 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
Archive of Issues
| Total articles in the database: | | 13025 |
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| In English (Mech. Solids): | | 4915 |
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| P.M. Shkapov, V.D. Sulimov, and A.V. Sulimov, "Jacobi Stability and Restoration of Parameters of the Nonlinear Double Pendulum," Mech. Solids. 59 (6), 3336-3346 (2024) |
| Year |
2024 |
Volume |
59 |
Number |
6 |
Pages |
3336-3346 |
| DOI |
10.1134/S0025654424604178 |
| Title |
Jacobi Stability and Restoration of Parameters of the Nonlinear Double Pendulum |
| Author(s) |
P.M. Shkapov (Bauman Moscow State Technical University, Moscow, 105005 Russia, spm@bmstu.ru)
V.D. Sulimov (Bauman Moscow State Technical University, Moscow, 105005 Russia)
A.V. Sulimov (Bauman Moscow State Technical University, Moscow, 105005 Russia; Sevastopol Branch of Moscow State University, Sevastopol, 299001 Russia) |
| 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. |
| Keywords |
nonlinear double pendulum, Jacobi stability, Kosambi–Cartan–Chern theory, geometric invariant, configuration space, restoration of parameters, global optimization |
| Received |
06 June 2024 | Revised |
06 July 2024 | Accepted |
09 July 2024 |
| Link to Fulltext |
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