The paper describes asymptotic properties of a strongly nonlinear system , . The existence of an parametric family of solutions tending to zero is proved. Conditions posed on the system try to be independent of its linear approximation.
The paper gives the answer to the question of the number and qualitative character of stationary points of an autonomous detailed balanced kinetical system.
In the theory of autonomous perturbations of periodic solutions of ordinary differential equations the method of the Poincaré mapping has been widely used. For the analysis of properties of this mapping in the case of two-dimensional systems, a result first obtained probably by Diliberto in 1950 is sometimes used. In the paper, this result is (partially) extended to a certain class of autonomous ordinary differential equations of higher dimension.
The aim of the paper is to give some preliminary information concerning a class of nonlinear differential equations often used in physical chemistry and biology. Such systems are often very large and it is well known that where studying properties of such systems difficulties rapidly increase with their dimension. One way how to get over the difficulties is to use special forms of such systems.
The paper is devoted to the question whether some kind of additional information makes it possible to determine the fundamental matrix of variational equations in . An application concerning computation of a derivative of a scalar Poincaré mapping is given.
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