# American Institute of Mathematical Sciences

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:
 [1] R. E., Jr. O'Malley, "Introduction to Singular Perturbations,", Academic Press, (1974). [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). [3] V. A. Sobolev, Integral manifolds and decomposition of singularly perturbed systems,, System and Control Lett., 5 (1984), 169. [4] E. F. Mishchenko, Yu. S. Kolesov, A. Yu. Kolesov and N. Kh. Rozov, "Asymptotic Methods in Singularly Perturbed Systems,", Plenum Press, (1995). [5] E. Shchepakina and V. Sobolev, Integral manifolds, canards and black swans,, Nonlinear Analysis, 44 (2001), 897. [6] , "Singular Perturbations and Hysteresis,", (eds. M. P. Mortell, (2005). [7] S. Baer and T. Erneux, Singular Hopf bifurcation to relaxation oscillations,, SIAM J. Appl. Math., 46 (1986), 721. [8] B. Braaksma, "Critical Phenomena in Dynamical Systems of van der Pol Type,", PhD thesis, (1993). [9] J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields,", Springer-Verlag, (1983). [10] J. Hale and H. Koçak, "Dynamics and Bifurcations,", Springer-Verlag, (1996). [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. [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. [13] M. Brφns and J. Sturis, Explosion of limit cycles and chaotic waves in a simple nonlinear chemical system,, Physical Review. E, 64 (2001). [14] J. Moehlis, Canards in a surface oxidation reaction,, J. Nonlinear Sci., 12 (2002), 319. [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. [16] E. F. Mishchenko and N. Kh. Rozov, "Differential Equations with Small Parameters and Relaxation Oscillations,", Plenum Press, (1980). [17] J. D. Murray, "Mathematical Biology,", Springer-Verlag, (2003). [18] J. Grasman, "Asymptotic Methods for Relaxation Oscillations and Applications,", Springer-Verlag, (1987). [19] M. Diener, "Nessie et Les Canards,", (French) [Nessie and canards], (1979). [20] E. Benoit, J. L. Calot, F. Diener and M. Diener, Chasse au canard,, (French) [The duck shooting], 31-32 (1981), 31. [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. [22] W. Eckhaus, Relaxation oscillations including a standart chase on French ducks,, Lect. Notes in Math., 985 (1983), 449. [23] A. K. Zvonkin and M. A. Shubin, Non-standard analysis and singular perturbations of ordinary differential equations,, Russian Math. Surveys, 39 (1984), 69. [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. [25] G. N. Gorelov and V. A. Sobolev, Duck-trajectories in a thermal explosion problem,, Appl. Math. Lett., 5 (1992), 3. [26] E. Shchepakina and V. Sobolev, Black swans and canards in laser and combustion models,, in, (2005), 207. [27] V. Sobolev and E. Shchepakina, Duck trajectories in a problem of combustion theory,, Differential Equations, 32 (1996), 1177. [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). [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. [30] E. Shchepakina, Black swans and canards in self-ignition problem,, Nonlinear Analysis: Real Word Applications, 4 (2003), 45. [31] B. Peng, V. Gáspár and K. Showalter, False bifurcations in chemical systems: Canards,, Phil. Trans. R. Soc. Lond. Ser. A, 337 (1991), 275. [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. [33] N. Kakiuchi and K. Tchizawa, On an explicit duck solution and delay in the Fitzhugh-Nagumo equation,, J. Diff. Eq., 141 (1997), 327. [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. [35] K. Schneider, E. Shchepakina and V. Sobolev, New type of travelling wave solutions,, Mathematical Methods in the Applied Sciences, 26 (2003), 1349. [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. [37] J. E. Marsden and M. McCracken, "Hopf Bifurcation and its Applications,", Springer, (1976). [38] J. Carr, "Applications of Centre Manifold Theory,", Springer-Verlag, (1981). [39] P. N. V. Tu, "Dynamical Systems: An Introduction with Applications in Economics and Biology,", Springer-Verlag, (1994). [40] E. Shchepakina and V. Sobolev, Exchange of stability of slow regimes in chemical systems,, Report 01-003, (2001), 01. [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. [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. [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). [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. [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. [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.

