Canard cascades

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March 2013, 18(2): 513-521. doi: 10.3934/dcdsb.2013.18.513

Canard cascades

Vladimir Sobolev ^1,

1.	Samara State Aerospace University, Molodogvardeiskaya, 151, Samara, 443001, Russian Federation

Received October 2011 Revised April 2012 Published November 2012

Abstract
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The existence of canard cascades is studied in the paper as a problem of gluing of stable and unstable one-dimensional slow invariant manifolds at turning points. This way of looking is made feasible to establish the existence of canard cascades, that can be considered as a generalization of canards. A further development of this approach, with applications to the van der Pol equation and a problem of population dynamics, is contained in the paper.

Keywords: Canards, population dynamics, van der Pol equation., singular perturbations.

Mathematics Subject Classification: 34C26, 34D15, 37G15, 92D2.

Citation: Vladimir Sobolev. Canard cascades. Discrete and Continuous Dynamical Systems - B, 2013, 18 (2) : 513-521. doi: 10.3934/dcdsb.2013.18.513

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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