International Journal of Mathematics Trends and Technology
Volume 70 | Issue 10 | Year 2024 | Article Id. IJMTT-V70I10P104 | DOI : https://doi.org/10.14445/22315373/IJMTT-V70I10P104
| Received | Revised | Accepted | Published |
|---|---|---|---|
| 11 Aug 2024 | 20 Sep 2024 | 11 Oct 2024 | 30 Oct 2024 |
KCC theory offers a robust geometric framework for analyzing the stability of dynamical systems described by second order differential equations. Its use of KCC invariants and Jacobi stability analysis provides insights into the behavior of nonlinear systems across various fields, including cosmology, mechanics, biology, and control theory. By transforming stability analysis into a geometric problem, the KCC theory enables a deeper understanding of the conditions under which systems maintain or lose stability, thereby offering practical insights into real-world applications. The main ideas of the KCC theory are examined in this study, along with how it is used for Jacobi stability analysis in specific systems. Jacobi stability for various dynamic systems, including the Rititake, Rossler, Chua circuit, RF, and tumor growth models, as well as the KCC theory and its constituent parts, is explained.
KCC- geometric theory, Jacobian stability, Deviation curvature tensor, Cartan tensor, Finsler connection.
[1] John L. Synge, “II. On the Geometry of Dynamics,” Philosophical Transactions of the Royal Society of London. Series A, vol. 226, no.
636-646, pp. 31-106, 1927.
[CrossRef] [Google Scholar] [Publisher Link]
[2] M.S. Knebelman, “Collineations and Motions in Generalized Spaces,” American Journal of Mathematics, vol. 51, no. 4, pp. 527-564,
1929.
[CrossRef] [Google Scholar] [Publisher Link]
[3] Jesse Douglas, “The General Geometry of Paths,” Annals of Mathematics, vol. 29, no. 1/4, pp. 143-169, 1928.
[CrossRef] [Google Scholar] [Publisher Link]
[4] D.D. Kosambi, “Parallelism and Path-Spaces,” Mathematische Zeitschrift, vol. 37, pp. 608-618, 1933.
[Google Scholar] [Publisher Link]
[5] E. Cartan, “Observations on the Previous Memorandum,” Mathematical Zeitschrift, vol. 37, pp. 619-622, 1933.
[Google Scholar] [Publisher Link]
[6] S.S. Chern, “On the Geometry of Systems of Differential Equations,” Bulletin of Mathematical Sciences, vol. 63, pp. 206-249, 1939.
[Google Scholar]
[7] P.L. Antonelli, Roman S. Ingarden, and M. Matsumoto, The Theory of Sprays and Finsler Spaces with Applications in Physics and
Biology, Springer Netherlands, pp. 1-312, 1993.
[Google Scholar] [Publisher Link]
[8] C.G. Bohmer, T. Harko, and S.V. Sabau, “Jacobi Stability Analysis of Dynamical Systems Applications in Gravitation and
Cosmology,” Advances in Theoretical and Mathematical Physics, vol. 16, no. 4, pp. 1145-1196, 2012.
[Google Scholar] [Publisher Link]
[9] P.L. Antonelli, Equivalence Problem for Systems of Second Order Ordinary Differential Equations, Encyclopedia of Mathematics, 2000.
[Google Scholar] [Publisher Link]
[10] Peter L. Antonelli, Handbook of Finsler Geometry, Kluwer Academic Publishers, vol. 2, pp. 1-1437, 2003.
[Google Scholar] [Publisher Link]
[11] Vasile Sorin Sabău, “Systems Biology and Deviation Curvature Tensor,” Nonlinear Analysis: Real World Applications, vol. 6, no. 3, pp.
563-587, 2005.
[CrossRef] [Google Scholar] [Publisher Link]
[12] Tiberiu Harko et al., “Jacobi Stability Analysis of the Lorenz System,” International Journal of Geometric Methods in Modern Physics,
vol. 12, no. 7, 2015.
