# American Institute of Mathematical Sciences

March  2013, 18(2): 495-512. doi: 10.3934/dcdsb.2013.18.495

## Canard explosion in chemical and optical systems

 1 Department of Technical Cybernetics, Samara State Aerospace University, Molodogvardeiskaya 151, Samara 443001, Russian Federation 2 Department of Physics, University of Miami, 1320 Campo Sano Drive, Coral Gables, FL 33146, United States

Received  October 2011 Revised  April 2012 Published  November 2012

The paper deals with the study of the relation between the Andronov--Hopf bifurcation, the canard explosion and the critical phenomena for the van der Pol's type system of singularly perturbed differential equations. Sufficient conditions for the limit cycle birth bifurcation in the case of the singularly perturbed systems are investigated. We use the method of integral manifolds and canards techniques to obtain the conditions under which the system possesses the canard cycle. Through the application to some chemical and optical models it is shown that the canard point should be considered as the critical value of the control parameter.
Citation: Elena Shchepakina, Olga Korotkova. Canard explosion in chemical and optical systems. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 495-512. doi: 10.3934/dcdsb.2013.18.495
##### References:
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Braaksma, "Critical Phenomena in Dynamical Systems of van der Pol Type,", PhD thesis, (1993).   Google Scholar [9] J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields,", Springer-Verlag, (1983).   Google Scholar [10] J. Hale and H. Koçak, "Dynamics and Bifurcations,", Springer-Verlag, (1996).   Google Scholar [11] M. Brφns, K. Bar-Eli, Canard explosion and excitation in a model of the Belousov-Zhabotinsky reaction,, J. Phys. Chem., 95 (1991), 8706.  doi: 10.1021/j100175a053.  Google Scholar [12] M. Brφns and K. Bar-Eli, Asymptotic analysis of canards in the EOE equations and the role of the inflection line,, Proc. R. Soc. London: Mathematical and Physical Sciences, 445 (1994), 305.   Google Scholar [13] M. Brφns and J. Sturis, Explosion of limit cycles and chaotic waves in a simple nonlinear chemical system,, Physical Review. E, 64 (2001).   Google Scholar [14] J. Moehlis, Canards in a surface oxidation reaction,, J. Nonlinear Sci., 12 (2002), 319.   Google Scholar [15] M. Sekikawa, N. Inaba and T. Tsubouchi, Chaos via duck solution breakdown in a piecewise linear van der Pol oscillator driven by an extremely small periodic perturbation,, Physica D, 194 (2004), 227.   Google Scholar [16] E. F. Mishchenko and N. Kh. Rozov, "Differential Equations with Small Parameters and Relaxation Oscillations,", Plenum Press, (1980).   Google Scholar [17] J. D. Murray, "Mathematical Biology,", Springer-Verlag, (2003).   Google Scholar [18] J. Grasman, "Asymptotic Methods for Relaxation Oscillations and Applications,", Springer-Verlag, (1987).   Google Scholar [19] M. Diener, "Nessie et Les Canards,", (French) [Nessie and canards], (1979).   Google Scholar [20] E. Benoit, J. L. Calot, F. Diener and M. Diener, Chasse au canard,, (French) [The duck shooting], 31-32 (1981), 31.   Google Scholar [21] E. Benoit, Systèmes lents-rapides dans $R^3$ et leurs canards,, (French) [Slow/fast systems in $R^3$ with canards], 109-110 (1983), 109.   Google Scholar [22] W. Eckhaus, Relaxation oscillations including a standart chase on French ducks,, Lect. Notes in Math., 985 (1983), 449.   Google Scholar [23] A. K. Zvonkin and M. A. Shubin, Non-standard analysis and singular perturbations of ordinary differential equations,, Russian Math. Surveys, 39 (1984), 69.   