American Institute of Mathematical Sciences

Advanced
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, (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), 109.   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., 445 (1994), 305.   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.   Google Scholar

[6]

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

[7]

M. Diener, "Nessie et Les Canards,", Publication IRMA, (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., 452 (1996), 2103.   Google Scholar

[9]

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

[10]

G. N. Gorelov and V. A. Sobolev, Mathematical modeling of critical phenomena in thermal explosion theory,, Combust. Flame, 87 (1991), 203.  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 \textbf{56} (2006), 56 (2006), 143.  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, (1995).   Google Scholar

[13]

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

[14]

R. E. O'Malley, "Singular Perturbation Methods for Ordinary Differential Equations,", Appl. Math. Sci. \textbf{89}, 89 (1991).   Google Scholar

[15]

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

[17]

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

[18]

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

[19]

V. A. Sobolev and E. A. Shchepakina, Duck trajectories in a problem of combustion theory,, Differential Equations, 32 (1996), 1177.   Google Scholar

[20]

V. A. Sobolev and E. A. Shchepakina, Self-ignition of dusty medium,, J. Combustion, 29 (1993), 378.   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, (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), 109.   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., 445 (1994), 305.   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.   Google Scholar

[6]

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

[7]

M. Diener, "Nessie et Les Canards,", Publication IRMA, (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., 452 (1996), 2103.   Google Scholar

[9]

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

[10]

G. N. Gorelov and V. A. Sobolev, Mathematical modeling of critical phenomena in thermal explosion theory,, Combust. Flame, 87 (1991), 203.  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 \textbf{56} (2006), 56 (2006), 143.  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, (1995).   Google Scholar

[13]

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

[14]

R. E. O'Malley, "Singular Perturbation Methods for Ordinary Differential Equations,", Appl. Math. Sci. \textbf{89}, 89 (1991).   Google Scholar

[15]

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

[17]

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

[18]

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

[19]

V. A. Sobolev and E. A. Shchepakina, Duck trajectories in a problem of combustion theory,, Differential Equations, 32 (1996), 1177.   Google Scholar

[20]

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

[1]

Stefan Siegmund. Normal form of Duffing-van der Pol oscillator under nonautonomous parametric perturbations. Conference Publications, 2001, 2001 (Special) : 357-361. doi: 10.3934/proc.2001.2001.357
[2]

Zhaosheng Feng. Duffing-van der Pol-type oscillator systems. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1231-1257. doi: 10.3934/dcdss.2014.7.1231
[3]

Xiaoqin P. Wu, Liancheng Wang. Hopf bifurcation of a class of two coupled relaxation oscillators of the van der Pol type with delay. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 503-516. doi: 10.3934/dcdsb.2010.13.503
[4]

Zhaosheng Feng, Guangyue Gao, Jing Cui. Duffing--van der Pol--type oscillator system and its first integrals. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1377-1391. doi: 10.3934/cpaa.2011.10.1377
[5]

Qiaolin He, Chang Liu, Xiaoding Shi. Numerical study of phase transition in van der Waals fluid. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4519-4540. doi: 10.3934/dcdsb.2018174
[6]

Cristina Anton, Alan Yong. Stochastic dynamics and survival analysis of a cell population model with random perturbations. Mathematical Biosciences & Engineering, 2018, 15 (5) : 1077-1098. doi: 10.3934/mbe.2018048
[7]

Shu-Yi Zhang. Existence of multidimensional non-isothermal phase transitions in a steady van der Waals flow. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 2221-2239. doi: 10.3934/dcds.2013.33.2221
[8]

Qun Liu, Daqing Jiang. Dynamics of a multigroup SIRS epidemic model with random perturbations and varying total population size. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1089-1110. doi: 10.3934/cpaa.2020050
[9]

Christian Lax, Sebastian Walcher. Singular perturbations and scaling. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 1-29. doi: 10.3934/dcdsb.2019170
[10]

Guy V. Norton, Robert D. Purrington. The Westervelt equation with a causal propagation operator coupled to the bioheat equation.. Evolution Equations & Control Theory, 2016, 5 (3) : 449-461. doi: 10.3934/eect.2016013
[11]

Zvi Artstein. Invariance principle in the singular perturbations limit. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3653-3666. doi: 10.3934/dcdsb.2018309
[12]

Michel Chipot, Senoussi Guesmia. On the asymptotic behavior of elliptic, anisotropic singular perturbations problems. Communications on Pure & Applied Analysis, 2009, 8 (1) : 179-193. doi: 10.3934/cpaa.2009.8.179
[13]

Senoussi Guesmia, Abdelmouhcene Sengouga. Some singular perturbations results for semilinear hyperbolic problems. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 567-580. doi: 10.3934/dcdss.2012.5.567
[14]

Claudio Marchi. On the convergence of singular perturbations of Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1363-1377. doi: 10.3934/cpaa.2010.9.1363
[15]

Chiara Zanini. Singular perturbations of finite dimensional gradient flows. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 657-675. doi: 10.3934/dcds.2007.18.657
[16]

Canela Jordi. Singular perturbations of Blaschke products and connectivity of Fatou components. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3567-3585. doi: 10.3934/dcds.2017153
[17]

Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control & Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017
[18]

Sebastián Ferrer, Martin Lara. Families of canonical transformations by Hamilton-Jacobi-Poincaré equation. Application to rotational and orbital motion. Journal of Geometric Mechanics, 2010, 2 (3) : 223-241. doi: 10.3934/jgm.2010.2.223
[19]

Manuel de León, Juan Carlos Marrero, David Martín de Diego. Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic mechanics. Journal of Geometric Mechanics, 2010, 2 (2) : 159-198. doi: 10.3934/jgm.2010.2.159
[20]

Thierry Horsin, Peter I. Kogut. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. I. Existence result. Mathematical Control & Related Fields, 2015, 5 (1) : 73-96. doi: 10.3934/mcrf.2015.5.73

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (17)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences