Jacobi Stability of the Nonlinear Double Pendulum and its Trajectories in the Configuration Space

Abstract

The paper considers problems of the Jacobi stability analysis in regard to a dynamic system, i.e., the non-linear double pendulum. Based on the Kosambi --- Cartan --- Chern theory, it introduces geometric description of the system evolution in time making it possible to determine the five geometric invariants. Eigenvalues of the second invariant called the deviation curvature tensor and provide an estimate of the system Jacobi stability. Such research is relevant in applications, where it is necessary to identify regions with the Lyapunov stability and the Jacobi stability appearing simultaneously. The paper investigates the in-time evolution of a system consisting of two identical mathematical pendulums connected in series. The deviation curvature tensor eigenvalues dependence on the initial conditions is presented. The MATLAB computing environment was used in integrating the motion nonlinear differential equations. Dependence of the behavior nature of the regular motion or global chaos system on the initial conditions is determined. The system regular or chaotic behavior is represented in the configuration space and is characterized by alterations in the generalized coordinates and in the deflection curvature tensor eigenvalues. Examples are provided of the system trajectory types in the configuration space depending on the initial conditions. The paper demonstrates effectiveness of the implemented approach in determination of the system Jacobi stability

Please cite this article in English as:

Shkapov P.M., Sulimov V.D., Danich M.A. Jacobi stability of the nonlinear double pendulum and its trajectories in the configuration space. Herald of the Bauman Moscow State Technical University, Series Natural Sciences, 2024, no. 4 (115), pp. 21--34 (in Russ.). EDN: UPBGEX

References

[1] Hafstein S.F., Valfells A. Efficient computation of Lyapunov functions for nonlinear systems by integrating numerical solutions. Nonlinear Dyn., 2019, vol. 97, no. 3, pp. 1895--1910. DOI: https://doi.org/10.1007/s11071-018-4729-5

[2] Abolghasem H. Liapunov stability versus Jacobi stability. JDSGT, 2012, vol. 10, iss. 1, pp. 13--32. DOI: https://doi.org/10.1080/1726037X.2012.10698604

[3] Bohmer C.G., Harko T., Sabau S.V. Jacobi stability analysis of dynamical systems --- applications in gravitation and cosmology. Adv. Theor. Math. Phys., 2012, vol. 16, no. 4, pp. 1145--1196. DOI: https://dx.doi.org/10.4310/ATMP.2012.v16.n4.a2

[4] Harko T., Pantaragphong P., Sabau S.V. Kosambi --- Cartan --- Chern (KCC) theory for higher order dynamical systems. Int. J. Geom. Methods Mod. Phys., 2016, vol. 13, no. 2, art. 1650014. DOI: https://doi.org/10.1142/S0219887816500146

[5] Lorenz E.I. Deterministic nonperiodic flow. J. Atmos. Sci., 1963, vol. 20, no. 2, pp. 130--141. DOI: https://doi.org/10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2

[6] Cattani M., Caldas I.L., de Souza S.L., et al. Deterministic chaos theory: basic concepts. Rev. Bras. de Ensino de Fis., 2017, vol. 39, no. 1, art. e1309. DOI: http://dx.doi.org/10.1590/1806-9126-RBEF-2016-0185

[7] Velarde C., Robledo A. Manifestations of the onset of chaos in condensed matter and complex systems. Eur. Phys. J. Spec. Top., 2018, vol. 227, no. 5-6, pp. 645--660. DOI: https://doi.org/10.1140/epjst/e2018-00128-9

[8] Wang F., Liu T., Kuznetsov N.V., et al. Jacobi stability analysis and the onset of chaos in a two-degree-of-freedom mechanical system. Int. J. Bifurcat. Chaos, 2021, vol. 31, no. 5, art. 2150075. DOI: https://doi.org/10.1142/S0218127421500759

[9] Levien R.B., Tan S.M. Double pendulum: an experiment in chaos. Am. J. Phys., 1993, vol. 61, iss. 11, pp. 1038--1044. DOI: https://doi.org/10.1119/1.17335

[10] Liang Y., Feeny B.F. Parametric identification of a chaotic base-excited double pendulum experiment. Nonlinear Dyn., 2008, vol. 52, no. 1-2, pp. 181--197. DOI: https://doi.org/10.1007/s11071-007-9270-x

[11] Fuentes M.A., Sato Y., Tsallis C. Sensitivity to initial conditions, entropy production, and escape rate at the onset of chaos. Phys. Lett. A, 2011, vol. 375, iss. 33, pp. 2988--2991. DOI: https://doi.org/10.1016/j.physleta.2011.06.039

[12] Yao Y. Numerical study on the influence of initial conditions on quasi-periodic oscillation of double pendulum system. J. Phys.: Conf. Ser., 2020, vol. 1437, art. 012093. DOI: https://doi.org/10.1088/1742-6596/1437/1/012093

[13] Watson P.A.G. Applying machine learning to improve simulations of a chaotic dynamical system using empirical error correction. JAMES, 2019, vol. 11, iss. 5, pp. 1402--1417. DOI: https://doi.org/10.1029/2018MS001597

[14] D’Alessio S. An analytical, numerical and experimental study of the double pendulum. Eur. J. Phys., 2023, vol. 44, no. 1, art. 015002. DOI: https://doi.org/10.1088/1361-6404/ac986b

[15] Oiwa S., Yajima T. Jacobi stability analysis and chaotic behavior of nonlinear double pendulum. Int. J. Geom. Methods Mod. Phys., 2017, vol. 14, no. 12, art. 1750176. DOI: https://doi.org/10.1142/S0219887817501766

[16] Monteanu F., Grin A., Musafirov E., et al. About the Jacobi stability of a generalized Hopf --- Langford system through the Kosambi --- Cartan --- Chern geometric theory. Symmetry, 2023, vol. 15, no. 3, art. 598. DOI: https://doi.org/10.3390/sym15030598

[17] Ye K., Hu S. Inverse eigenvalue problem for tensors. Commun. Math. Sci., 2017, vol. 15, no. 6, pp. 1627--1649. DOI: https://dx.doi.org/10.4310/CMS.2017.v15.n6.a7

[18] Sulimov V.D., Shkapov P.M., Sulimov A.V. Jacobi stability and updating parameters of dynamical systems using hybrid algorithms. IOP Conf. Ser.: Mater. Sci. Eng., 2018, vol. 468, art. 012040. DOI: https://doi.org/10.1088/1757-899X/468/1/012040

[19] 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, no. 4 (97), pp. 40--56 (in Russ.). DOI: http://dx.doi.org/10.18698/1812-3368-2021-4-40-56

[20] Sulimov A.V. Jacobi stability analysis and restoring parameters of the damped double pendulum. Inzhenernyy zhurnal: nauka i innovatsii [Engineering Journal: Science and Innovation], 2023, no. 7 (in Russ.). DOI: http://dx.doi.org/10.18698/2308-6033-2023-7-2287