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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

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 & 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. Google Scholar

[2]

E. Benoit, J. L. Callot, F. Diener and M. Diener, Chasse au canard,, Collect. Math., 31-32 (): 31.   Google Scholar

[3]

E. Benoit, Systèmes lents-rapides dans $ R^3$ et leurs canards, Société Mathématique de France. Astérisque, 109-110 (1983), 159-191.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[6]

M. Brøns and R. Kaasen, Canards and mixed-mode oscillations in a forest pest model, Theor Popul Biol., 77 (2010), 238-242. Google Scholar

[7]

M. Diener, "Nessie et Les Canards," Publication IRMA, Strasbourg, 1979. Google Scholar

[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. Google Scholar

[9]

G. N. Gorelov and V. A. Sobolev, Duck-trajectories in a thermal explosion problem, Appl. Math. Lett., 5 (1992), 3-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

E. F. Mishchenko and N. Kh. Rozov, "Differential Equations with Small Parameters and Relaxation Oscillations," Plenum Press, New York, 1980.  Google Scholar

[14]

R. E. O'Malley, "Singular Perturbation Methods for Ordinary Differential Equations," Appl. Math. Sci. 89, Springer-Verlag, New York, 1991.  Google Scholar

[15]

M. P. Mortell, R. E. O'Malley, A. Pokrovskii and V. A. Sobolev, "Singular Perturbation and Hysteresis," SIAM, Philadelphia, 2005.  Google Scholar

[16]

A. Pokrovskii1, E. Shchepakina and V. Sobolev, Canard doublet in a Lotka-Volterra type model, J. Phys.: Conf. Ser., 138 (2008), 012019. Google Scholar

[17]

E. Shchepakina and V. Sobolev, Integral manifolds, canards and black swans, Nonlinear Analysis A, 44 (2001), 45-50.  Google Scholar

[18]

E. Shchepakina, Black swans and canards in self-ignition problem, Nonlinear Anal.: Real World Appl., 4 (2003), 45-50.  Google Scholar

[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.  Google Scholar

[20]

V. A. Sobolev and E. A. Shchepakina, Self-ignition of dusty medium, J. Combustion, Explosion and Shock Waves, 29 (1993), 378-381. Google Scholar

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. Google Scholar

[2]

E. Benoit, J. L. Callot, F. Diener and M. Diener, Chasse au canard,, Collect. Math., 31-32 (): 31.   Google Scholar

[3]

E. Benoit, Systèmes lents-rapides dans $ R^3$ et leurs canards, Société Mathématique de France. Astérisque, 109-110 (1983), 159-191.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[6]

M. Brøns and R. Kaasen, Canards and mixed-mode oscillations in a forest pest model, Theor Popul Biol., 77 (2010), 238-242. Google Scholar

[7]

M. Diener, "Nessie et Les Canards," Publication IRMA, Strasbourg, 1979. Google Scholar

[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. Google Scholar

[9]

G. N. Gorelov and V. A. Sobolev, Duck-trajectories in a thermal explosion problem, Appl. Math. Lett., 5 (1992), 3-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

E. F. Mishchenko and N. Kh. Rozov, "Differential Equations with Small Parameters and Relaxation Oscillations," Plenum Press, New York, 1980.  Google Scholar

[14]

R. E. O'Malley, "Singular Perturbation Methods for Ordinary Differential Equations," Appl. Math. Sci. 89, Springer-Verlag, New York, 1991.  Google Scholar

[15]

M. P. Mortell, R. E. O'Malley, A. Pokrovskii and V. A. Sobolev, "Singular Perturbation and Hysteresis," SIAM, Philadelphia, 2005.  Google Scholar

[16]

A. Pokrovskii1, E. Shchepakina and V. Sobolev, Canard doublet in a Lotka-Volterra type model, J. Phys.: Conf. Ser., 138 (2008), 012019. Google Scholar

[17]

E. Shchepakina and V. Sobolev, Integral manifolds, canards and black swans, Nonlinear Analysis A, 44 (2001), 45-50.  Google Scholar

[18]

E. Shchepakina, Black swans and canards in self-ignition problem, Nonlinear Anal.: Real World Appl., 4 (2003), 45-50.  Google Scholar

[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.  Google Scholar

[20]

V. A. Sobolev and E. A. Shchepakina, Self-ignition of dusty medium, J. Combustion, Explosion and Shock Waves, 29 (1993), 378-381. Google Scholar

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