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8.8 Guide to the Literature
Anosov, D.V. (1960), On limit cycles in systems of differential equations with a small parameter in the highest derivatives, Mat. Sb. 50(92), pp. 299–334; translated in AMS Trans.. Ser. 2, vol. 33, pp. 233–276.
Flaherty, J.E. and O’Malley, R.E. (1980), Analysis and numerical methods for nonlinear singular singularly perturbed initial value problems, SIAM J. Appl. Math. 38, pp. 225–248.
Flatto, L. and Levinson, N (1955), Periodic solutions of singularly perturbed equations, J. Math. Mech. 4, pp. 943–950.
Grasman, J. (1987), Asymptotic Methods of Relaxation Oscillations and Applications, Applied Mathematical Sciences 63, Springer-Verlag, New York.
Hale, J.K. (1963), Oscillations in Nonlinear Systems, McGraw-Hill, New York, reprinted Dover, New York (1992). Chapter 5
Hoppensteadt, F. (1967), Stability in systems with parameters, J. Math. Anal. Appl. 18, pp. 129–134.
Hoppensteadt, F. (1969), Asymptotic series solutions for nonlinear ordinary differential equations with a small parameter, J. Math. Anal. Appl. 25, pp. 521–536.
Jones, C.K.R.T. (1994), Geometric singular perturbation theory, in Dynamical Systems, Montecatini Terme 1994 (Johnson, R., ed.), Lecture Notes in Mathematics 1609, pp. 44–118, Springer-Verlag, Berlin.
Kaper, T.J. (1999), An introduction to geometric methods and dynamical systems theory for singular perturbation problems, in Proceedings Symposia Applied Mathematics 56: Analyzing Multiscale Phenomena Using Singular Perturbation Methods, (Cronin, J. and O’Malley, Jr., R.E., eds.). pp. 85–131, American Mathematical Society, Providence, RI.
Kaper, T.J. and Jones, C.K.R.T. (2001), A primer on the exchange lemma for fast-slow systems, IMA Volumes in Mathematics and its Applications 122: Multiple-Time-Scale Dynamical Systems, (Jones, C.K.R.T., and Khibnik, A.I., eds.). Springer-Verlag, New York.
Lebovitz, N.R. and Schaar, R.J. (1975), Exchange of stabilities in autonomous systems, Stud. in Appl. Math. 54, pp. 229–260.
Lebovitz, N.R. and Schaar, R.J. (1977), Exchange of stabilities in autonomous systems-II. Vertical bifurcation., Stud. in Appl. Math. 56, pp. 1–50.
Mishchenko, F.F. and Rosov, N.Kh. (1980), Differential Equations with Small Parameters and Relaxation Oscillations, Plenum Press, New York.
Neishtadt, A.I. (1991), Averaging and passage through resonances, Proceedings International Congress Mathematicians 1990, Math. Soc. Japan, Kyoto, pp. 1271–1283, Springer-Verlag, Tokyo; see also the paper and references in Chaos 1, pp. 42–48 (1991).
O’Malley, Jr., R.E. (1968), Topics in singular perturbations, Adv. Math. 2, pp. 365–470.
O’Malley, Jr., R.E. (1971), Boundary layer methods for nonlinear initial value problems, SIAM Rev. 13, pp. 425–434.
O’Malley, Jr., R.E. (1991), Singular Perturbation Methods for Ordinary Differential Equations, Applied Mathematical Sciences 89, Springer-Verlag, New York.
Shiskova, M.A. (1973), Examination of a system of differential equations with a small parameter in the highest derivatives, Dokl. Akad. Nauk 209(3), pp. 576–579, transl. in Sov. Math. Dokl. 14, pp. 483–487.
Tikhonov, A.N. (1952), Systems of differential equations containing a small parameter multiplying the derivative (in Russian), Mat. Sb. 31(73), pp. 575–586.
Vasil’eva, A.B. (1963), Asymptotic behaviour of solutions to certain problems involving nonlinear differential equations containing a small parameter multiplying the highest derivatives, Russ. Math. Surv. 18, pp. 13–84.
Vasil’eva, A.B., Butuzov, V.F. and Kalachev, L.V. (1995), The Boundary Function Method for Singular Perturbation Problems, SIAM Studies in Applied Mathematics 14, SIAM, Philadeplhia.
Verhulst, F. (1976), Matched asymptotic expansions in the two-body problem with quick loss of mass, J. Inst. Math. Appl. 18, pp. 87–98.
Wasow, W. (1965), Asymptotic Expansions for Ordinary Differential Equations, Interscience, New York.
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(2005). Boundary Layers in Time. In: Methods and Applications of Singular Perturbations. Texts in Applied Mathematics, vol 50. Springer, New York, NY. https://doi.org/10.1007/0-387-28313-7_8
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