|Dopov. Nac. akad. nauk Ukr. 2019, 9:12-19|
|Section: Information Science and Cybernetics|
A general model of the origin and evolution of chaotic wave processes in complex systems based on the propo sed method of matrix decomposition of operators of nonlinear systems is developed. The proposed model shows that the effect of self-organization in complex systems of different physical nature (for example, hydrodynamic, electronic, and physiological ones) is based on the interaction of nonlinear processes of higher orders leading to the stabilization (to a finite value) of the amplitude of the chaotic wave process. Mathematically, this means the synchronous “counteraction” of nonlinear processes of even and odd orders in a general vector-matrix model of a complex system, being in the chaotic mode. The implementation of the vector-matrix decomposition by means of computational experiments shows that the model of L.D. Landau describes the scenario of the occurrence of chaotic modes in complex systems quite well. It is noted that the regime of hard self-excitation of nonlinear oscillations in complex systems leads to the appearance of a chaotic attractor in the state-space. Moreover, the proposed vector-matrix model allows one to find more general conditions for the origin and evolution of chaotic wave processes and, as a result, to explain the appearance of coherent nonlinear phenomena in complex systems.
|Keywords: chaotic attractor, complex nonlinear dynamical system, general vector-matrix model of chaotic wave processes, matrix series in state-space, mode of hard self-excitation of nonlinear oscillations, stabilization of the amplitude of chaotic process, state-space|
1. Landau, L. D. (1944). To the problem of turbulence. Dokl. Akad. nauk SSSR [Reports of the Academy of Sciences USSR], 44, No. 8, pp. 339-342 (in Russian).
2. Krot, A. M. (1990). On a class of discrete quasistationary linear dynamic systems. Sov. Phys. Dokl. 35, No. 8, pp. 711-713.
3. Lorenz, E. N. (1963). Deterministic nonperiodic flow. J. Atmosph. Sci., 20, March, pp. 130-141. doi: https://doi.org/10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2
4. Ruelle, D.&Takens, F. (1971). On the nature of turbulence. Communications in Math. Phys. 20, pp. 167-192. doi: https://doi.org/10.1007/BF01646553
5. Nicolis, G. & Prigogine, I. (1977). Self-organization in Nonequilibrium Systems: from Dissipative Structures to Order through Fluctuation. New York: John Willey&Sons.
6. Bergé, P., Pomeau, Y. & Vidal, C. (1988). L’ordre dans le chaos: Vers une approche déterministe de la turbulence. Paris: Hermann.
7. Krot, A. M. (2000). The decomposition of vector functions in vector-matrix series into state-space of nonlinear dynamic system. EUSIPCO-2000: Proc. X Eur. Signal Proc. Conf. (Tampere, Finland, September 4–8, 2000). Vol. 3. Tampere, pp. 2453-2456.
8. Krot, A. M. (2001). Matrix decompositions of vector functions and shift operators on the trajectories of a nonlinear dynamical system. Nonlinear Phenomena in Complex Systems. 4, No. 2, pp. 106-115. ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2019. № 9
9. Krot, A. M. (2004). Analysis of attractors of complex nonlinear dynamical systems based on matrix series in the state-space. Informatica, No. 1, pp. 7-16 (in Russian).
10. Krot, A. M. (2009). The development of matrix decomposition theory for nonlinear analysis of chaotic attractors of complex systems and signals. DSP-2009: Proc. 16th IEEE Int. Conf. on Digital Signal Proc. (Thira, Santorini, Greece, July 5-7, 2009). Santorini, pp. 1-5. doi: https://doi.org/10.1109/ICDSP.2009.5201123