Lyapunov-like Conditions of Forward Invariance and Boundedness for a Class of Unstable Systems

We provide Lyapunov-like characterizations of boundedness and convergence of non-trivial solutions for a class of systems with unstable invariant sets. Examples of systems to which the results may apply include interconnections of stable subsystems with one-dimensional unstable dynamics or critically stable dynamics. Systems of this type arise in problems of nonlinear output regulation, parameter estimation and adaptive control. In addition to providing boundedness and convergence criteria the results allow to derive domains of initial conditions corresponding to solutions leaving a given neighborhood of the origin at least once. In contrast to other works addressing convergence issues in unstable systems, our results require neither input-output characterizations for the stable part nor estimates of convergence rates. The results are illustrated with examples, including the analysis of phase synchronization of neural oscillators with heterogenous coupling

Mathematical model of movement of a single spherical lupine particle in the extractor using low-frequency mechanical vibrations

In our case, the solid body is the raw material of plant origin-lupine, crushed into grits, and the extractant is the cheese whey. The turbulent situation in the apparatus was created by the imposition of low-frequency mechanical vibrations, which have a significant impact on the characteristics of hydro-mechanical, mass transfer and thermal processes. This feature must be taken into account in the calculation of the extraction apparatus. The basic assumptions for the solution of the problem are formulated. The equation of motion of a single particle, which is contained in a number of works (Sow, an introduction, Chen, Protodyakonov, etc.). It is true in the instant values of the parameters. A simpler equation describing the motion of the dispersed particle and time correlation tensors with their subsequent decomposition into the Fourier integral are written. Further, taking into account the definition of tensors, the dependences for the calculation of the intensity of the chaotic motion of continuous and dispersed phases are shown, and the final expression is obtained, showing the ratio of the intensities of the phases. The coefficient of turbulent diffusion of each phase is proportional to the intensity of the chaotic motion of the corresponding phase. Therefore, the written finite equation for the phase ratio allows to estimate the ratio of the turbulent diffusion coefficients of the liquid and dispersed phases in the extraction apparatus. In our case, the ratio of the density of Hg / Hg is 1.1. Since the density of lupine and cheese whey differ quantitatively, we should expect some increase in the relative velocity of the phases, which will increase the rate of mass transfer. The intensities of the phases chaotic motion will not be the same, as well as the coefficients of turbulent diffusion. Thus, the case of motion of a single particle in a turbulent flow is complex and can be solved only under sufficiently serious assumptions formulated below