Dynamics of decisions of quadratic systems on the plane.

*(Ukrainian. English summary)*Zbl 1363.93191Summary: The nonlinear run-time systems described by ordinary differential equalizations with quadratic right part found wide application in the mathematical models of population, economy, biology. They display an sufficiently adequate level the dynamics of leak processes, take into account the limited nature of area of residence of population, reverse effects in economic processes and other. As known, solving systems of linear stationary differential equations can be done using an exponential matrix, and the rate decision using the extreme eigenvalues of symmetric positive definite matrices, included in the Lyapunov matrix equation. For nonlinear dependency of general form similar systems do not exist. In this work, two nonlinear differential equations with quadratic nonlinearity of the general form are considered:

Systems recorded in the universal vector matrix form. It is assumed that the linear part of the system is asymptotically stable. Calculation of “guaranteed stability of the region” and estimates of the convergence of solutions in this area are carried out using a quadratic Lyapunov function. The total derivative of a quadratic Lyapunov function is calculated. The inequality for the derivative is of the form is similar to the scalar Riccati equation. The solution of the inequality and the rate of inequality for differential equations with quadratic right-hand side of the general form of the plane are presented.

In the second part, we consider a system of differential equations of “quasi-linear type”. Systems of this kind are the “extension” of linear time-invariant systems. In ecological models they are used for studies in a limited area of populations residing. Estimates of the convergence of solutions of “quasi linear” systems with initial data from a small neighborhood of the zero equilibrium position are given. Basic results are got with the use of the second Lyapunov method.

Systems recorded in the universal vector matrix form. It is assumed that the linear part of the system is asymptotically stable. Calculation of “guaranteed stability of the region” and estimates of the convergence of solutions in this area are carried out using a quadratic Lyapunov function. The total derivative of a quadratic Lyapunov function is calculated. The inequality for the derivative is of the form is similar to the scalar Riccati equation. The solution of the inequality and the rate of inequality for differential equations with quadratic right-hand side of the general form of the plane are presented.

In the second part, we consider a system of differential equations of “quasi-linear type”. Systems of this kind are the “extension” of linear time-invariant systems. In ecological models they are used for studies in a limited area of populations residing. Estimates of the convergence of solutions of “quasi linear” systems with initial data from a small neighborhood of the zero equilibrium position are given. Basic results are got with the use of the second Lyapunov method.

##### MSC:

93D05 | Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory |

34D20 | Stability of solutions to ordinary differential equations |

93C15 | Control/observation systems governed by ordinary differential equations |