FLOW CURVATURE APPLIED TO MODELLING OF CRITICAL PHENOMENA
- Authors: Balabaev M.O.1
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Affiliations:
- Samara National Research University
- Issue: Vol 25, No 2 (2019)
- Pages: 92-99
- Section: Articles
- URL: https://journals.ssau.ru/est/article/view/7480
- DOI: https://doi.org/10.18287/2541-7525-2019-25-2-92-99
- ID: 7480
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Abstract
Modeling of critical phenomena is a very important problem, which has direct applied application in many branches of science and technology. In this paper we regard a modification of the low curvature method applied to construction of invariant manifolds of autonomous fast-slow dynamic systems. We compared a new method with original ones via finding duck-trajectories and their multidimensional analogues surfaces with variable stability. Comparison was used a three-dimensional autocatalytic reaction model and a model of the burning problem.
About the authors
M. O. Balabaev
Samara National Research University
Author for correspondence.
Email: morenov@ssau.ru
ORCID iD: 0000-0001-6761-104X
postgraduate student of the Department of Differential Equations and Control Theory
References
- Sobolev V.A., Shchepakina E.A. Reduktsiya modelei i kriticheskie yavleniya v makrokinetike . M.: Fizmatlit, 2010, 320 p. Available at: http://www.studentlibrary.ru/book/ISBN9785922112697.html .
- Benoit E., Calot J.L., Diener M. Chasse au Canard. Collectanea Mathematica, 1981, V. 31–32, pp. 37–119. Available at: https://www.academia.edu/19856616/Chasse_au_canard.
- .
- Diener M. Nessie et les Canards. Strasbourg: Publication IRMA, 1979. .
- Gorelov G.N., Sobolev V. A. Duck-trajectories in a Thermal Explosion Problem. Applied Mathematical Letters, 1992, V. 5, pp. 3–6. doi: 10.1016/0893-9659(92)90002-Q
- Sobolev V.A., Shchepakina E.A. Self-ignition of Dusty Media. Combustion: Explosion and Shock Waves, 1993, V. 29, pp. 378–381. doi: 10.1007/BF00797664 .
- Goldshtein V., Zinoviev A., Sobolev V., Shchepakina E. Criterion for Thermal Explosion with Reactant Consumption in a Dusty Gas. Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences, 1996, V. 452, pp. 2103–2119. doi: 10.1098/rspa.1996.0111 .
- Sobolev V.A., Shchepakina E.A. Duck Trajectories in a Problem of Combustion Theory. Differential Equations, 1996, V. 32, pp. 1177–1186. Available at: https://elibrary.ru/item.asp?id=13231308
- Gavin C., Pokrovskii A., Prentice M., Sobolev V. Dynamics of a Lotka-Volterra Type Model with Applications to Marine Phage Population Dynamics. Journal of Physics: Conference Series, 2006, V. 55, pp. 80–93. doi: 10.1088/1742-6596/55/1/008 .
- Shchepakina E., Korotkova O. Condition for Canard Explosion in a Semiconductor Optical Amplier. Journal of the Optical Society of America B: Optical Physics, 2011, V. 28, pp. 1988–1993. doi: 10.1364/JOSAB.28.001988 .
- Shchepakina E., Korotkova O. Canard Explosion in Chemical and Optical Systems. Discrete and Continuous Dynamical Systems (series B), 2013, V. 18, pp. 495–512. doi: 10.3934/dcdsb.2013.18.495 .
- Shchepakina E., Sobolev V. Integral Manifolds, Canards and Black Swans. Nonlinear Analysis. Ser. A: Theory Methods, 2001, V. 44, pp. 897–908. doi: 10.1016/S0362-546X(99)00312-0 .
- Shchepakina E.A., Sobolev V.A., Mortell M.P. Introduction to system order reduction methods with applications. Lecture notes in math, 2014, Vol. 2114, 201 p. Available at: https://pl.b-ok.cc/book/2466101/06d4ed
- Sobolev V. A., Shchepakina E. A. Standard Chase on Black Swans and Canards. Preprint N 426. Berlin: WIAS, 1998. Available at: https://pdfs.semanticscholar.org/ec52/384507ac225541417f856ce8a02de7680a66.pdf?_ga= 2.63816171.510207900.1570716853-18694247.1570716853 .
- Shchepakina E. Canards and Black Swans in Model of a 3-D Autocatalator. Journal of Physics: Conference Series, 2005, V. 22, pp. 194–207. doi: 10.1088/1742-6596/22/1/013 .
- Balabaev M. Black swan and curvature in an autocatalator model. Procedia Engineering, 2017, V. 201, pp. 561–566. doi: 10.1016/j.proeng.2017.09.614 .
- Balabaev M. Metod krivisni potoka v zadache goreniya . Sbornik trudov IV mezhdunarodnoi konferentsii i molodezhnoi shkoly ”Informatsionnye tekhnologii i nanotekhnologii” (ITNT-2018) . Samara: Novaya tekhnika, 2018, p. 1996–2000. Available at: http://repo.ssau.ru/ bitstream/Informacionnye-tehnologii-i-nanotehnologii/Metod-krivizny-potoka-v-zadache-goreniya-69411/1/ paper_269.pdf .
- Bogolyubov N.N., Mitropolsky Yu.A. Asimptoticheskie metody v teorii nelineinykh kolebanii . М.: Nauka, 1974, 504 p. Available at: http://bookre.org/reader?file=544039 .
- Darboux J.G. Memoire sur les equations differentielles dlgebriques du premier ordre et du premier degre. Bulletin des Sciences Mathematiques et Astronomiques, Serie 2, V. 2(1878), pp. 151–200. Available at: http://www.numdam.org/article/BSMA_1878_2_2_1_151_1.pdf .
- Demazure M. Catastrophes et bifurcations. Paris: Ellipses., 1989. Available at: http://bookre.org/ reader?file=1329312 .
- Ginoux J.M. Differential geometry applied to dynamical systems. Singapore: World Scientic, 2009. Vol. 3. Available at: http://bookre.org/reader?file=700337 .
- Rossetto B. Singular Approximation of Chaotic Slow-fast Dynamical Systems. Lecture Notes in Physics, 1986, V. 278, pp. 12–14. doi: 10.1007/3-540-17894-5_306