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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 
References:
[1] 
R. E., Jr. O'Malley, "Introduction to Singular Perturbations," Academic Press, New York, 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), 169179. 
[4] 
E. F. Mishchenko, Yu. S. Kolesov, A. Yu. Kolesov and N. Kh. Rozov, "Asymptotic Methods in Singularly Perturbed Systems," Plenum Press, New York, 1995. 
[5] 
E. Shchepakina and V. Sobolev, Integral manifolds, canards and black swans, Nonlinear Analysis, 44 (2001), 897908. 
[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), 721739. 
[8] 
B. Braaksma, "Critical Phenomena in Dynamical Systems of van der Pol Type," PhD thesis, University of Utrecht, 1993. 
[9] 
J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields," SpringerVerlag, New York. 1983. 
[10] 
J. Hale and H. Koçak, "Dynamics and Bifurcations," SpringerVerlag, New York. 1996. 
[11] 
M. Brφns, K. BarEli, Canard explosion and excitation in a model of the BelousovZhabotinsky reaction, J. Phys. Chem., 95 (1991), 87068713. doi: 10.1021/j100175a053. 
[12] 
M. Brφns and K. BarEli, 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), 305322. 
[13] 
M. Brφns and J. Sturis, Explosion of limit cycles and chaotic waves in a simple nonlinear chemical system, Physical Review. E, 64 (2001), 026209. 
[14] 
J. Moehlis, Canards in a surface oxidation reaction, J. Nonlinear Sci., 12 (2002), 319345. 
[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), 227249. 
[16] 
E. F. Mishchenko and N. Kh. Rozov, "Differential Equations with Small Parameters and Relaxation Oscillations," Plenum Press, New York, 1980. 
[17] 
J. D. Murray, "Mathematical Biology," SpringerVerlag, Berlin, 2003. 
[18] 
J. Grasman, "Asymptotic Methods for Relaxation Oscillations and Applications," SpringerVerlag, New York, 1987. 
[19] 
M. Diener, "Nessie et Les Canards," (French) [Nessie and canards], Publication IRMA, Strasbourg, 1979. 
[20] 
E. Benoit, J. L. Calot, F. Diener and M. Diener, Chasse au canard, (French) [The duck shooting], Collectanea Mathematica, 3132 (1981), 37119. (MR 653890) (85g:58062d) 
[21] 
E. Benoit, Systèmes lentsrapides dans $ R^3$ et leurs canards, (French) [Slow/fast systems in $ R^3$ with canards], Société Mathématique de France, Astérisque, 109110 (1983), 159191. 
[22] 
W. Eckhaus, Relaxation oscillations including a standart chase on French ducks, Lect. Notes in Math., 985 (1983), 449494. 
[23] 
A. K. Zvonkin and M. A. Shubin, Nonstandard analysis and singular perturbations of ordinary differential equations, Russian Math. Surveys, 39 (1984), 69131. 
[24] 
G. N. Gorelov and V. A. Sobolev, Mathematical modeling of critical phenomena in thermal explosion theory, Combust. Flame, 87 (1991), 203210. doi: 10.1016/00102180(91)90170G. 
[25] 
G. N. Gorelov and V. A. Sobolev, Ducktrajectories in a thermal explosion problem, Appl. Math. Lett., 5 (1992), 36. 
[26] 
E. Shchepakina and V. Sobolev, Black swans and canards in laser and combustion models, in "Singular Perturbations and Hysteresis'' (eds. M. P. Mortell, R. E. O'Malley, A. Pokrovskii and V. A. Sobolev), SIAM, (2005), 207255. 
[27] 
V. Sobolev and E. Shchepakina, Duck trajectories in a problem of combustion theory, Differential Equations, 32 (1996), 11771186. (99a:34141) 
[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), 074104. 
[29] 
M. Desroches, B. Krauskopf and H. M. Osinga, Numerical continuation of canard orbits in slowfast dynamical systems, Nonlinearity, 23 (2010), 739765. doi: 10.1088/09517715/23/3/017. 
[30] 
E. Shchepakina, Black swans and canards in selfignition problem, Nonlinear Analysis: Real Word Applications, 4 (2003), 4550. 
[31] 
B. Peng, V. Gáspár and K. Showalter, False bifurcations in chemical systems: Canards, Phil. Trans. R. Soc. Lond. Ser. A, 337 (1991), 275289. 
[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), 21032119. 