Google Scholar [24] 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 [25] G. N. Gorelov and V. A. Sobolev, Duck-trajectories in a thermal explosion problem,, Appl. Math. Lett., 5 (1992), 3.   Google Scholar [26] E. Shchepakina and V. Sobolev, Black swans and canards in laser and combustion models,, in, (2005), 207.   Google Scholar [27] V. Sobolev and E. Shchepakina, Duck trajectories in a problem of combustion theory,, Differential Equations, 32 (1996), 1177.   Google Scholar [28] F. Marino, F. Marino, S. Balle and O. Piro, Chaotically spiking canards in an excitable system with 2D inertial fast manifolds,, Phys. Rev. Lett., 98 (2007).   Google Scholar [29] M. Desroches, B. Krauskopf and H. M. Osinga, Numerical continuation of canard orbits in slow-fast dynamical systems,, Nonlinearity, 23 (2010), 739.  doi: 10.1088/0951-7715/23/3/017.  Google Scholar [30] E. Shchepakina, Black swans and canards in self-ignition problem,, Nonlinear Analysis: Real Word Applications, 4 (2003), 45.   Google Scholar [31] B. Peng, V. Gáspár and K. Showalter, False bifurcations in chemical systems: Canards,, Phil. Trans. R. Soc. Lond. Ser. A, 337 (1991), 275.   Google Scholar [32] V. Gol'dshtein, A. Zinoviev, V. Sobolev and E. Shchepakina, Criterion for thermal explosion with reactant consumption in a dusty gas,, Proc. of London R. Soc. Ser. A., 452 (1996), 2103.   Google Scholar [33] N. Kakiuchi and K. Tchizawa, On an explicit duck solution and delay in the Fitzhugh-Nagumo equation,, J. Diff. Eq., 141 (1997), 327.   Google Scholar [34] E. Freire, E. Gamero and A. J. Rodriguez-Luis, First-order approximation for canard periodic orbits in a van der Pol electronic oscillator,, Appl. Math. Let., 12 (1999), 73.   Google Scholar [35] K. Schneider, E. Shchepakina and V. Sobolev, New type of travelling wave solutions,, Mathematical Methods in the Applied Sciences, 26 (2003), 1349.   Google Scholar [36] M. Brφns, Relaxation oscillations and canards in a nonlinear model of discontinuous plastic deformation in metals at very low temperatures,, Proc. R. Soc. A, 461 (2005), 2289.  doi: 10.1098/rspa.2005.1486.  Google Scholar [37] J. E. Marsden and M. McCracken, "Hopf Bifurcation and its Applications,", Springer, (1976).   Google Scholar [38] J. Carr, "Applications of Centre Manifold Theory,", Springer-Verlag, (1981).   Google Scholar [39] P. N. V. Tu, "Dynamical Systems: An Introduction with Applications in Economics and Biology,", Springer-Verlag, (1994).   Google Scholar [40] E. Shchepakina and V. Sobolev, Exchange of stability of slow regimes in chemical systems,, Report 01-003, (2001), 01.   Google Scholar [41] V. V. Strygin and V. A. Sobolev, Effect of geometric and kinetic parameters and energy dissipation on orientation stability of satellites with double spin,, Cosmic Research, 14 (1976), 331.   Google Scholar [42] V. M. Gol'dshtein, V. A. Sobolev and G. S. Yablonskii, Relaxation self-oscillations in chemical kinetics: A model, conditions for realization,, Chem. Eng. Sci., 41 (1986), 2761.   Google Scholar [43] F. Marino, G. Catalán, P. Sánchez, S. Balle and O. Piro, Thermo-optical "canard orbits" and excitable limit cycles,, Phys. Rev. Lett., 92 (2004).   Google Scholar [44] E. Shchepakina and O. Korotkova, Condition for canard explosion in a semiconductor optical amplifier,, JOSA B, 28 (2011), 1988.  doi: 10.1364/JOSAB.28.001988.  Google Scholar [45] A. W. L. Chan, K. L. Lee and C. Shu, Self-starting photonic clock using semiconductor optical amplifier based Mach-Zehnder interferometer,, Electronics Letters, 40 (2004), 827.  doi: 10.1049/el:20040513.  Google Scholar [46] E. I. Volkov, E. Ullner, A. A. Zaikin and J. Kurths, Oscillatory amplification of stochastic resonance in excitable systems,, Phys. Rev. E, 68 (2003).  doi: 10.1103/PhysRevE.68.026214.  Google Scholar

