TY - JOUR

T1 - Existence of an Invariant Foliation Near a Locally Integral Surface of Neutral Type

AU - Il’in, Yu.

N1 - Il’in, Y. Existence of an Invariant Foliation Near a Locally Integral Surface of Neutral Type. Lobachevskii J Math 42, 3438–3450 (2021). https://doi.org/10.1134/S199508022202010X

PY - 2022

Y1 - 2022

N2 - We consider a system of essentially nonlinear differential equations that does not have linear terms on the right-hand side in a neighbourhood of the rest point. Earlier, for this system, the author proved the existence of two locally integral surfaces of so-called ‘‘stable’’ and ‘‘neutral’’ types. In this article, we prove the existence of a foliation into surfaces of stable type in some neighborhood of a neutral surface under the additional assumption that the zero solution on this surface is Lyapunov uniformly stable. This result generalizes the well-known one for quasilinear systems of ODEs. Instead of assumptions on the eigenvalues of the linear approximation, we use conditions on the logarithmic norms of the Jacobi matrices of the right-hand sides. The result obtained is important for describing the behavior of integral curves of complicated systems in a neighborhood of a stationary point, for the theory of stability of solutions, for local equivalence of ODEs.

AB - We consider a system of essentially nonlinear differential equations that does not have linear terms on the right-hand side in a neighbourhood of the rest point. Earlier, for this system, the author proved the existence of two locally integral surfaces of so-called ‘‘stable’’ and ‘‘neutral’’ types. In this article, we prove the existence of a foliation into surfaces of stable type in some neighborhood of a neutral surface under the additional assumption that the zero solution on this surface is Lyapunov uniformly stable. This result generalizes the well-known one for quasilinear systems of ODEs. Instead of assumptions on the eigenvalues of the linear approximation, we use conditions on the logarithmic norms of the Jacobi matrices of the right-hand sides. The result obtained is important for describing the behavior of integral curves of complicated systems in a neighborhood of a stationary point, for the theory of stability of solutions, for local equivalence of ODEs.

KW - essentially nonlinear differential equations

KW - locally integral manifolds

KW - invariant foliation

KW - logarithmic norms

KW - stability

U2 - 10.1134/S199508022202010X

DO - 10.1134/S199508022202010X

M3 - Article

VL - 43

SP - 378

EP - 390

JO - Lobachevskii Journal of Mathematics

JF - Lobachevskii Journal of Mathematics

SN - 1995-0802

IS - 2

ER -