Abstract
For an autonomous system of \(N\) nonlinear ordinary differential equations considered on a semi-infinite interval \({{T}_{0}} \leqslant t < \infty \) and having a (pseudo)hyperbolic equilibrium point, the paper considers an \(n\)-dimensional \((0 < n < N)\) stable solution manifold, or a manifold of conditional Lyapunov stability, which, for each sufficiently large \(t\), exists in the phase space of the system’s variables in the neighborhood of its saddle point. A smooth separatrix saddle surface for such a system is described by solving a singular Lyapunov-type problem for a system of quasilinear first-order partial differential equations with degeneracy in the initial data. An application of the results to the correct formulation of boundary conditions at infinity and their transfer to the end point for an autonomous system of nonlinear equations is given, and the use of this approach in some applied problems is indicated.
REFERENCES
N. B. Konyukhova, “On the stationary Lyapunov problem for a system of first-order quasilinear partial differential equations,” Differ. Equations 30 (8), 1284–1294 (1994).
N. B. Konyukhova, “Stable Lyapunov manifolds for autonomous systems of nonlinear ordinary differential equations,” Comput. Math. Math. Phys. 34 (10), 1179–1195 (1994).
N. B. Konyukhova, “Smooth Lyapunov manifolds and singular boundary value problems,” in Reports on Applied Mathematics of Computing Center of the Russian Academy of Sciences (Vychisl. Tsentr Ross. Akad. Nauk, Moscow, 1996) [in Russian].
Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory (Springer-Verlag, New York, 1995).
A. M. Lyapunov, General Problem of the Stability of Motion (Princeton Univ. Press, Princeton, 1947).
S. Lefschetz, Differential Equations: Geometric Theory (Interscience, New York, 1963).
I. M. Gelfand and S. V. Fomin, Calculus of Variations (Fizmatlit, Moscow, 1961; Prentice Hall, Englewood Cliffs, N.J., 1963).
A. A. Abramov, “On the boundary conditions at a singular point for linear ordinary differential equations,” USSR Comput. Math. Math. Phys. 11 (1), 363–367 (1971).
A. A. Abramov and N. B. Konyukhova, “Transfer of admissible boundary conditions from a singular point of linear ordinary differential equations,” Sov. J. Numer. Anal. Math. Model. 1 (4), 245–265 (1986).
A. A. Abramov and N. B. Konyukhova, “Admissible boundary conditions at infinity or at a singular point for systems of linear ordinary differential equations,” Numer. Anal. Math. Model. 24, 181–198 (1990).
E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, New York, 1955).
B. P. Demidovich, Lectures on Mathematical Stability Theory (Nauka, Moscow, 1957) [in Russian].
Ju. L. Daleckiĭ and M. G. Kreĭn, Stability of Solutions of Differential Equations in Banach Space (Nauka, Moscow, 1970; Am. Math. Soc., Providence, 1974).
N. B. Konyukhova, “On the existence and uniqueness of solutions to singular Cauchy problems for systems of nonlinear functional-differential equations,” Dokl. Akad. Nauk SSSR 295 (4), 798–801 (1987).
N. B. Konyukhova, “On the existence of stable initial manifolds for systems of nonlinear functional-differential equations,” Dokl. Akad. Nauk SSSR 306 (3), 535–540 (1989).
N. B. Konyukhova, “On stable initial manifolds for systems of nonlinear functional-differential equations,” in Analytical and Numerical Methods for Solving Problems in Mathematical Physics (Vychisl. Tsentr Akad. Nauk SSSR, Moscow, 1989), pp. 136–154 [in Russian].
N. B. Konyukhova, “Singular Cauchy problems for some systems of nonlinear functional-differential equations,” Differ. Uravn. 31 (8), 1340–1347 (1995).
N. B. Konyukhova, “Singular Cauchy problems for systems of ordinary differential equations,” USSR Comput. Math. Math. Phys. 23 (3), 72–82 (1983).
P. Guan and Y. Y. Li, “\({{C}^{{1,1}}}\) estimates for solutions of a problem of Alexandrov,” Commun. Pure Appl. Math. 50, 789–811 (1997).
F. R. Gantmacher, The Theory of Matrices (Chelsea, New York, 1966).
Kh. D. Ikramov, Numerical Solution of Matrix Equations (Nauka, Moscow, 1984) [in Russian].
A. I. Zadorin, “Numerical solution of an equation with a small parameter on an infinite interval,” Comput. Math. Math. Phys. 38 (10), 1602–1614 (1998).
A. I. Zadorin, “Transfer of a boundary condition from infinity in the numerical solution of second-order equations with a small parameter,” Sib. Zh. Vychisl. Mat. 2 (1), 21–35 (1999).
N. B. Konyukhova and S. V. Kurochkin, “Singular nonlinear problems for self-similar solutions of boundary-layer equations with zero pressure gradient: Analysis and numerical solution,” Comput. Math. Math. Phys. 61 (10), 1603–1629 (2021).
N. B. Konyukhova and A. I. Sukov, “Smooth Lyapunov manifolds and correct mathematical simulation of nonlinear singular problems in mathematical physics,” Mathematical Modeling: Problems, Methods, Applications (Kluwer Academic/Plenum, New York, 2001), pp. 205–217.
N. B. Konyukhova and A. I. Sukov, “On correct statement of singular BVPs for autonomous systems of nonlinear ODEs with the applications to hydrodynamics,” Proceedings of the International Seminar “Day on Diffraction 2003,” St. Petersburg, Russia, June 24–27, 2003, Ed. by I. V. Andronov (IEEE Xplore, Digital Library, 2003), pp. 99–109. https://doi.org/10.1109/DD.2003.238181
Funding
This work was supported by ongoing institutional funding. No additional grants to carry out or direct this particular research were obtained.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
The author of this work declares that she has no conflicts of interest.
Additional information
Translated by E. Chernokozhin
Publisher’s Note.
Pleiades Publishing remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Konyukhova, N.B. Smooth Lyapunov Manifolds for Autonomous Systems of Nonlinear Ordinary Differential Equations and Their Application to Solving Singular Boundary Value Problems. Comput. Math. and Math. Phys. 64, 217–236 (2024). https://doi.org/10.1134/S0965542524020064
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0965542524020064