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Methods of modeling anomalous modes of dynamic processes based on energy estimation

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The problem of forecasting methodology for special modes of dynamic processes the nonlinear effect that occurs in the marine environment, called “rogue waves”, is considered. Rogue waves are waves that occur in the ocean, as a rule, suddenly, exist for a short period of time and have a huge destructive potential. There are many directions in the study of this phenomenon based on the application of computer modeling and numerical methods. At the same time, there is a tendency to search for rogue waves not only in hydrodynamics, but also in other subject areas, in which, when constructing models of the phenomena and processes under study, the apparatus for solving the corresponding initial boundary value problems for systems of differential equations is used. As a rule, the authors try to find solutions to differential equations, based on which it is possible to demonstrate the occurrence of abnormally high waves. It should be noted that the search for analytical solutions for some differential equations is an extremely difficult task or even impossible to solve. An alternative approach is proposed that makes it possible to prove the existence of the possibility of an anomaly without the need to solve the corresponding system of differential equations, and a model of a dynamic system is constructed similar to the formalism of Koopman theory which takes into account the asymptotic growth rate of the image of a dynamic operator in the energy space, on the basis of which an ordered hierarchy of classes of dynamic operators arises. The definition of an anomaly in the formalism of the mathematical apparatus under consideration is proposed, while the phenomenon of a rogue wave is interpreted as a special case of the occurrence of an anomalous phenomenon in a hydrodynamic system with a sufficiently high average value of the wave background. Within the framework of the proposed approach, it is possible to formulate the necessary conditions for the occurrence of an abnormal phenomenon and sufficient conditions for the absence of anomalies. A time series processing method is proposed that considers the hypothesis of the frequency of occurrence of anomalous phenomena. The existence of anomalies in magnetohydrodynamic processes is demonstrated, which is proved by constructing a model of magnetic field inversion, and the solution of the corresponding dispersion equation is carried out using a modification of the numerical Ivanisov-Polishchuk method consisting in combining the Ivanisov-Polishchuk algorithm and the Adam optimization method. The results obtained may be in demand for further development of the study of the structure of dynamic systems and for identifying more interdisciplinary connections that allow constructive transfer of some of the results from one subject area to another.

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