Abstract

A model for the semiconductor optical amplifier (SOA) consisting of two coupled, nonlinear, first-order differential equations is analytically explored on the basis of the geometric theory of singularly perturbed dynamical systems. The value of the control parameter μ, accounting for the stimulated emission, for which the solution exhibits the phenomenon of “canard explosion” is determined depending on the rest of the SOA’s parameters. Such value is represented by a power series of a small parameter ε of the singularly perturbed system. An example is considered where the canard explosion is numerically evaluated for a typical set of the SOA’s parameters. The importance of rigorous determination of the critical regime in the SOA for optical synchronization and photonic clocking is outlined.

© 2011 Optical Society of America

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  1. R. E. O’Malley, Jr., Introduction to Singular Perturbations (Academic, 1974).
  2. A. B. Vasil’eva, V. F. Butuzov, and L. V. Kalachev, The Boundary Function Method for Singular Perturbation Problems (SIAM, 1995), Vol.  14.
    [CrossRef]
  3. F. Marino, G. Catalán, P. Sánchez, S. Balle, and O. Piro, “Thermo-optical “canard orbits” and excitable limit cycles,” Phys. Rev. Lett. 92, 073901 (2004).
    [CrossRef] [PubMed]
  4. S. Baer and T. Erneux, “Singular Hopf bifurcation to relaxation oscillations,” SIAM J. Appl. Math. 46, 721–739 (1986).
    [CrossRef]
  5. B. Braaksma, “Critical phenomena in dynamical systems of van der Pol type,” Ph.D. thesis (University of Utrecht, 1993).
  6. E. F. Mishchenko, Y. S. Kolesov, A. Y. Kolesov, and N. K. Rozov, Asymptotic Methods in Singularly Perturbed Systems(Plenum, 1995).
  7. M.P.Mortell, R.E.O’Malley, A.Pokrovskii, and V.A.Sobolev, eds. Singular Perturbations and Hysteresis (SIAM, 2005).
    [CrossRef]
  8. A. W. L. Chan, K. L. Lee, and C. Shu, “Self-starting photonic clock using semiconductor optical amplifier based Mach-Zehnder interferometer,” Electron. Lett. 40, 827–828 (2004).
    [CrossRef]
  9. E. I. Volkov, E. Ullner, A. A. Zaikin, and J. Kurths, “Oscillatory amplification of stochastic resonance in excitable systems,” Phys. Rev. E 68, 026214 (2003).
    [CrossRef]
  10. M. Brøns and K. Bar-Eli, “Canard explosion and excitation in a model of the Belousov-Zhabotinsky reaction,” J. Phys. Chem. 95, 8706–8713 (1991).
    [CrossRef]
  11. M. Brøns and K. Bar-Eli, “Asymptotic analysis of canards in the EOE equations and the role of the inflection line,” Proc. R. Soc. A 445, 305–322 (1994).
    [CrossRef]
  12. M. Brøns and J. Sturis, “Explosion of limit cycles and chaotic waves in a simple nonlinear chemical system,” Phys. Rev. E 64, 026209 (2001).
    [CrossRef]
  13. J. Moehlis, “Canards in a surface oxidation reaction,” J. Nonlinear Sci. 12, 319–345 (2002).
    [CrossRef]
  14. 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, 227–249 (2004).
    [CrossRef]
  15. M. Brøns, “Relaxation oscillations and canards in a nonlinear model of discontinuous plastic deformation in metals at very low temperatures,” Proc. R. Soc. A 461, 2289–2302 (2005).
    [CrossRef]
  16. E. F. Mishchenko and N. K. Rozov, Differential Equations with Small Parameters and Relaxation Oscillations (Plenum, 1980).
  17. J. D. Murray, Mathematical Biology (Springer-Verlag, 2003).
  18. J. Grasman, Asymptotic Methods for Relaxation Oscillations and Applications (Springer-Verlag, 1987).
    [CrossRef]
  19. M. Diener, Nessie et Les Canards (Publication IRMA, 1979).
  20. E. Benoit, J. L. Calot, F. Diener, and M. Diener, “Chasse au canard,” Collectanea Mathematica 31–32, 37–119 (1981).
  21. E. Benoit, “Systèmes lents-rapides dans R3 et leurs canards,” Astérisque 109–110, 159–191 (1983).
  22. W. Eckhaus, “Relaxation oscillations including a standart chase on French ducks,” Lect. Notes Math. 985, 449–494(1983).
    [CrossRef]
  23. A. K. Zvonkin and M. A. Shubin, “Non-standard analysis and singular perturbations of ordinary differential equations,” Russ. Math. Surv. 39, 69–131 (1984).
    [CrossRef]
  24. G. N. Gorelov and V. A. Sobolev, “Mathematical modeling of critical phenomena in thermal explosion theory,” Combust. Flame 87, 203–210 (1991).
    [CrossRef]
  25. G. N. Gorelov and V. A. Sobolev, “Duck-trajectories in a thermal explosion problem,” Appl. Math. Lett. 5, 3–6 (1992).
    [CrossRef]
  26. E. Shchepakina and V. Sobolev, “Black swans and canards in laser and combustion models,” in Singular Perturbations and Hysteresis, M.P.Mortell, R.E.O’Malley, A.Pokrovskii, and V.A.Sobolev, eds. (SIAM, 2005), pp. 207–255.
    [CrossRef]
  27. V. Sobolev and E. Shchepakina, “Duck trajectories in a problem of combustion theory,” Differ. Equ. 32, 1177–1186 (1996).
  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, 074104 (2007).
    [CrossRef] [PubMed]
  29. M. Desroches, B. Krauskopf, and H. M. Osinga, “Numerical continuation of canard orbits in slow-fast dynamical systems,” Nonlinearity 23, 739–765 (2010).
    [CrossRef]
  30. 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 Res. 14, 331–335 (1976).
  31. S. Balle, Institut Mediterrani d’Estudis Avançats, CSIC-UIB, E-07071, Palma de Mallorca, Spain (personal communication, 2011).

