REFERENCES
H. Amann, “Existence and stability of solutions for semi-linear parabolic systems and applications to some diffusion-reaction equations,” Proc. Roy. Soc. Edinburgh, Sec. A, 81, 35–47 (1978).
E. Benoît (Ed.), Dynamic Bifurcation, Lect. Notes Math., 1493, Springer-Verlag, New York (1991).
V. F. Butuzov, “Singularly perturbed parabolic equations in case of intersecting roots of the degenerate equation,” Russ. J. Math. Phys., 9, 50–59 (2002).
V. F. Butuzov and E. A. Gromova, “Limit theorem for a Tikhonov system of equations,” Comput. Math. Math. Phys., 40, 669–679 (2000).
V. F. Butuzov and E. A. Gromova, “A boundary-value problem for a system of fast and slow second-order equations in the case of intersecting roots of the degenerate equation,” Zh. Vychisl. Mat. Mat. Fiz., 41, 1165–1179 (2001).
V. F. Butuzov and N. N. Nefedov, “Singularly perturbed boundary-value problems for a second-order equation in case of exchange of stability,” Mat. Zametki, 63, 354–362 (1998).
V. F. Butuzov, N. N. Nefedov, and K. R. Schneider, Singularly perturbed boundary-value problems in case of exchange of stabilities,” J. Math. Anal. Appl., 229, 543–562 (1999).
V. F. Butuzov, N. N. Nefedov, and K. R. Schneider,” “Singularly perturbed boundary-value problems for systems of Tikhonov's type in case of exchange of stabilities,” J. Differ. Equations, 159, 427–446 (1999).
V. F. Butuzov, N. N. Nefedov, and K. R. Schneider, “Singularly perturbed reaction-diffusion systems in cases of exchange of stabilities,” Nat. Resour. Model., 13, 247–269 (2000).
V. F. Butuzov, N. N. Nefedov, and K. R. Schneider, Singularly perturbed partly dissipative reaction-diffusion systems in case of exchange of stabilities, Preprint No. 572, Weierstraß-Inst. Angew. Anal. Stochastik, Berlin (2000).
V. F. Butuzov, N. N. Nefedov, and K. R. Schneider, “Singularly perturbed elliptic problems in the case of exchange of stabilities,” J. Differ. Equations, 169, 373–395 (2001).
V. F. Butuzov, N. N. Nefedov, and K. R. Schneider, On a class of singularly perturbed partly dissipative reaction-diffusion systems, Preprint No. 646, Weierstraß-Inst. Angew. Anal. Stochastik, Berlin (2001).
V. F. Butuzov and A. V. Nesterov, “On some singularly perturbed problems with nonsmoooth boundary functions,” Dokl. Akad. Nauk SSSR, 263, 786–789 (1982).
V. F. Butuzov and I. Smurov, “Initial-boundary-value problem for a singularly perturbed parabolic equation in case of exchange of stability,” J. Math. Anal. Appl., 234, 183–192 (1999).
V. F. Butuzov and A. B. Vasil'eva, “Singularly perturbed problems with boundary and interior layers: theory and applications,” Adv. Chem. Phys., 97, 47–179 (1997).
V. F. Butuzov, A. B. Vasil'eva, and M. V. Fedoryuk, “Asymptotic methods in the theory of ordinary differential equations,” In: Itogi Nauki Tekh., Ser. Mat. Anal., All-Union Institute for Scientific and Technical Information, Moscow (1967).
J. L. Callot, F. Diener, and M. Diener, “Le probleme de la 'chasse au canard',” C. R. Acad. Sci., Paris, Ser. A, 286, 1059–1061 (1978).
K. W. Chang and F. A. Howes, Nonlinear Singular Perturbation Phenomena: Theory and Application, Springer-Verlag, New York (1984).
S. A. Chaplygin, A New Method for the Integration of Differential Equations [in Russian], GITL, Moscow-Leningrad (1950).
J. Cronin, Mathematical Aspects of Hodgkin-Huxley Neural Theory, Cambridge Stud. Math. Biology, 7, Cambridge Univ. Press, Cambridge (1987).
