Jacobi stability and restoration of parameters of the nonlinear double pendulum

Cover Page

Cite item

Full Text

Open Access Open Access
Restricted Access Access granted
Restricted Access Subscription Access

Abstract

The Jacobi stability analysis of the nonlinear dynamical system on base of Kosambi–Cartan–Chern theory is considered. Geometric description of time evolution of the system is introduced, that makes it possible to determine five geometric invariants. Eigenvalues of the second invariant (the deviation curvature tensor) give an estimate of Jacobi stability of the system. This approach is relevant in applications where it is required to identify the areas of Lyapunov and Jacobi stability simultaneously. For the nonlinear system – the double pendulum – the dependence of the Jacobi stability on initial conditions is investigated. The components of the deviation curvature tensor corresponding to the initial conditions and the eigenvalues of the tensor are defined explicitly. The boundary of the deterministic system transition from regular motion to chaotic one determined by the initial conditions has been found. The formulation of the inverse eigenvalue problem for the deviation curvature tensor associated with the restoration of significant parameters of the system is proposed. The solution of the formulated inverse problem has been obtained with the use of optimization approach. Numerical examples of restoring the system parameters for cases of its regular and chaotic behavior are given.

Full Text

Restricted Access

About the authors

P. M. Shkapov

Bauman MSTU

Author for correspondence.
Email: spm@bmstu.ru
Russian Federation, Moscow

V. D. Sulimov

Bauman MSTU

Email: spm@bmstu.ru
Russian Federation, Moscow

A. V. Sulimov

Bauman MSTU; Lomonosov MSU, Branch in Sevastopol

Email: spm@bmstu.ru
Russian Federation, Moscow; Sevastopol

References

  1. Hafstein S.F., Valfells A. Efficient computation of Lyapunov functions for non-linear systems by integrating numerical solutions // Nonlinear Dynamics. 2019. V. 97, № 3. P. 1895–1910. https://doi.org/10.1007/s11071-018-4729-5
  2. Abolghasem H. Liapunov stability versus Jacobi stability // Journal of Dynamical Systems and Geometric Theories. 2012. V. 10, № 1. P. 13–32.
  3. Blaga C., Blaga P., Harko T. Jacobi and Lyapunov stability analysis of circular geodesics around a spherically symmetric dilation black holl // Symmetry. 2023. V. 15, № 2, 329. P. 1–23. https://doi.org/10.48550/arXiv.2301.07678
  4. Böhmer C.G., Harko T., Sabau S.V. Jacobi stability analysis of dynamical systems – applications in gravitation and cosmology // Advances in Theoretical and Mathematical Physics, 2012. V. 16, № 4. P. 1145–1196.
  5. Punzi R., Wohlfarth M.N.R. Geometry and stability of dynamical systems // Physical Review E. 2009. V. 79, № 4. 046606. P. 1–11.
  6. Harko T., Pantaragphong P., Sabau S.V. Kosambi–Cartan–Chern (KCC) theory for higher order dynamical systems // International Journal of Geometric Methods in Modern Physics. 2016. V. 13. № 2. 1656014. P. 1–24.
  7. Munteanu F., Grin A., Musafirov E., et al. About the Jacobi and stability of a generalized Hopf–Langford system through the Kosambi–Cartan–Chern theory // Symmetry. 2023. V. 15, № 2, 598. P. 1–13. https://doi.org/10.3390/sym15030598
  8. Yamasaki K., Yajima T. Kosambi – Karatan – Chern analysis of the nonequilibrium singular point in one-dimensional elementary catastrophe // International Journal of Bifurcation and Chaos. 2022. V. 32. № 4. 2250053. P. 1–17. https://doi.org/10.1142/S021812742250016X
  9. Zhang X. When Shimizu–Morioka model meets Jacobi stability analysis: detecting chaos // International Journal of Geometric Methods in Modern Physics. 2023. V. 20. № 2. 2350023. P. 1–14. https://doi.org/10.1142/S0219887823500330
  10. Cattani M., Caldas I.L., de Souza S.L., Iarosz K.C. Deterministic chaos theory: basic concepts // Revista Brasileira de Ensino de Fisica. 2017. V. 39. № 1. e1309. P. 1–13.