show all references

##### References:
 [1] R. E., Jr. O'Malley, "Introduction to Singular Perturbations,", Academic Press, (1974).   Google Scholar [2] A. B. Vasil'eva, V. F. Butuzov and L. V. Kalachev, "The Boundary Function Method for Singular Perturbation Problems,", SIAM Studies in Appl. Math., 14 (1995).   Google Scholar [3] V. A. Sobolev, Integral manifolds and decomposition of singularly perturbed systems,, System and Control Lett., 5 (1984), 169.   Google Scholar [4] E. F. Mishchenko, Yu. S. Kolesov, A. Yu. Kolesov and N. Kh. Rozov, "Asymptotic Methods in Singularly Perturbed Systems,", Plenum Press, (1995).   Google Scholar [5] E. Shchepakina and V. Sobolev, Integral manifolds, canards and black swans,, Nonlinear Analysis, 44 (2001), 897.   Google Scholar [6] , "Singular Perturbations and Hysteresis,", (eds. M. P. Mortell, (2005).   Google Scholar [7] S. Baer and T. Erneux, Singular Hopf bifurcation to relaxation oscillations,, SIAM J. Appl. Math., 46 (1986), 721.   Google Scholar [8] B. Braaksma, "Critical Phenomena in Dynamical Systems of van der Pol Type,", PhD thesis, (1993).   Google Scholar [9] J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields,", Springer-Verlag, (1983).   Google Scholar [10] J. Hale and H. Koçak, "Dynamics and Bifurcations,", Springer-Verlag, (1996).   Google Scholar [11] M. Brφns, K. Bar-Eli, Canard explosion and excitation in a model of the Belousov-Zhabotinsky reaction,, J. Phys. Chem., 95 (1991), 8706.  doi: 10.1021/j100175a053.  Google Scholar [12] M. Brφns and K. Bar-Eli, Asymptotic analysis of canards in the EOE equations and the role of the inflection line,, Proc. R. Soc. London: Mathematical and Physical Sciences, 445 (1994), 305.   Google Scholar [13] M. Brφns and J. Sturis, Explosion of limit cycles and chaotic waves in a simple nonlinear chemical system,, Physical Review. E, 64 (2001).   Google Scholar [14] J. Moehlis, Canards in a surface oxidation reaction,, J. Nonlinear Sci., 12 (2002), 319.   Google Scholar [15] M. Sekikawa, N. Inaba and T. Tsubouchi, Chaos via duck solution breakdown in a piecewise linear van der Pol oscillator driven by an extremely small periodic perturbation,, Physica D, 194 (2004), 227.   Google Scholar [16] E. F. Mishchenko and N. Kh. Rozov, "Differential Equations with Small Parameters and Relaxation Oscillations,", Plenum Press, (1980).   Google Scholar [17] J. D. Murray, "Mathematical Biology,", Springer-Verlag, (2003).   Google Scholar [18] J. Grasman, "Asymptotic Methods for Relaxation Oscillations and Applications,", Springer-Verlag, (1987).   Google Scholar [19] M. Diener, "Nessie et Les Canards,", (French) [Nessie and canards], (1979).   Google Scholar [20] E. Benoit, J. L. Calot, F. Diener and M. Diener, Chasse au canard,, (French) [The duck shooting], 31-32 (1981), 31.   Google Scholar [21] E. Benoit, Systèmes lents-rapides dans $R^3$ et leurs canards,, (French) [Slow/fast systems in $R^3$ with canards], 109-110 (1983), 109.   Google Scholar [22] W. Eckhaus, Relaxation oscillations including a standart chase on French ducks,, Lect. Notes in Math., 985 (1983), 449.   Google Scholar [23] A. K. Zvonkin and M. A. Shubin, Non-standard analysis and singular perturbations of ordinary differential equations,, Russian Math. Surveys, 39 (1984), 69.   Google Scholar [24] 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 [25] G. N. Gorelov and V. A. Sobolev, Duck-trajectories in a thermal explosion problem,, Appl. Math. Lett., 5 (1992), 3.   Google Scholar [26] E. Shchepakina and V. Sobolev, Black swans and canards in laser and combustion models,, in, (2005), 207.   