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Yu. A. Mitropol'skiî and O. B. Lykova, “On the integral manifold of differential equations with slow and fast motions,” Ukrain. Mat. Zh.,16, No. 2, 157–163 (1964).
V. V. Strygin and V. A. Sobolev, Separation of Motions by the Method of Integral Manifolds [in Russian], Nauka, Moscow (1988).
V. M. Gol'dshteîn and V. A. Sobolev, Qualitative Analysis of Singularly-Perturbed Systems [in Russian], Inst. Mat. (Novosibirsk), Novosibirsk (1988).
J. K. Hale, Ordinary Differential Equations, Waley Interscience, New York (1969).
A. B. Vasil'eva and V. F. Butuzov, Singularly-Perturbed Equations in Critical Cases [in Russian], Moscow Univ., Moscow (1978).
V. F. Butuzov and A. B. Vasil'eva, Asymptotic Methods in the Theory of Singular Perturbations [in Russian], Vysshaya Shkola, Moscow (1990).
Z.-M. Gu, N. N. Nefëdov, and R. E. O'Malley Jr., “On singular singularly perturbed initial value problems,” SIAM J. Appl. Math.,49, No. 1, 1–25 (1989).
V. I. Babushok, V. M. Gol'dshtein, and V. A. Sobolev, “Critical conditions for thermal explosion with reactant consumption,” Combust. Sci. Tech.,70, 81–89 (1990).
G. N. Gorelov and V. A. Sobolev, “Mathematical modeling of critical phenomena in thermal explosion theory,” Combust. Flame,87, 203–210 (1991).
G. N. Gorelov and V. A. Sobolev, “Duck-trajectories in a thermal explosion problem,” Appl. Math. Lett.,5, No. 6, 3–6 (1992).
P. V. Kokotović, H. K. Khatil, and J. O'Reily, Singular Perturbation Methods in Control: Analysis and Design, Academic Press, New York (1986).
V. A. Sobolev, “Integral manifolds and decomposition of singularly perturbed systems,” Systems Control Lett.,5, No. 4, 169–179 (1984).
J.-L. Callot, F. Diener, and M. Diener, “Le problème de la ‘chasse au canard’,” C. R. Acad. Sci. Paris Sér. I Math.,286, No. 22, 1059–1061 (1978).
A. K. Zvonkin and M. A. Shubin, “Nonstandard analysis and singular perturbations of ordinary differential equations,” Uspekhi Mat. Nauk,39, No. 2, 77–127 (1984).
V. I. Arnol'd, V. S. Afraîmovich, Yu. S. Il'yashenko, and L. P. Shil'nikov, “The theory of bifurcations,” Contemporary Problems of Mathematics. Fundamental Trends, Vol. 5 (Itogi Nauki i Tekhniki) [in Russian], VINITI, Moscow, 1986, pp. 5–218.
K. I. Chernyshëv, “On the asymptotic behavior of a solution to a linear reduced equation,” Sibirsk. Mat. Zh.,32, No. 3, 192–200 (1991).
M. A. Krasnosel'skiî, G. M. Vaînikko, P. P. Zabreîko, Ya. B. Rutitskiî, and V. Ya. Stetsenko, Approximate Solution of Operator Equations [in Russian], Nauka, Moscow (1969).
M. M. Vaînberg and V. A. Trenogin, The Branching Theory of Solutions to Nonlinear Equations [in Russian], Nauka, Moscow (1969).
V. M. Gol'dshteîn, L. I. Kononenko, M. Z. Lazman, V. A. Sobolev, and G. S. Yablonskiî, “Qualitative analysis of dynamical properties for an isothermal catalytic reactor of an ideal mixture,” in: Mathematical Problems in Chemical Kinetics [in Russian], Nauka, Novosibirsk, 1989, pp. 176–204.
K.-K. D. Young, P. V. Kokotović, and V. I. Utkin, “A singular perturbation analysis of high-gain feedback systems,” IEEE Trans. Automatic Control,22, No. 6, 931–938 (1977).
R. E. O'Malley Jr., “Singular perturbations and optimal control,” Lecture Notes in Math.,680, 171–218 (1978).
V. A. Sobolev, “Singular perturbations in a linearly quadratic problem of optimal control,” Avtomatika i Telemekhanika, No. 2, 53–64 (1991).
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Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 35, No. 6, pp. 1264–1278, November–December, 1994.
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Kononenko, L.I., Sobolev, V.A. Asymptotic decomposition of slow integral manifolds. Sib Math J 35, 1119–1132 (1994). https://doi.org/10.1007/BF02104713
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DOI: https://doi.org/10.1007/BF02104713