Regularization of numerical estimation of the sets of solutions of ODEs in stability problems on a finite time interval

Abstract: The sets of ODE solutions, with initial data belonging to the initial data regions, have complex boundaries (boundary surfaces in the dimension space). For the boundaries of the sets of solutions (surfaces in the space of solutions), it is impossible to choose formulas of functions with the help of which it was possible to describe the boundaries. As a result, there are two possibilities — either to describe the values of the boundary surfaces in a set of discrete points (on a grid), or to calculate their estimates of the maximum values in the directions of the coordinate axes, or the maximum in any chosen direction. The paper investigates and further uses the injectivity property of solutions to ODEs. For linear systems of ODEs the shift operator is linear and monomorphic (i.e., injective). These properties are also possessed by the resolving operator, which associates with the initial value the solution of the corresponding Cauchy problem (the entire solution, not its value at a point) as an element of space.

For nonlinear ODE systems that have unique solutions in a certain region of initial data, the boundaries of the regions of initial data pass into the boundaries of the regions of solutions at each specific moment in time. The class of such nonlinear ODE systems consists of systems whose solutions are uniformly bounded (Lagrange stable). Preliminarily, it is useful to construct a regularization of estimates for the boundaries of the solution sets, passing to the linear approximation of the original system. Regularization is understood as finding information about sets of exact solutions. This regularization establishes the values of compression / expansion in the given directions, offset along the time axis, and rotation through some angle. Examples of stability studies on a finite time interval are given.

classical analysis and ODEsdynamical systemsnumerical analysisnonlinear sciences

Audience: researchers in the topic

Mathematical models and integration methods