To the problem of existence of integral manifolds systems of differential equations not solved with respect to derivatives

M. I. kuptsov ; M. T. Teryokhin ; V. V. Tenyaev

Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 19 (2017) no. 2, pp. 76-84

Résumé

The issue of the article is finding a local non-zero integral manifold of a nonlinear $(n+m)$-dimensional system of ordinary differential equations that is not solved with respect to derivatives. It is assumed that the examined system has $n$-dimensional trivial integral manifold for all parameter values and that corresponding linear subsystem has the $m$-parametric family of periodic solutions. In particular it means that the linear system does not have the property of exponential dichotomy. It is allowed for the linear approximation matrix to be a function of independent variable when the parameter value is zero. The problem of existence of integral manifolds is reduced to the issue of operator equations’ solution in the space of bounded Lipschitz-continuous periodic vector functions. Linearization is used to prove the existence of integral manifolds of the original system; the method of transforming matrix is used here. This method may be extended on the case of absence of a linearity in the parameter of the operator equations members. The sufficient conditions of existence of $n$-dimensional nonzero periodic integral manifold in the neighborhood of the equilibrium state of the system were obtained.

Détail

Keywords: the method of transforming matrix, integral manifold, ordinary differential equations’ system, operator equation, dimensional reduction of phase space.