[33] 
N. Kakiuchi and K. Tchizawa, On an explicit duck solution and delay in the FitzhughNagumo equation, J. Diff. Eq., 141 (1997), 327339. 
[34] 
E. Freire, E. Gamero and A. J. RodriguezLuis, Firstorder approximation for canard periodic orbits in a van der Pol electronic oscillator, Appl. Math. Let., 12 (1999), 7378. 
[35] 
K. Schneider, E. Shchepakina and V. Sobolev, New type of travelling wave solutions, Mathematical Methods in the Applied Sciences, 26 (2003), 13491361. 
[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), 22892302. doi: 10.1098/rspa.2005.1486. 
[37] 
J. E. Marsden and M. McCracken, "Hopf Bifurcation and its Applications," Springer, New York, 1976. 
[38] 
J. Carr, "Applications of Centre Manifold Theory," SpringerVerlag, New York, 1981. 
[39] 
P. N. V. Tu, "Dynamical Systems: An Introduction with Applications in Economics and Biology," SpringerVerlag, Berlin, New York, 1994. 
[40] 
E. Shchepakina and V. Sobolev, Exchange of stability of slow regimes in chemical systems, Report 01003, March 2001. Institute for Nonlinear Science, Cork, Ireland, 2001. 
[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), 331335. 
[42] 
V. M. Gol'dshtein, V. A. Sobolev and G. S. Yablonskii, Relaxation selfoscillations in chemical kinetics: A model, conditions for realization, Chem. Eng. Sci., 41 (1986), 27612766. 
[43] 
F. Marino, G. Catalán, P. Sánchez, S. Balle and O. Piro, Thermooptical "canard orbits" and excitable limit cycles, Phys. Rev. Lett., 92 (2004), 073901. 
[44] 
E. Shchepakina and O. Korotkova, Condition for canard explosion in a semiconductor optical amplifier, JOSA B, 28 (2011), 19881993. doi: 10.1364/JOSAB.28.001988. 
[45] 
A. W. L. Chan, K. L. Lee and C. Shu, Selfstarting photonic clock using semiconductor optical amplifier based MachZehnder interferometer, Electronics Letters, 40 (2004), 827828. 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), 026214. doi: 10.1103/PhysRevE.68.026214. 
show all references
References:
[1] 
R. E., Jr. O'Malley, "Introduction to Singular Perturbations," Academic Press, New York, 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), 169179. 
[4] 
E. F. Mishchenko, Yu. S. Kolesov, A. Yu. Kolesov and N. Kh. Rozov, "Asymptotic Methods in Singularly Perturbed Systems," Plenum Press, New York, 1995. 
[5] 
E. Shchepakina and V. Sobolev, Integral manifolds, canards and black swans, Nonlinear Analysis, 44 (2001), 897908. 
[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), 721739. 
[8] 
B. Braaksma, "Critical Phenomena in Dynamical Systems of van der Pol Type," PhD thesis, University of Utrecht, 1993. 
[9] 
J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields," SpringerVerlag, New York. 1983. 
[10] 
J. Hale and H. Koçak, "Dynamics and Bifurcations," SpringerVerlag, New York. 1996. 
[11] 
M. Brφns, K. BarEli, Canard explosion and excitation in a model of the BelousovZhabotinsky reaction, J. Phys. Chem., 95 (1991), 87068713. doi: 10.1021/j100175a053. 
[12] 
M. Brφns and K. BarEli, 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), 305322. 
[13] 
M. Brφns and J. Sturis, Explosion of limit cycles and chaotic waves in a simple nonlinear chemical system, Physical Review. E, 64 (2001), 026209. 
[14] 
J. Moehlis, Canards in a surface oxidation reaction, J. Nonlinear Sci., 12 (2002), 319345. 
[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), 227249. 
[16] 
E. F. Mishchenko and N. Kh. Rozov, "Differential Equations with Small Parameters and Relaxation Oscillations," Plenum Press, New York, 1980. 
[17] 
J. D. Murray, "Mathematical Biology," SpringerVerlag, Berlin, 2003. 
[18] 
J. Grasman, "Asymptotic Methods for Relaxation Oscillations and Applications," SpringerVerlag, New York, 1987. 
[19] 
M. Diener, "Nessie et Les Canards," (French) [Nessie and canards], Publication IRMA, Strasbourg, 1979. 
[20] 
E. Benoit, J. L. Calot, F. Diener and M. Diener, Chasse au canard, (French) [The duck shooting], Collectanea Mathematica, 3132 (1981), 37119. (MR 653890) (85g:58062d) 
[21] 
E. Benoit, Systèmes lentsrapides dans $ R^3$ et leurs canards, (French) [Slow/fast systems in $ R^3$ with canards], Société Mathématique de France, Astérisque, 109110 (1983), 159191. 
