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Canard explosion in chemical and optical systems
Canard cascades
1. | Samara State Aerospace University, Molodogvardeiskaya, 151, Samara, 443001, Russian Federation |
References:
[1] |
V. I. Arnold, V. S. Afraimovich, Yu. S. Il'yashenko and L. P.Shil'nikov, Theory of bifurcations, in "Dynamical Systems" (ed. V. Arnold), 5. Encyclopedia of Mathematical Sciences, Springer Verlag, New York, 1994. |
[2] |
E. Benoit, J. L. Callot, F. Diener and M. Diener, Chasse au canard, Collect. Math., 31-32 (1981-1982), 37-119. |
[3] |
E. Benoit, Systèmes lents-rapides dans $ R^3$ et leurs canards, Société Mathématique de France. Astérisque, 109-110 (1983), 159-191. |
[4] |
M. Brøns and K. Bar-Eli, Asymptotic Analysis of Canards in the EOE Equations and the Role of the Inflection Line, Proc. London Roy Soc., Ser. A, 445 (1994), 305-322. |
[5] |
M. Brøns and K. Bar-Eli, Canard explosion and excitation in a model of the Belousov-Zhabotinsky reaction, Journal of Physical Chemistry, 95 (1991), 8706-8713. |
[6] |
M. Brøns and R. Kaasen, Canards and mixed-mode oscillations in a forest pest model, Theor Popul Biol., 77 (2010), 238-242. |
[7] |
M. Diener, "Nessie et Les Canards," Publication IRMA, Strasbourg, 1979. |
[8] |
V. Gol'dshtein, A. Zinoviev, V. Sobolev and E. Shchepakina, Criterion for thermal explosion with reactant consumption in a dusty gas, Proc. London Roy. Soc., Ser. A, 452 (1996), 2103-2119. |
[9] |
G. N. Gorelov and V. A. Sobolev, Duck-trajectories in a thermal explosion problem, Appl. Math. Lett., 5 (1992), 3-6. |
[10] |
G. N. Gorelov and V. A. Sobolev, Mathematical modeling of critical phenomena in thermal explosion theory, Combust. Flame, 87 (1991), 203-210.
doi: 10.1016/0010-2180(91)90170-G. |
[11] |
G. N. Gorelov, E. A. Shchepakina and V. A. Sobolev, Canards and critical behavior in autocatalytic combustion models, J Eng Math 56 (2006), 143-160.
doi: 10.1007/s10665-006-9047-0. |
[12] |
E. F. Mishchenko, Yu. S. Kolesov, A. Yu. Kolesov and N. Kh. Rozov, "Asymptotic Methods in Singularly Perturbed Systems," Plenum Press, New York, 1995. |
[13] |
E. F. Mishchenko and N. Kh. Rozov, "Differential Equations with Small Parameters and Relaxation Oscillations," Plenum Press, New York, 1980. |
[14] |
R. E. O'Malley, "Singular Perturbation Methods for Ordinary Differential Equations," Appl. Math. Sci. 89, Springer-Verlag, New York, 1991. |
[15] |
M. P. Mortell, R. E. O'Malley, A. Pokrovskii and V. A. Sobolev, "Singular Perturbation and Hysteresis," SIAM, Philadelphia, 2005. |
[16] |
A. Pokrovskii1, E. Shchepakina and V. Sobolev, Canard doublet in a Lotka-Volterra type model, J. Phys.: Conf. Ser., 138 (2008), 012019. |
[17] |
E. Shchepakina and V. Sobolev, Integral manifolds, canards and black swans, Nonlinear Analysis A, 44 (2001), 45-50. |
[18] |
E. Shchepakina, Black swans and canards in self-ignition problem, Nonlinear Anal.: Real World Appl., 4 (2003), 45-50. |
[19] |
V. A. Sobolev and E. A. Shchepakina, Duck trajectories in a problem of combustion theory, Differential Equations, 32 (1996), 1177-1186, translation from Differ. Uravn., 32 (1996), 1175-1184. |
[20] |
