Simplification of weakly nonlinear systems and analysis of cardiac activity using them

<p style='text-indent:20px;'>The paper deals with the transformation of a weakly nonlinear system of differential equations in a special form into a simplified form and its relation to the normal form and averaging. An original method of simplification is proposed, that is, a way to determine the coefficients of a given nonlinear system in order to simplify it. We call this established method the degree equalization method, it does not require integration and is simpler and more efficient than the classical Krylov-Bogolyubov method of normalization. The method is illustrated with several examples and provides an application to the analysis of cardiac activity modelled using van der Pol equation.</p>

Конкуренция режимов колебаний плохо обтекаемого тела в воздушном потоке

Based on well-known mathematical models describing vibrations in the gas flow of a bluff body with one degree of freedom, a model of vibrations of a body with two degrees of freedom is proposed. The equations of transverse translational vibrations and rotational vibrations of an elastically fixed body around an axis perpendicular to the velocity vector of the incoming flow are obtained. Using the Krylov-Bogolyubov method in the first approximation, the equations are reduced to equations for slowly varying amplitudes and frequencies of vibrations. It turned out that the differential equations written for the squares of dimensionless amplitudes of translational and rotational vibrations coincide with the well-known Lotka-Volterra equations describing competition between two species of animals that eat the same food. The coefficients of the equations depend on the velocity of the incoming flow. The model is verified in the experiments in the wind tunnel.

Temperature and velocity relaxation in plasma. Spectral theory approach

The electron temperature and velocity relaxation of completely ionized plasma is studied on the basis of kinetic equation obtained from the Landau equation in a generalized Lorentz model. In this model contrary to the standard one ions form an equilibrium subsystem. Relaxation processes in the system are studied on the basis of spectral theory of the collision integral operator. This leads to an exact theory of relaxation processes of component temperatures and velocities equalizing. The relation of the developed theory with the Bogolyubov method of the reduced description of nonequilibrium systems is established, because the theory contains a proof of the relevant functional hypothesis, the idea of which is the basis of the Bogolyubov method. The temperature and velocity relaxation coefficients as eigenvalues of the collision integral operator are calculated by the method of truncated expansion of its eigenfunctions in the Sonine orthogonal polynomials. The coefficients are found in one- and two-polynomial approximation. As one can expect, convergence of this expansion is slow.