Google Scholar [27] V. Sobolev and E. Shchepakina, Duck trajectories in a problem of combustion theory,, Differential Equations, 32 (1996), 1177.   Google Scholar [28] F. Marino, F. Marino, S. Balle and O. Piro, Chaotically spiking canards in an excitable system with 2D inertial fast manifolds,, Phys. Rev. Lett., 98 (2007).   Google Scholar [29] M. Desroches, B. Krauskopf and H. M. Osinga, Numerical continuation of canard orbits in slow-fast dynamical systems,, Nonlinearity, 23 (2010), 739.  doi: 10.1088/0951-7715/23/3/017.  Google Scholar [30] E. Shchepakina, Black swans and canards in self-ignition problem,, Nonlinear Analysis: Real Word Applications, 4 (2003), 45.   Google Scholar [31] B. Peng, V. Gáspár and K. Showalter, False bifurcations in chemical systems: Canards,, Phil. Trans. R. Soc. Lond. Ser. A, 337 (1991), 275.   Google Scholar [32] V. Gol'dshtein, A. Zinoviev, V. Sobolev and E. Shchepakina, Criterion for thermal explosion with reactant consumption in a dusty gas,, Proc. of London R. Soc. Ser. A., 452 (1996), 2103.   Google Scholar [33] N. Kakiuchi and K. Tchizawa, On an explicit duck solution and delay in the Fitzhugh-Nagumo equation,, J. Diff. Eq., 141 (1997), 327.   Google Scholar [34] E. Freire, E. Gamero and A. J. Rodriguez-Luis, First-order approximation for canard periodic orbits in a van der Pol electronic oscillator,, Appl. Math. Let., 12 (1999), 73.   Google Scholar [35] K. Schneider, E. Shchepakina and V. Sobolev, New type of travelling wave solutions,, Mathematical Methods in the Applied Sciences, 26 (2003), 1349.   Google Scholar [36] M. Brφns, Relaxation oscillations and canards in a nonlinear model of discontinuous plastic deformation in metals at very low temperatures,, Proc. R. Soc. A, 461 (2005), 2289.  doi: 10.1098/rspa.2005.1486.  Google Scholar [37] J. E. Marsden and M. McCracken, "Hopf Bifurcation and its Applications,", Springer, (1976).   Google Scholar [38] J. Carr, "Applications of Centre Manifold Theory,", Springer-Verlag, (1981).   Google Scholar [39] P. N. V. Tu, "Dynamical Systems: An Introduction with Applications in Economics and Biology,", Springer-Verlag, (1994).   Google Scholar [40] E. Shchepakina and V. Sobolev, Exchange of stability of slow regimes in chemical systems,, Report 01-003, (2001), 01.   Google Scholar [41] V. V. Strygin and V. A. Sobolev, Effect of geometric and kinetic parameters and energy dissipation on orientation stability of satellites with double spin,, Cosmic Research, 14 (1976), 331.   Google Scholar [42] V. M. Gol'dshtein, V. A. Sobolev and G. S. Yablonskii, Relaxation self-oscillations in chemical kinetics: A model, conditions for realization,, Chem. Eng. Sci., 41 (1986), 2761.   Google Scholar [43] F. Marino, G. Catalán, P. Sánchez, S. Balle and O. Piro, Thermo-optical "canard orbits" and excitable limit cycles,, Phys. Rev. Lett., 92 (2004).   Google Scholar [44] E. Shchepakina and O. Korotkova, Condition for canard explosion in a semiconductor optical amplifier,, JOSA B, 28 (2011), 1988.  doi: 10.1364/JOSAB.28.001988.  Google Scholar [45] A. W. L. Chan, K. L. Lee and C. Shu, Self-starting photonic clock using semiconductor optical amplifier based Mach-Zehnder interferometer,, Electronics Letters, 40 (2004), 827.  doi: 10.1049/el:20040513.  Google Scholar [46] E. I. Volkov, E. Ullner, A. A. Zaikin and J. Kurths, Oscillatory amplification of stochastic resonance in excitable systems,, Phys. Rev. E, 68 (2003).  doi: 10.1103/PhysRevE.68.026214.  Google Scholar
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