V. A. Sobolev and E. A. Shchepakina, Self-ignition of dusty medium, J. Combustion, Explosion and Shock Waves, 29 (1993), 378-381. |
show all references
References:
[1] |
V. I. Arnold, V. S. Afraimovich, Yu. S. Il'yashenko and L. P.Shil'nikov, Theory of bifurcations, in "Dynamical Systems" (ed. V. Arnold), 5. Encyclopedia of Mathematical Sciences, Springer Verlag, New York, 1994. |
[2] |
E. Benoit, J. L. Callot, F. Diener and M. Diener, Chasse au canard, Collect. Math., 31-32 (1981-1982), 37-119. |
[3] |
E. Benoit, Systèmes lents-rapides dans $ R^3$ et leurs canards, Société Mathématique de France. Astérisque, 109-110 (1983), 159-191. |
[4] |
M. Brøns and K. Bar-Eli, Asymptotic Analysis of Canards in the EOE Equations and the Role of the Inflection Line, Proc. London Roy Soc., Ser. A, 445 (1994), 305-322. |
[5] |
M. Brøns and K. Bar-Eli, Canard explosion and excitation in a model of the Belousov-Zhabotinsky reaction, Journal of Physical Chemistry, 95 (1991), 8706-8713. |
[6] |
M. Brøns and R. Kaasen, Canards and mixed-mode oscillations in a forest pest model, Theor Popul Biol., 77 (2010), 238-242. |
[7] |
M. Diener, "Nessie et Les Canards," Publication IRMA, Strasbourg, 1979. |
[8] |
V. Gol'dshtein, A. Zinoviev, V. Sobolev and E. Shchepakina, Criterion for thermal explosion with reactant consumption in a dusty gas, Proc. London Roy. Soc., Ser. A, 452 (1996), 2103-2119. |
[9] |
G. N. Gorelov and V. A. Sobolev, Duck-trajectories in a thermal explosion problem, Appl. Math. Lett., 5 (1992), 3-6. |
[10] |
G. N. Gorelov and V. A. Sobolev, Mathematical modeling of critical phenomena in thermal explosion theory, Combust. Flame, 87 (1991), 203-210.
doi: 10.1016/0010-2180(91)90170-G. |
[11] |
G. N. Gorelov, E. A. Shchepakina and V. A. Sobolev, Canards and critical behavior in autocatalytic combustion models, J Eng Math 56 (2006), 143-160.
doi: 10.1007/s10665-006-9047-0. |
[12] |
E. F. Mishchenko, Yu. S. Kolesov, A. Yu. Kolesov and N. Kh. Rozov, "Asymptotic Methods in Singularly Perturbed Systems," Plenum Press, New York, 1995. |
[13] |
E. F. Mishchenko and N. Kh. Rozov, "Differential Equations with Small Parameters and Relaxation Oscillations," Plenum Press, New York, 1980. |
[14] |
R. E. O'Malley, "Singular Perturbation Methods for Ordinary Differential Equations," Appl. Math. Sci. 89, Springer-Verlag, New York, 1991. |
[15] |
M. P. Mortell, R. E. O'Malley, A. Pokrovskii and V. A. Sobolev, "Singular Perturbation and Hysteresis," SIAM, Philadelphia, 2005. |
[16] |
A. Pokrovskii1, E. Shchepakina and V. Sobolev, Canard doublet in a Lotka-Volterra type model, J. Phys.: Conf. Ser., 138 (2008), 012019. |
[17] |
E. Shchepakina and V. Sobolev, Integral manifolds, canards and black swans, Nonlinear Analysis A, 44 (2001), 45-50. |
[18] |
E. Shchepakina, Black swans and canards in self-ignition problem, Nonlinear Anal.: Real World Appl., 4 (2003), 45-50. |
[19] |
V. A. Sobolev and E. A. Shchepakina, Duck trajectories in a problem of combustion theory, Differential Equations, 32 (1996), 1177-1186, translation from Differ. Uravn., 32 (1996), 1175-1184. |
[20] |
V. A. Sobolev and E. A. Shchepakina, Self-ignition of dusty medium, J. Combustion, Explosion and Shock Waves, 29 (1993), 378-381. |
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