  11. Lorenz E.I. Deterministic nonperiodic flow // Journal of the Atmospheric Sciences. 1963. V. 20. № 2. P. 130–141.
  12. Stachowiak T., Okada T. A numerical analysis of chaos in the double pendulum // Chaos, Solitons & Fractals. 2006. V. 29. № 3. P. 417–422.
  13. D’Alessio S. An analytical, numerical and experimental study of the double pendulum // European Journal of Physics. 2023. V. 44. 015002. P. 1–20. https://doi.org/١٠.١٠٨٨/١٣٦١-٦٤٠٤/ac٩٨٦b
  14. Yao Y. Numerical study on the influence of initial conditions on quasi-periodic oscillation of double pendulum system // Journal of Physics: Conference Series. 2020. V. 1437. 012093. P. 1–8. https://doi.org/10.1088/1742-6596/1437/1/012093
  15. Oiwa S., Yajima T. Jacobi stability analysis and chaotic behavior of nonlinear double pendulum // International Journal of Geometric Methods in Modern Physics. 2017. V. 14. № 12. 1750176. P. 1–19. https://doi.org/10.1142/S0219887817501766
  16. Wang F., Liu T., Kuznetsov N.V., Wei Z. Jacobi stability analysis and the onset of chaos in a two-degree-of-freedom mechanical system // International Journal of Bifurcation and Chaos. 2021. V. 31. № 05. 2150075. P. 1–15. https://doi.org/10.1142/S0218127421500759
  17. Ye K., Hu S. Inverse eigenvalue problem for tensors // Communications in Mathematical Sciences. 2017. V. 15. № 6. P. 1627–1649.
  18. Hu S., Ye K. Multiplicities of eigenvalues of tensors // Communications in Mathematical Sciences. 2016. V. 14, № 4. P. 1049–1071.
  19. Нелинейные некорректные задачи: монография / А.Н. Тихонов, А.С. Леонов, А.Г. Ягола. – М.: КУРС, 2017. – 400 с.
  20. Benning M., Burger M. Modern regularization methods for inverse problems // Acta Numerica. 2018. V. 27. P. 1–111.
  21. Xia Y., Wang L., Yang M. A fast algorithm for globally solving Tikhonov regularized total least squares problem // Journal of Global Optimization. 2019. V. 73. № 2. P. 311–330.
  22. Li H., Schwab J., Antholzer S., Haltmeier M. NETT: solving inverse problems with deep neural networks // Inverse Problems. 2020. V. 36. № 6. 065005. P. 1–23. https://doi.org/10.1088/1361-6420/ab6d57
  23. Torres R.H., Campos Velho H.F., da Luz E.F.P. Enhancement of the Multi–Particle Collision Algorithm by mechanisms derived from the opposition-based optimization // Selecciones Matemáticas. 2019. V. 06 (2). P. 156–177. https://doi.org/10.17268/sel.mat.2019.02.03
  24. Sulimov V.D., Shkapov P.M., Sulimov A.V. Jacobi stability and updating parameters of dynamical systems using hybrid algorithms // IOP Conference Series: Material Science and Engineering. 2018. V. 468. 012040. P 1–11. . https://doi.org/١٠.١٠٨٨/١٧٥٧-٨٩٩X/468/1/012040
  25. Shkapov P.M., Sulimov A.V., Sulimov V.D. Computational diagnostics of Jacobi unstable dynamical systems with the use of hybrid algorithms of global optimization // Herald of the Bauman Moscow State Technical University. Series Natural Sciences. 2021. № 4 (97). P. 40–56. https://doi.org/10.18698/1812-3368-2021-4-40-56
  26. Sulimov V.D., Sulimov A.V., Shkapov P.M. Programma dlya EVM, realizuyuschaya gibridnyi algoritm globalnoy nedifferentsiruemoy optimizatsii QOM-PCALMSI // Svidetelstvo o gosudarstvennoy registratsii programmy dlya EVM № 2022664841. Zayavka № 2022663517. Data gosudarstvennoy registratsii v Reestre program dlya EVM 05 avgusta 2022.

Supplementary files

Supplementary Files
Action
1. JATS XML
Download
2. Fig. 1. Nonlinear double pendulum.

Download (6KB)
Indexing metadata
3. Fig. 2. Change in eigenvalues ​​and tensor at . On the graph in deg.

Download (21KB)
Indexing metadata
4. Fig. 3. Trajectories of the system in the configuration space at . Along the axes in deg.

Download (10KB)
Indexing metadata
5. Fig. 4. Trajectories of the system in the configuration space at . Along the axes in deg.

Download (12KB)
Indexing metadata
6. Fig. 5. Trajectories of the system in the configuration space at . Along the axes in deg.

Download (14KB)
Indexing metadata

Copyright (c) 2024 Russian Academy of Sciences