[22] 
W. Eckhaus, Relaxation oscillations including a standart chase on French ducks, Lect. Notes in Math., 985 (1983), 449494. 
[23] 
A. K. Zvonkin and M. A. Shubin, Nonstandard analysis and singular perturbations of ordinary differential equations, Russian Math. Surveys, 39 (1984), 69131. 
[24] 
G. N. Gorelov and V. A. Sobolev, Mathematical modeling of critical phenomena in thermal explosion theory, Combust. Flame, 87 (1991), 203210. doi: 10.1016/00102180(91)90170G. 
[25] 
G. N. Gorelov and V. A. Sobolev, Ducktrajectories in a thermal explosion problem, Appl. Math. Lett., 5 (1992), 36. 
[26] 
E. Shchepakina and V. Sobolev, Black swans and canards in laser and combustion models, in "Singular Perturbations and Hysteresis'' (eds. M. P. Mortell, R. E. O'Malley, A. Pokrovskii and V. A. Sobolev), SIAM, (2005), 207255. 
[27] 
V. Sobolev and E. Shchepakina, Duck trajectories in a problem of combustion theory, Differential Equations, 32 (1996), 11771186. (99a:34141) 
[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), 074104. 
[29] 
M. Desroches, B. Krauskopf and H. M. Osinga, Numerical continuation of canard orbits in slowfast dynamical systems, Nonlinearity, 23 (2010), 739765. doi: 10.1088/09517715/23/3/017. 
[30] 
E. Shchepakina, Black swans and canards in selfignition problem, Nonlinear Analysis: Real Word Applications, 4 (2003), 4550. 
[31] 
B. Peng, V. Gáspár and K. Showalter, False bifurcations in chemical systems: Canards, Phil. Trans. R. Soc. Lond. Ser. A, 337 (1991), 275289. 
[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), 21032119. 
[33] 
N. Kakiuchi and K. Tchizawa, On an explicit duck solution and delay in the FitzhughNagumo equation, J. Diff. Eq., 141 (1997), 327339. 
[34] 
E. Freire, E. Gamero and A. J. RodriguezLuis, Firstorder approximation for canard periodic orbits in a van der Pol electronic oscillator, Appl. Math. Let., 12 (1999), 7378. 
[35] 
K. Schneider, E. Shchepakina and V. Sobolev, New type of travelling wave solutions, Mathematical Methods in the Applied Sciences, 26 (2003), 13491361. 
[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), 22892302. doi: 10.1098/rspa.2005.1486. 
[37] 
J. E. Marsden and M. McCracken, "Hopf Bifurcation and its Applications," Springer, New York, 1976. 
[38] 
J. Carr, "Applications of Centre Manifold Theory," SpringerVerlag, New York, 1981. 
[39] 
P. N. V. Tu, "Dynamical Systems: An Introduction with Applications in Economics and Biology," SpringerVerlag, Berlin, New York, 1994. 
[40] 
E. Shchepakina and V. Sobolev, Exchange of stability of slow regimes in chemical systems, Report 01003, March 2001. Institute for Nonlinear Science, Cork, Ireland, 2001. 
[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), 331335. 
[42] 
V. M. Gol'dshtein, V. A. Sobolev and G. S. Yablonskii, Relaxation selfoscillations in chemical kinetics: A model, conditions for realization, Chem. Eng. Sci., 41 (1986), 27612766. 
[43] 
F. Marino, G. Catalán, P. Sánchez, S. Balle and O. Piro, Thermooptical "canard orbits" and excitable limit cycles, Phys. Rev. Lett., 92 (2004), 073901. 
[44] 
E. Shchepakina and O. Korotkova, Condition for canard explosion in a semiconductor optical amplifier, JOSA B, 28 (2011), 19881993. doi: 10.1364/JOSAB.28.001988. 
[45] 
A. W. L. Chan, K. L. Lee and C. Shu, Selfstarting photonic clock using semiconductor optical amplifier based MachZehnder interferometer, Electronics Letters, 40 (2004), 827828. 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), 026214. doi: 10.1103/PhysRevE.68.026214. 
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