J. Cronin and R. E. O'Malley, Jr. (Eds.), Analyzing Multiscale Phenomena Using Singular Perturbation Methods, Proc. Symp. Appl. Math., 56, Am. Math. Soc., Providence, Rhode Island (1999).
M. Diener, “Deux nouveaux 'phenomenes-canard,'” C. R. Acad. Sci., Paris, Ser. A, 290, 541–544 (1980).
F. Dumortier and B. Smits, “Transition time analysis in singularly perturbed boundary-value problems,” Trans. Am. Math. Soc., 347, 4129–4145 (1995).
F. Dumortier and R. Roussarie, Canard Cycles and Center Manifolds, Mem. Am. Math. Soc., 577, Am. Math. Soc., Providence, Rhode Island (1996).
W. Eckhaus, “Relaxation oscillations including a standard chase on french ducks,” Lect. Notes Math., 985, 449–494 (1983).
W. Eckhaus and E. M. de Jager (Eds.), Theory and Applications of Singular Perturbations, Lect. Notes Math., 942, Springer-Verlag, Berlin (1982).
T. Erneux and P. Mandel, “Imperfect bifurcation with a slowly-varying control parameter,” SIAM J. Appl. Math., 46, 1–15 (1986).
T. Erneux and P. Mandel, “The slow passage through a steady state bifurcation: Delay and memory effects,” J. Stat. Phys., 48, 1059–1069 (1987).
P. Fabrie and C. Galusinski, “Exponential attractors for a partially dissipative reaction system,” Asympt. Anal., 12, 329–354 (1996).
M. V. Fedoryuk, “Equations with rapidly oscillating solutions,” In: Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. Fundam. Napravl., 34, All-Union Institute for Scientific and Technical Information, Moscow (1988), pp. 5–56.
N. Fenichel, “Geometric singular perturbation theory for ordinary differential equations,” J. Differ. Equations, 31, 53–98 (1979).
L. S. Frank, Singular Perturbations in Elasticity Theory, Anal. Its Appl., 1, IOS Press, Amsterdam (1997).
J. Grasman, Asymptotic Methods for Relaxation Oscillations and Applications, Appl. Math. Sci., 63, Springer-Verlag, New York (1987).
J. Guckenheimer and Yu. Ilyashenko, “The duck and the devil: Canards on the staircase,” Moscow Math. J., 1, 27–47 (2001).
V. V. Gudkov, Yu. A. Klokov, Ya. Lepin, and V. D. Ponomarev, Two-Point Boundary-Value Problems for Ordinary Differential Equations, [in Russian], Zinatne, Riga (1973).
R. Haberman, “Slowly varying jump and transition phenomena associated with algebraic bifurcation problems,” SIAM J. Appl. Math., 37, 69–106 (1979).
K. P. Hadeler, “Quasimonotone systems and convergence to equilibrium in a population genetic model,” J. Math. Anal. Appl., 95, 297–303 (1983).
M. W. Hirsch, “Systems of differential equations which are competitive or cooperative, I. Limit sets,” SIAM J. Math. Anal., 13, 167–179 (1982).
S. L. Hollis and J. J. Morgan, “Partly dissipative reaction-diffusion systems and a model of phosphorus diffusion in silicon,” Nonlin. Anal. Theory Methods Appl., 19, 427–440 (1992).
F. Hoppensteadt, “Properties of solutions of ordinary differential equations with small parameters,” Commun. Pure Appl. Math., 24, 807–840 (1971).
F. C. Hoppenstedt and E. M. Izhikevich, Weakly Connected Neural Networks, Appl. Math. Sci., 126, Springer-Verlag, New York (1997).
A. M. Il'in, “The boundary layer,” In: Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. Fundam. Napravl., 34, All-Union Institute for Scientific and Technical Information, Moscow (1988), pp. 175–213.
L. K. Jackson, “Sub-functions and second-order ordinary differential equations,” Adv. Math., 2, 308–363 (1968).
S. Karimov, “The asymptotics of the solutions of some classes of differential equations with a small parameter at the highest derivatives in case of exchange of stability of the equilibrium point in the plane of fast motions,” Differ. Equations, 21, 1136–1139 (1985)
J. P. Keener, Principles of Applied Mathematics: Transformation and Approximation, Addison-Wesley, Redwood City (1988).
J. K. Kevorkian and J. D. Cole, Multiple Scale and Singular Perturbation Methods, Appl. Math. Sci., 114, Springer-Verlag, Berlin (1996).
