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

Kuptsov, M. I.; Terekhin, M. T.; Tenyaev, Victor

doi:10.15507/2079-6900.19.201701.076-084

Cited by 1 publication

(3 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Application of these methods made it possible to obtain new sufficient conditions for the existence of both stable and unstable local integral manifolds of the system (1). The results obtained develop and generalize the results of [3][4][5][6][7][8].…”

Section: Introduction Consider the Following System Of Ordinary Diffe...supporting

confidence: 83%

“…We omit the proof of Theorem 1 (for detail, see [7,8]). Note that for the integral manifold Ψ(φ 0 , t) whose existence is stated in Theorem 1, the following equality holds:…”

Section: Existence Of An Integral Manifoldmentioning

confidence: 99%

“…Then Eqs. (7) and (8) respectively can be rewritten in the form φ = sin t + ε 1 (φ 0 ) + ε 2 (φ 0 ) + F (φ 0 , t) cos φ + F 2 (φ 0 , t) sin 2 φ, ẋ = ε 1 (φ 0 ) − ε 2 1 (φ 0 ) + F 2 (φ 0 , t) x, and the fundamental matrix of Eq. ( 8) is as follows:…”

Section: Definitionmentioning

confidence: 99%

See 2 more Smart Citations

On the Stability of Integral Manifolds of a System of Ordinary Differential Equations in the Critical Case

et al. 2022

View full text Add to dashboard Cite

In this paper, we consider the stability problem for nonzero integral manifolds of a nonlinear, finite-dimensional system of ordinary differential equations whose right-hand side is a vector-valued function containing a parameter and periodic in an independent variable. We assume that the system possesses a trivial integral manifold for all values of the parameter and the corresponding linear subsystem does not possess the exponential dichotomy property. We find sufficient conditions for the existence of a nonzero integral manifold in a neighborhood of the equilibrium of the system and conditions for its stability or instability. For this purpose, based of the ideas of the Lyapunov method and the method of transform matrices, we construct operators that allow one to reduce the solution of this problem to the search for fixed points.

show abstract

Section: Introduction Consider the Following System Of Ordinary Diffe...supporting

confidence: 83%

“…We omit the proof of Theorem 1 (for detail, see [7,8]). Note that for the integral manifold Ψ(φ 0 , t) whose existence is stated in Theorem 1, the following equality holds:…”

Section: Existence Of An Integral Manifoldmentioning

confidence: 99%