[CrossRef] [Google Scholar] [Publisher Link]
[13] Tiberiu Harko, Praiboon Pantaragphong, and Sorin V. Sabau, “Kosambi–Cartan–Chern (KCC) Theory for Higher-Order Dynamical
Systems,” International Journal of Geometric Methods in Modern Physics, vol. 13, no. 2, 2016.
[CrossRef] [Google Scholar] [Publisher Link]
[14] Kazuhito Yamasaki, and Takahiro Yajima, “Lotka–Volterra System and KCC Theory: Differential Geometric Structure of Competitions
and Predations,” Nonlinear Analysis: Real World Applications, vol. 14, no. 4, pp. 1845-1853, 2013.
[CrossRef] [Google Scholar] [Publisher Link]
[15] M.K. Gupta, and C.K. Yadav, “Jacobi Stability Analysis of Modified Chua Circuit System,” International Journal of Geometric Methods
in Modern Physics, vol. 14, no. 6, 2017.
[CrossRef] [Google Scholar] [Publisher Link]
[16] M.K. Gupta, and C.K. Yadav, “Rabinovich-Fabrikant System in View Point of KCC Theory in Finsler Geometry,” Journal of
Interdisciplinary Mathematics, vol. 22, no. 3, pp. 219-241, 2019.
[CrossRef] [Google Scholar] [Publisher Link]
[17] Hossein Abolghasem, “Stability of Circular Orbits in Schwarzschild Spacetime,” International Journal of Differential Equations and
Applications, vol. 12, no. 3, pp. 131-147, 2013.
[Google Scholar] [Publisher Link]
[18] Hossein Abolghasem, “Jacobi Stability of Hamiltonian Systems,” International Journal of Pure and Applied Mathematics, vol. 87, no. 1,
pp. 181-194, 2013.
[Google Scholar] [Publisher Link]
[19] Cristina Blaga, Paul Blaga, and Tiberiu Harko, “Jacobi and Lyapunov Stability Analysis of Circular Geodesics around a Spherically
Symmetric Dilaton Black Hole,” Symmetry, vol. 15, no. 2, pp. 1-23, 2023.
[CrossRef] [Google Scholar] [Publisher Link]
[20] Joseph Grifone, “Near-Tangent Structure and Connections II,” Annals of the Fourier Institute, vol. 22, no. 3, 291-338, 1972.
[CrossRef] [Google Scholar] [Publisher Link]
[21] Tsuneji Rikitake, “Oscillations of a System of Disk Dynamos,” Mathematical Proceedings of the Cambridge Philosophical Society, vol.
54, no. 1, pp. 89-105, 1958.
[CrossRef] [Google Scholar] [Publisher Link]
[22] O.E. Rössler, “An Equation for Continuous Chaos,” Physics Letters A, vol. 57, no. 5, pp. 397-398, 1976.
[CrossRef] [Google Scholar] [Publisher Link]
[23] Otto E. Rössler, “Different Types of Chaos in Two Simple Differential Equations,” Zeitschrift Für Naturforschung A, vol. 31, no. 12, pp.
1664-1670, 1976.
[CrossRef] [Google Scholar] [Publisher Link]
[24] Raluca Eftimie, Jonathan L. Bramson, and David J. D. Earn, “Interactions between the Immune System and Cancer: A Brief Review of
Non-spatial Mathematical Models,” Bulletin of Mathematical Biology, vol. 73, pp. 2-32, 2011.
[CrossRef] [Google Scholar] [Publisher Link]
[25] Mikhail I. Rabinovich, and Anatoly L. Fabrikant, “Stochastic Wave Self-Modulation in Nonequilibrium Media,” Journal of Experimental
and Theoretical Physics, vol. 77, pp. 617-629, 1979.
[Google Scholar] [Publisher Link]
Nand Kishor Kumar, "Kosambi-Carton-Chern (KCC) Theory for Jacobi Stability Analysis in Certain Systems," International Journal of Mathematics Trends and Technology (IJMTT), vol. 70, no. 10, pp. 22-29, 2024. Crossref, https://doi.org/10.14445/22315373/IJMTT-V70I10P104
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