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

Canard cascades

1. 

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

Received  October 2011 Revised  April 2012 Published  November 2012

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.
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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