadav, C.K. (2016), Jacobi stability analysis of Rikitake system, International Journal of Geometric Methods in Modern Physics, 13(7), 1650098.
  14. [14] Huang, Q., Liu, A., and Liu, Y. (2019), Jacobi Stability Analysis of the Chen System, International Journal of Bifurcation and Chaos, 29(10), 01-15.
  15. [15] Lu, J.H., Chen, G., Cheng, D.Z., and Celikovsky, S. (2002), Bridge the gap between the Lorenz system and the Chen system, International Journal of Bifurcation and Chaos, 12, 2917-2926.
  16. [16] Bohmer, C.G., Harko, T., and Sabau, S.V. (2012), Jacobi stability analysis of dynamical systems application in gravitation and cosmology, Advances in Theoretical and Mathematical Physics, 16, 1145-1196.
  17. [17] Sabau, S.V. (2005), Some remarks on Jacobi stability, Nonlinear Analysis, 63, 143-153.
  18. [18] Sulimov, V.D., Shkapov, P.M., and Sulimov, A.V. (2018), Jacobi stability and updating parameters of dynamical systems using hybrid algorithms, Fundamental and Applied Problems of Mechanics-2017, IOP Conf. Series: Materials Science and Engineering.
  19. [19] Abolghasem, H. (2012), Liapunov stability versus Jacobi stability, Journal of Dynamical Systems and Geometric Theories, 10(1), 13-32.
  20. [20] 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.
  21. [21] Harko, T. and Sabau, V.S. (2008), Jacobi stability of the vacuum in the static spherically symmetric brane world models, Physical Review D, 77, 104009.
  22. [22] Tigan, G. (2005), Analysis of a dynamical system derived from the Lorenz system, Science Bulleton Politehnica Univversity Timisoara Tomul, 50(64), 61-72.
  23. [23] Tigan, G. (2004), Bifurcation and the stability in a system derived from the Lorenz system, Third International Colloquium-Mathematics in Engineering and Numerical Physics, 265-272.
  24. [24] Tigan, G. and Opris, D. (2006), Analysis of a 3D chaotic system, Chaos Solitons $\&$ Fractals, 36, 1315-1319.
  25. [25] Antonelli, P.L., Ingarden, R.S., and Matsumoto, M. (1993), The theories of sprays and Finsler spaces with application in physics and biology, Kluwer Academic Publishers, Dordrecht/Boston/London.
  26. [26] Antonelli, P.L. (2003), Handbook of Finsler geometry I, II Kluwer Academic Publishers,Boston.
  27. [27] Skokos, C. (2010), The Lyapunov characteristic exponents and their computation, Lecture Notes Physics, 790, 63-135.
  28. [28] Robinson, C. (1995), Dynamical Systems: Stability, Symbolic Dynamics and Chaos, CRC Press.

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