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

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

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] Kvasov D.E., Sergeyev Ya.D. Lipschitz global optimization methods in control problems. Autom. Remote Control, 2013, vol. 74, no. 9, pp. 1435--1448. DOI: https://doi.org/10.1134/S0005117913090014

[3] Velishchanskiy M.A., Krishchenko A.P., Tkachev S.B. Synthesis of spacecraft reorientation algorithms using the concept of the inverse dynamic problem. J. Comput. Syst. Sci. Int., 2003, vol. 42, no. 5, pp. 811--818.

[4] Colombo L.A. A variational-geometric approach for the optimal control of nonholonomic systems. Int. J. Dynam. Control, 2018, vol. 6, no. 2, pp. 652--662. DOI: https://doi.org/10.1007/s40435-017-0326-6

[5] Bloch A.M., Gupta F., Kolmanovsky I.V. Neighboring external optimal control for mechanical systems on Riemannian manifolds. J. Geom. Mech., 2016, vol. 8, iss. 3, pp. 257--272. DOI: http://dx.doi.org/10.3934/jgm.2016007

[6] 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: http://dx.doi.org/10.4310/ATMP.2012.v16.n4.a2

[7] Harko T., Pantaragphong P., Sabau S.V. Kosambi --- Cartan --- Chern (KCC) theory for higher order dynamical systems. Int. J. Geom. Methods M., 2016, vol. 13, no. 2, art. 1656014. DOI: https://doi.org/10.1142/S0219887816500146

[8] Abolghasem H. Liapunov stability versus Jacobi stability. J. Dyn. Syst. Geom. Theor., 2012, vol. 10, iss. 1, pp. 13--32. DOI: https://doi.org/10.1080/1726037X.2012.10698604

[9] Lavrent’yev M.M., Zharinov S.Yu., Zekal’ S.M., et al. Computational diagnostics of surface characteristics of lengthy cylindrical objects by means of active location. Sibirskiy zhurnal industrial’noy matematiki, 2002, vol. 5, no. 1, pp. 105--113 (in Russ.).

[10] Charles C., Bish A., Boswell R.W., et al. A short review of experimental and computational diagnostics for radiofrequency plasma micro-thrusters. Plasma Chem. Plasma Process., 2016, vol. 36, no. 1, pp. 29--44. DOI: https://doi.org/10.1007/s11090-015-9654-5

[11] Oulas A., Minadakis G., Zachariou M., et al. Systems Bioinformatics: increasing precision of computational diagnostics and therapeutics through network-based approaches. Brief. Bioinformatics, 2019, vol. 20, iss. 3, pp. 806--824. DOI: https://doi.org/10.1093/bib/bbx151

[12] Wang Y., Yang C., Yagola A.G. (eds). Optimization and regularization for computational inverse problems and applications. Berlin, Heidelberg, Springer, 2010. DOI: https://doi.org/10.1007/978-3-642-13742-6

[13] Andreev R. Tikhonov and Landweber convergence rates: characterization by interpolation spaces. Inverse Probl., 2015, vol. 31, no. 10, art. 105007. DOI: https://doi.org/10.1088/0266-5611/31/10/105007

[14] Sprung B., Hohage T. Higher order convergence rates for Bregman iterated variational regularization of inverse problems. Numer. Math., 2019, vol. 141, no. 2, pp. 215--252. DOI: https://doi.org/10.1007/s00211-018-0987-x

[15] Xu Y.-T., Zhang Y., Wang S.-G. A modified tunneling function method for non-smooth global optimization and its applications in artificial neural network. Appl. Math. Model., 2015, vol. 39, iss. 21, pp. 6438--6450. DOI: https://doi.org/10.1016/j.apm.2015.01.059

[16] Harko T., Ho C.Y., Leung C.S., et al. Jacobi stability analysis of the Lorenz system. Int. J. Geom. Methods M., 2015, vol. 12, no. 7, art. 1550081. DOI: https://doi.org/10.1142/S0219887815500814

[17] Yajima T., Yamasaki K. Jacobi stability for dynamical systems of two-dimensional second-order differential equations and application to overhead crane system. Int. J. Geom. Methods Mod. Phys., 2016, vol. 13, no. 4, art. 1650045. DOI: https://doi.org/10.1142/S0219887816500456

[18] Custodio A.L., Madeira J.F.A. GLODS: global and local optimization using direct search. J. Glob. Optim., 2015, vol. 62, no. 1, pp. 1--28. DOI: https://doi.org/10.1007/s10898-014-0224-9

[19] Du Y., Ruszczynski A. Rate of convergence of the bundle method. J. Optim. Theory Appl., 2017, vol. 173, no. 3, pp. 908--922. DOI: https://doi.org/10.1007/s10957-017-1108-1

[20] Davis D., Drusvyatskiy D., MacPhee K.J., et al. Subgradient methods for sharp weakly convex functions. J. Optim. Theory Appl., 2018, vol. 179, no. 3, pp. 962--982. DOI: https://doi.org/10.1007/s10957-018-1372-8

[21] Curtis F.E., Robinson D.P., Zhou B. A self-correcting variable-metric algorithm framework for nonsmooth optimization. IMA J. Numer. Anal., 2020, vol. 40, iss. 2, pp. 1154--1187. DOI: https://doi.org/10.1093/imanum/drz008

[22] Floudas C.A., Gounaris C.E. A review of recent advances in global optimization. J. Glob. Optim., 2009, vol. 45, no. 1, pp. 3--38. DOI: https://doi.org/10.1007/s10898-008-9332-8

[23] Zilinskas A., Zhigljavsky A. Stochastic global optimization: a review on the occasion of 25 years of Informatica. Informatica, 2016, vol. 27, no. 2, pp. 229--256.

[24] Liu J., Zhang S., Wu C., et al. A hybrid approach to constraint global optimization. Appl. Soft Comput., 2016, vol. 47, pp. 281--294. DOI: https://doi.org/10.1016/j.asoc.2016.05.021

[25] Torres R.H., de Campos Velho H.F., da Luz E.F.P. Enhancement of the Multi-Particle Collision Algorithm by mechanisms derived from the opposition-based optimization. Sel. Mat., 2019, vol. 6, no. 2, pp. 156--177.

[26] Torres R.H., de Campos Velho H.F. Rotation-based sampling Multi-Particle Collision Algorithm with Hooke --- Jeeves. Proc. Ser. Brazil. Soc. Comput. Appl. Math., 2017, vol. 5, no. 1, art. 01473. DOI: https://doi.org/10.5540/03.2017.005.01.0473

[27] Xu J., Yan F., Yun K., et al. Dynamically dimensioned search embedded with piecewise opposition-based learning for global optimization. Sci. Program., 2019, vol. 2019, art. 2401818. DOI: https://doi.org/10.1155/2019/2401818

[28] Sulimov V.D., Shkapov P.M., Sulimov A.V. Optimization of singular values of parameter dependent matrices using hybrid algorithms. Herald of the Bauman Moscow State Technical University, Series Natural Sciences, 2016, no. 5 (68), pp. 46--66 (in Russ.). DOI: https://doi.org/10.18698/1812-3368-2016-5-46-66