Jacobi stability analysis in regard to the Lorenz dynamical system
Authors: Shkapov P.M., Sulimov V.D., Sulimov A.V.
Published in issue: #7(151)/2024
DOI: 10.18698/2308-6033-2024-7-2368
Category: Mechanics | Chapter: Theoretical Mchanics, Machine Dynamics
The paper considers problems of the Jacobi stability analysis and recovery of the Lorenz dynamic system free parameters according to the indirect approximately specified information. In the context of the Kosambi — Cartan — Chern theory, it presents geometric description of the system evolution in time and defines the five geometric invariants. Eigenvalues of the second invariant (deviation curvature tensor) are assessing the system Jacobi stability. Such a study is of interest in applications, where it is required to establish domains with both the Liapunov and the Jacobi stabilities. The paper formulates the inverse problem on recovering the system parameters according to the approximately specified eigenvalues of the second invariant. The regularized inverse problem is solved using the optimization approach. The scalar criterion functions are assumed to be continuous, multidimensional, multi-extremal, locally Lipschitzian, and not necessarily differentiable anywhere. A new hybrid algorithm is introduced in searching for the global solutions. It integrates the stochastic algorithm scanning the variables space and the local minimization deterministic method. In the local search phase, the two-parameter smoothing approximations of the criterial functions are introduced. Numerical example of the Lorenz system parameters recovery is provided.
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