Chaos in Hydrodynamic System with Delay

The effect of delay on a dynamic of the cross-waves in hydrodynamic systems is investigated. An explicit fifth-order hybrid method for systems of delay differential equations for the variable step of numerical integration is constructed. This algorithm is based on the most commonly used explicit fifth-order Runge–Kutta methods for ordinary systems of differential equations and the construction of Newton polynomials for the prehistory of the model and Taylor formula. The explicit hybrid method constructed by us for systems of delay differential equations is convenient for programming, has a high speed of calculations of numerical solution, in comparison with implicit methods for such models of problems. Also, this method has no restrictions on the magnitude of the delay, in contrast to the hybrid methods with decomposition in Taylor series for delay. An accurate estimate of the local error of numerical integration by this method is made. The hybrid obtained method is applied to cross-waves study in hydrodynamic systems with delay. Regular and chaotic dynamics of the cross-waves are investigated.