P. Kokotovic, A. Bensoussan, and G. Blankenship (Eds.), Singular Perturbations and Asymptotic Analysis in Control Systems, Lect. Notes Control Information Sci., 90, Springer-Verlag, Berlin (1987).
P. Kokotovic, H. K. Khalil, and J. O'Reilly, Singular Perturbation Methods in Control: Analysis and Design, Classics in Appl. Math., 25, SIAM, Philadelphia (1999).
A. Yu. Kolesov, E. F. Mishchenko, and N. Kh. Rozov, “Solution to singularly perturbed boundary-value problems by the duck hunting method,” Proc. Steklov Inst. Math., 224, 169–188 (1999).
A. Yu. Kolesov and N. Kh. Rozov, “'Chase on ducks' in the investigation of singularity perturbed boundary-value problems,” Differ. Equations, 35, 1374–1383 (1999).
Yu. S. Kolesov and N. Kh. Rozov, “Duck cycles of difference-differential equations,” Mosc. Univ. Math. Bull., 50,No. 1, 10–14 (1995).
M. Krupa and P. Szmolyan, “Relaxation oscillation and canard explosion,” J. Differ. Equations, 174, 312–368 (2001).
M. Krupa and P. Szmolyan, “Extending geometric singular perturbation theory to nonhyperbolic points—fold and canard points in two dimensions,” SIAM J. Math. Anal., 33, 286–314 (2001).
N. R. Lebovitz, “Bifurcation and unfolding in systems with two time scales,” Ann. N. Y. Acad. Sci., 617, 73–86 (1990).
N. R. Lebovitz and A. I. Pesci, “Dynamic bifurcation in Hamiltonian systems with one degree of freedom,” SIAM J. Appl. Math., 55, 1117–1133 (1995).
N. R. Lebovitz and R. J. Schaar, “Exchange of stabilities in autonomous systems,” Stud. Appl. Math., 54, 229–260 (1975).
N. R. Lebovitz and R. J. Schaar, “Exchange of stabilities in autonomous systems, II. Vertical bifurcation,” Stud. Appl. Math., 56, 1–50 (1977).
M. Levinson, “Perturbations of discontinuous solutions of nonlinear systems of differential equations,” Acta Math., 82, 71–106 (1951).
M. Marion, “Inertial manifolds associated to partly dissipative reaction-diffusion systems,” J. Math. Anal. Appl., 143, 295–326 (1989).
M. Marion, “Finite-dimensional attractors associated with partly dissipative reaction-diffusion systems,” SIAM J. Math. Anal., 20, 816–844 (1989).
E. F. Mishchenko and N. Kh. Rozov, Differential Equations with Small Parameter and Relaxation Oscillations, Plenum Press, New York (1980).
E. F. Mishchenko, Yu. S. Kolesov, A. Yu. Kolesov, and N. Kh. Rozov, Asymptotic Methods in Singularly Perturbed Systems, Consultants Bureau, New York (1994).
R. M. Miura (Ed.), Some Mathematical Questions in Biology and Neurobiology, Lect. Notes Math. Life Sci., 15, Am. Math. Soc., Providence, Rhode Island (1982).
M. Müller, “Über das Fundamentaltheorem in der Theorie der gewöhnlichen Differentialgleichungen,” Math. Z., 26, 619–645 (1926).
J. A. Murdock, Perturbations: Theory and Methods, Wiley, New York (1991).
J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin (1993).
M. Nagumo, “Über das Randwertproblem der nichtlinearen gewöhnlichen Differentialgleichungen zweiter Ordnung,” Proc. Phys.-Math. Soc. Japan, III. Ser., 24, 845–851 (1942).
N. N. Nefedov, “The method of differential inequalities for some singularly perturbed partial differential problems,” Differ. Uravn., 31, 719–722 (1993).
N. N. Nefedov and K. R. Schneider, Delayed exchange of stabilities in singularly perturbed systems, Preprint No. 270, Weierstraβ-Inst. Angew. Anal. Stochastik, Berlin (1996).
