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Computational Diagnostics of Jacobi Unstable Dynamical Systems with the Use of Hybrid Algorithms of Global Optimization

Authors: Shkapov P.M., Sulimov A.V., Sulimov V.D. Published: 26.08.2021
Published in issue: #4(97)/2021  
DOI: 10.18698/1812-3368-2021-4-40-56

 
Category: Mathematics and Mechanics | Chapter: Differential Equations and Mathematical Physics  
Keywords: dynamical system, control, Jacobi stability, geometrical invariant, computational diagnostics, global optimization, hybrid algorithm

The study focuses on the problems of restoration and analysis of free parameters of dynamical systems from indirect, approximately given information. In the context of the Kosambi --- Cartan --- Chern theory, a geometric description of the time-evolution of the system is introduced. Five geometric invariants are determined for the system under study. The eigenvalues of the second invariant estimate the Jacobi stability of the system. Such a study is of interest in practical applications, where it is required to identify the regions in which both Lyapunov stability and Jacobi stability occur simultaneously. The inverse problem of computational diagnostics of the system is formulated for approximately given eigenvalues of the second invariant. The solution to the regularized inverse problem is determined using an optimization approach. Scalar criterion functions are assumed to be continuous, multidimensional, locally Lipschitzian, and not necessarily everywhere differentiable. When searching for global solutions, we used new hybrid algorithms that integrate stochastic algorithms for scanning the space of variables and a deterministic local minimization procedure. The numerical scanning procedure is implemented with the use of two modified versions: quasi-opposition-based and rotation-based learning mechanisms. In the phase of local search, two-parameter smoothing approximations of criterion functions are introduced. Examples of solving problems of computational diagnostics of Jacobi unstable dynamical systems are given: the Lorentz system and a controllable elliptical pendulum

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