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##### References:
 [1] R. E., Jr. O'Malley, "Introduction to Singular Perturbations,", Academic Press, (1974). [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). [3] V. A. Sobolev, Integral manifolds and decomposition of singularly perturbed systems,, System and Control Lett., 5 (1984), 169. [4] E. F. Mishchenko, Yu. S. Kolesov, A. Yu. Kolesov and N. Kh. Rozov, "Asymptotic Methods in Singularly Perturbed Systems,", Plenum Press, (1995). [5] E. Shchepakina and V. Sobolev, Integral manifolds, canards and black swans,, Nonlinear Analysis, 44 (2001), 897. [6] , "Singular Perturbations and Hysteresis,", (eds. M. P. Mortell, (2005). [7] S. Baer and T. Erneux, Singular Hopf bifurcation to relaxation oscillations,, SIAM J. Appl. Math., 46 (1986), 721. [8] B. Braaksma, "Critical Phenomena in Dynamical Systems of van der Pol Type,", PhD thesis, (1993). [9] J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields,", Springer-Verlag, (1983). [10] J. Hale and H. Koçak, "Dynamics and Bifurcations,", Springer-Verlag, (1996). [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. [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. [13] M. Brφns and J. Sturis, Explosion of limit cycles and chaotic waves in a simple nonlinear chemical system,, Physical Review. E, 64 (2001). [14] J. Moehlis, Canards in a surface oxidation reaction,, J. Nonlinear Sci., 12 (2002), 319. [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. [16] E. F. Mishchenko and N. Kh. Rozov, "Differential Equations with Small Parameters and Relaxation Oscillations,", Plenum Press, (1980). [17] J. D. Murray, "Mathematical Biology,", Springer-Verlag, (2003). [18] J. Grasman, "Asymptotic Methods for Relaxation Oscillations and Applications,", Springer-Verlag, (1987). [19] M. Diener, "Nessie et Les Canards,", (French) [Nessie and canards], (1979). [20] E. Benoit, J. L. Calot, F. Diener and M. Diener, Chasse au canard,, (French) [The duck shooting], 31-32 (1981), 31. [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. [22] W. Eckhaus, Relaxation oscillations including a standart chase on French ducks,, Lect. Notes in Math., 985 (1983), 449. [23] A. K. Zvonkin and M. A. Shubin, Non-standard analysis and singular perturbations of ordinary differential equations,, Russian Math. Surveys, 39 (1984), 69. [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. [25] G. N. Gorelov and V. A. Sobolev, Duck-trajectories in a thermal explosion problem,, Appl. Math. Lett., 5 (1992), 3. [26] E. Shchepakina and V. Sobolev, Black swans and canards in laser and combustion models,, in, (2005), 207. [27] V. Sobolev and E. Shchepakina, Duck trajectories in a problem of combustion theory,, Differential Equations, 32 (1996), 1177. [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). [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. [30] E. Shchepakina, Black swans and canards in self-ignition problem,, Nonlinear Analysis: Real Word Applications, 4 (2003), 45. [31] B. Peng, V. Gáspár and K. Showalter, False bifurcations in chemical systems: Canards,, Phil. Trans. R. Soc. Lond. Ser. A, 337 (1991), 275. [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. [33] N. Kakiuchi and K. Tchizawa, On an explicit duck solution and delay in the Fitzhugh-Nagumo equation,, J. Diff. Eq., 141 (1997), 327. [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. [35] K. Schneider, E. Shchepakina and V. Sobolev, New type of travelling wave solutions,, Mathematical Methods in the Applied Sciences, 26 (2003), 1349. [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. [37] J. E. Marsden and M. McCracken, "Hopf Bifurcation and its Applications,", Springer, (1976). [38] J. Carr, "Applications of Centre Manifold Theory,", Springer-Verlag, (1981). [39] P. N. V. Tu, "Dynamical Systems: An Introduction with Applications in Economics and Biology,", Springer-Verlag, (1994). [40] E. Shchepakina and V. Sobolev, Exchange of stability of slow regimes in chemical systems,, Report 01-003, (2001), 01. [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. [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. [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). [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. [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. [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.
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