N. N. Nefedov and K. R. Schneider, “Immediate exchange of stabilities in singularly perturbed systems,” Differ. Int. Equations, 12, 583–599 (1999).
N. N. Nefedov, K. R. Schneider, and A. Schuppert, Jumping behavior in singularly perturbed systems modelling bimolecular reactions, Preprint No. 137, Weierstraβ-Inst. Angew. Anal. Stochastik, Berlin (1994).
N. N. Nefedov, K. R. Schneider, and A. Schuppert, “Jumping behavior of the reaction rate of fast bimolecular reactions,” Z. Angew. Math. Mech., 76,No. 2, 69–72 (1996).
A. I. Neishtadt, “On delayed stability loss under dynamic bifurcations, I,” Differ. Uravn., 23, 2060–2067 (1987).
A. I. Neishtadt, “On delayed stability loss under dynamic bifurcations, II,” Differ. Uravn., 24, 226–233 (1988).
R. E. O'Malley, Jr., Singular Perturbation Methods for Ordinary Differential Equations, Springer-Verlag, New York (1991).
D. Panazzolo, “On the existence of canard solutions,” Publ. Mat. Barc., 44, 503–592 (2000).
C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York-London (1992).
L. S. Pontryagin and E. F. Mishchenko, “Some questions in the theory of differential equations with a small parameter,” Proc. Steklov Inst. Math., 169,No. 4, 103–122 (1986).
S. Schecter, “Persistent unstable equilibria and closed orbits of singularly perturbed equations,” J. Differ. Equations, 60, 131–141 (1985).
K. R. Schneider, “On the existence of wave trains in partly dissipative systems,” In: Proc. Int. Conf. Differential Equations (C. Perello, C. Simo, and J. Sola-Morales, Eds.), 2, World Scientific, Singapore (1993), pp. 893–898.
Z. Shao, “Existence of inertial manifolds for partly dissipative reaction-diffusion systems in higher space dimensions,” J. Differ. Equations, 144, 1–43 (1998).
M. A. Shishkova, “Study of a system of differential equations with a small parameter at the highest derivatives,” Dokl. Akad. Nauk SSSR, 209, 576–579 (1973).
E. A. Shchepakina and V. A. Sobolev, “Integral manifolds, canards, and black swans,” Nonlin. Anal., Theory, Methods, Appl., 44, 897–908 (2001).
V. A. Sobolev and E. A. Shchepakina, “Integral surfaces of duck-trajectories with changing stability,” Izv. Ross. Akad. Estestv. Nauk, Ser. MMMIU, 1,No. 3, 151–175 (1997).
A. N. Tikhonov, “On the dependence of solutions of differential equations on a small parameter,” Mat. Sb., 64, 193–204 (1948).
A. N. Tikhonov, “Systems of differential equations containing small parameters,” Mat. Sb., 73, 575–586 (1952).
A. B. Vasil'eva and V. F. Butuzov, Asymptotic Expansions of Solutions of Singularly Perturbed Equations [in Russian], Nauka, Moscow (1973).
A. B. Vasil'eva and V. F. Butuzov, Asymptotic Methods in the Theory of Singular Perturbation [in Russian], Vysshaya Shkola, Moscow (1990).
A. B. Vasil'eva, V. F. Butuzov, and L. V. Kalachev, The Boundary Function Method for Singular Perturbation Problems, SIAM Stud. Appl. Math., Philadelphia (1995).
A. B. Vasil'eva and M. G. Dimitriev, “Singular perturbations in problems of optimal control,” in: Itogi Nauki Tekhn., Ser. Mat. Anal., 20, All-Union Institute for Scientific and Technical Information, Moscow (1982), pp. 3–77.
W. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Wiley, New York (1965).
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Butuzov, V.F., Nefedov, N.N. & Schneider, K.R. Singularly Perturbed Problems in Case of Exchange of Stabilities. Journal of Mathematical Sciences 121, 1973–2079 (2004). https://doi.org/10.1023/B:JOTH.0000021571.21423.52
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DOI: https://doi.org/10.1023/B:JOTH.0000021571.21423.52