American Institute of Mathematical Sciences

2013, 2013(special): 77-83. doi: 10.3934/proc.2013.2013.77

An iterative method for the canard explosion in general planar systems

 1 Department of Applied Mathematics and Computer Science, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark

Received  September 2012 Revised  April 2013 Published  November 2013

The canard explosion is the change of amplitude and period of a limit cycle born in a Hopf bifurcation in a very narrow parameter interval. The phenomenon is well understood in singular perturbation problems where a small parameter controls the slow/fast dynamics. However, canard explosions are also observed in systems where no such parameter can obviously be identified. Here we show how the iterative method of Roussel and Fraser, devised to construct regular slow manifolds, can be used to determine a canard point in a general planar system of nonlinear ODEs. We demonstrate the method on the van der Pol equation, showing that the asymptotics of the method is correct, and on a templator model for a self-replicating system.
Citation: Morten Brøns. An iterative method for the canard explosion in general planar systems. Conference Publications, 2013, 2013 (special) : 77-83. doi: 10.3934/proc.2013.2013.77
References:
 [1] E. Benoit, J. L. Callot, F. Diener, and M. Diener., Chasse au canard. Collectanea Mathematica, 32:37-119, 1981. [2] K. M. Beutel and E. Peacock-López., Complex dynamics in a cross-catalytic self-replication mechanism. Journal of Chemical Physics, 126:125104, 2007. [3] M. Brøns., Relaxation oscillations and canards in a nonlinear model of discontinuous plastic deformation in metals at very low temperatures. Proceedings of the Royal Society of London Series A, 461:2289-2302, 2005. [4] M. Brøns., Canard explosion of limit cycles in templator models of self-replication mechanisms. Journal of Chemical Physics, 134:144105, 2011. [5] M. Brøns and K. Bar-Eli., Asymptotic analysis of canards in the EOE equations and the role of the inflection line. Proceedings of the Royal Society of London Series A, 445:305-322, 1994. [6] M. Brøns and J. Sturis., Explosion of limit cycles and chaotic waves in a simple nonlinear chemical system. Physical Review E, 64:026209, 2001. [7] M. Desroches and M. R: Jeffrey., Canards and curvature: the 'smallness' of $\epsilon$ in slow-fast dynamics. Proceedings of the Royal Society of London Series A, 467:2404-2421, 2011. [8] W. Eckhaus., Relaxation oscillations including a standard chase on French ducks. In Asymptotic Analysis II, volume 985 of Lecture Notes in Mathematics, pages 449-494. Springer Verlag, New York/Berlin, 1983. [9] S. J. Fraser., The steady state and equilibrium approximations: A geometrical picture. Journal of Chemical Physics, 88(8):4732-4738, 1988. [10] M. Krupa and P. Szmolyan., Relaxation oscillations and canard explosion. Journal of Differential Equations, 174:312-368, 2001. [11] M. R. Roussel and S. J. Fraser., Geometry of the steady-state approximation: Perturbation and accelerated convergence methods. Journal of Chemical Physics, 93(2):1072-1081, 1990. [12] F. Verhulst., Methods and Applications of Singular Perturbations. Number 50 in Texts in Applied Mathematics. Springer, New York, 2005. [13] A. K. Zvonkin and M. A. Shubin., Non-standard analysis and singular perturbations of ordinary differential equations. Russian Mathematical Surveys, 39(2):77-127, 1984.

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References:
 [1] E. Benoit, J. L. Callot, F. Diener, and M. Diener., Chasse au canard. Collectanea Mathematica, 32:37-119, 1981. [2] K. M. Beutel and E. Peacock-López., Complex dynamics in a cross-catalytic self-replication mechanism. Journal of Chemical Physics, 126:125104, 2007. [3] M. Brøns., Relaxation oscillations and canards in a nonlinear model of discontinuous plastic deformation in metals at very low temperatures. Proceedings of the Royal Society of London Series A, 461:2289-2302, 2005. [4] M. Brøns., Canard explosion of limit cycles in templator models of self-replication mechanisms. Journal of Chemical Physics, 134:144105, 2011. [5] M. Brøns and K. Bar-Eli., Asymptotic analysis of canards in the EOE equations and the role of the inflection line. Proceedings of the Royal Society of London Series A, 445:305-322, 1994. [6] M. Brøns and J. Sturis., Explosion of limit cycles and chaotic waves in a simple nonlinear chemical system. Physical Review E, 64:026209, 2001. [7] M. Desroches and M. R: Jeffrey., Canards and curvature: the 'smallness' of $\epsilon$ in slow-fast dynamics. Proceedings of the Royal Society of London Series A, 467:2404-2421, 2011. [8] W. Eckhaus., Relaxation oscillations including a standard chase on French ducks. In Asymptotic Analysis II, volume 985 of Lecture Notes in Mathematics, pages 449-494. Springer Verlag, New York/Berlin, 1983. [9] S. J. Fraser., The steady state and equilibrium approximations: A geometrical picture. Journal of Chemical Physics, 88(8):4732-4738, 1988. [10] M. Krupa and P. Szmolyan., Relaxation oscillations and canard explosion. Journal of Differential Equations, 174:312-368, 2001. [11] M. R. Roussel and S. J. Fraser., Geometry of the steady-state approximation: Perturbation and accelerated convergence methods. Journal of Chemical Physics, 93(2):1072-1081, 1990. [12] F. Verhulst., Methods and Applications of Singular Perturbations. Number 50 in Texts in Applied Mathematics. Springer, New York, 2005. [13] A. K. Zvonkin and M. A. Shubin., Non-standard analysis and singular perturbations of ordinary differential equations. Russian Mathematical Surveys, 39(2):77-127, 1984.
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