# American Institute of Mathematical Sciences

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

 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 & 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

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##### 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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