Pechuk V. Regular and chaotic dynamics in mechanics of cross-waves

Dissertation on the receipt of scientific degree of Ph.D. by speciality a 131 «Applied mechanics»( 13 Mechanical engineering). - Kyiv national university of building and architecture, Kyiv 2024. In this work possibility of pumping over of energy is first shown from wavemaker to the cruciform superficial waves in an ideal liquid in the rectangular pool of eventual sizes from wave maker. It is shown analytically on the basis of application of method of superposition and it is tested experimentally. Conformities to the law of regular withstand superficial waves are set in a liquid in the rectangular pool of eventual sizes from wavemaker. Existence of the regular modes of vibrations of free surface of liquid is shown at one modes parametrical resonances. Parameters, at which chaotic proof vibrations are possible in the averaged dynamic system which describes the dynamics of amplitudes of cruciform waves, are resulted. A new numeral algorithm is built for the differential systems with a delay. This is an explicit hybrid method for the systems of differential equations with a delay on the basis of polynomials of Newton and formula of Teylora for approaching on the next step of numeral integration and classic methods of Runge-kutti. The explicit hybrid method of fifth order is got thus for the systems of differential equations with a delay, for example, with the coefficients of Kutti-nyustrema or coefficients of method of Dormana-prinsa, with the estimation of error of numeral integration, befits for research of the actual now systems with a delay on the large enough, comparatively with a delay, intervals of time. Numerical calculation of maximum exponent of Lyapunov from usage classic algorithm of Benettin not always is satisfactory in case of the dynamic system with dissipation of energy. It is done modification of this algorithm that gives opportunity to promote exactness of calculations and extend an application of method domain. Application of this modification for the model of excitation of cross-waves on the free surface of liquid in the rectangular channel of eventual length and depth enabled more precisely to identify the dynamical modes. The numeral calculation of maximum exponent of Lyapunov on the basis of classic algorithm of Benettina does not take into account running back and running approach of trajectory on chaotic attractor. An universal algorithm exactness of which does not depend on it and other features of chaotic атракторів is developed in work, such as a form, size and structure, their amount and location in phase space. A new algorithm allows to increase exactness of calculations and extend an application of the noted method domain. Application of the offered modification for the model of excitation of cross-waves on the free surface of liquid in the rectangular channel of eventual length and depth allows more exactly to identify the dynamic modes. First probed influencing of delay on the dynamics of cross-waves. Importance of account of factors of delay is set at research on the regular and chaotic dynamics of the system that describes cross-waves in the pool of complete sizes from wave maker.