Abstract
The basic elements of the theory of slow invariant manifolds and canard phenomena of singularly perturbed nonlinear differential equations in the context of thermal-explosion problems are outlined. The mathematical results are applied to the investigation of the critical phenomena in autocatalytic combustion models described by singularly perturbed differential equations with lumped and distributed parameters. Critical regimes are modeled by canards (one-dimensional stable-unstable slow invariant manifolds). The geometric approach in combination with asymptotic and numeric methods permits to explain the strong parametric sensitivity and to obtain asymptotic representations of the critical conditions of self-ignition.
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References
Semenov NN (1958) Some problems of chemical kinetics and reactivity. I. Pergamon Press, New York, London, Paris
Zeldovich YaB, Barenblatt GI, Librovich VB, Makhviladze GM (1985) Mathematical theory of combustion and explosions. Plenum Press, New York
Frank-Kamenetskii DA (1969) Diffusion and heat transfer in chemical kinetics. Plenum Press, New York
Todes OM, Melent’ev PV (1939) The theory of thermal explosion. Zhurnal Fizichaskoi Khimii 13(7):52–58 (in Russian)
Merzhanov AG, Dubovitsky FI (1960) Quasi-stationary theory of the thermal explosion of a self-accelerating reaction. Zhurnal Fizichaskoi Khimii 34:2235–2244 (in Russian)
Merzhanov AG Dubovitskii FI (1966) Present state of the theory of thermal explosions. Russ Chem Rev 35:278–292
Gray BF (1973) Critical behaviour in chemical reacting systems: 2. An exactly soluble model. Combust Flame 21:317–325
Benoit E, Callot JL, Diener F, Diener M (1981–1982) Chasse au canard. Collectanea Mathematica 31–32:37–119 (in French)
Brøns M, Bar-Eli K (1994) Asymptotic analysis of canards in the EOE equations and the role of the inflaction line. Proc R Soc London A 445:305–322
Gorelov GN, Sobolev VA (1992) Duck-trajectories in a thermal explosion problem. Appl Math Lett 5(6):3–6
Gorelov GN, Sobolev VA (1991) Mathematical modeling of critical phenomena in thermal explosion theory. Combust Flame 87:203–210
Shchepakina E (2003) Black swans and canards in self–ignition problem. Nonlin Anal: Real World Applic 4:45–50
Sobolev VA, Shchepakina EA (1996) Duck trajectories in a problem of combustion theory. Differ Equ 32:1177–1186
Sobolev VA, Shchepakina EA (1993) Self-ignition of dusty media. Combus Explosion Shock Waves 29:378–381
Mishchenko EF, Rozov NKh (1980) Differential equations with small parameters and relaxation oscillations. Plenum Press, New York
Arnold VI, Afraimovich VS, Il’yashenko YuS, Shil’nikov LP (1994) Theory of bifurcations. In: Arnold V (ed) Dynamical systems, 5. Encyclopedia of mathematical sciences. Springer Verlag, New York, pp 1–205
Sobolev VA (1984) Integral manifolds and decomposition of singularly perturbed system. System Control Lett 5:169–179
Semenov NN (1928) Zur theorie des verbrennungsprozesses. Z Physik Chem 48:571–581 (in German)
Babushok VI, Gol’dshtein VM, Romanov AS, Babkin VS (1992) Thermal explosion in an inert porous medium. Combust Explosion Shock Waves 28:319–325
Gol’dshtein V, Zinoviev A, Sobolev V, Shchepakina E (1996) Criterion for thermal explosion with reactant consumption in a dusty gas. Proc R Soc London A 452:2103–2119
Schneider K, Shchepakina E, Sobolev V (2003) A new type of travelling wave. Math Methods Appl Sci 26:1349–1361
Henry D (1981) Geometrical theory of semilinear parabolic equations. Lecture Notes in Mathematics, vol 840. Springer Verlag, New York
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Gorelov, G.N., Shchepakina, E.A. & Sobolev, V.A. Canards and critical behavior in autocatalytic combustion models. J Eng Math 56, 143–160 (2006). https://doi.org/10.1007/s10665-006-9047-0
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DOI: https://doi.org/10.1007/s10665-006-9047-0