2010 (1)

M. Desroches, B. Krauskopf, and H. M. Osinga, “Numerical continuation of canard orbits in slow-fast dynamical systems,” Nonlinearity 23, 739–765 (2010).
[CrossRef]

2007 (1)

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, 074104 (2007).
[CrossRef] [PubMed]

2005 (1)

M. Brøns, “Relaxation oscillations and canards in a nonlinear model of discontinuous plastic deformation in metals at very low temperatures,” Proc. R. Soc. A 461, 2289–2302 (2005).
[CrossRef]

2004 (3)

F. Marino, G. Catalán, P. Sánchez, S. Balle, and O. Piro, “Thermo-optical “canard orbits” and excitable limit cycles,” Phys. Rev. Lett. 92, 073901 (2004).
[CrossRef] [PubMed]

A. W. L. Chan, K. L. Lee, and C. Shu, “Self-starting photonic clock using semiconductor optical amplifier based Mach-Zehnder interferometer,” Electron. Lett. 40, 827–828 (2004).
[CrossRef]

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, 227–249 (2004).
[CrossRef]

2003 (1)

E. I. Volkov, E. Ullner, A. A. Zaikin, and J. Kurths, “Oscillatory amplification of stochastic resonance in excitable systems,” Phys. Rev. E 68, 026214 (2003).
[CrossRef]

2002 (1)

J. Moehlis, “Canards in a surface oxidation reaction,” J. Nonlinear Sci. 12, 319–345 (2002).
[CrossRef]

2001 (1)

M. Brøns and J. Sturis, “Explosion of limit cycles and chaotic waves in a simple nonlinear chemical system,” Phys. Rev. E 64, 026209 (2001).
[CrossRef]

1996 (1)

V. Sobolev and E. Shchepakina, “Duck trajectories in a problem of combustion theory,” Differ. Equ. 32, 1177–1186 (1996).

1994 (1)

M. Brøns and K. Bar-Eli, “Asymptotic analysis of canards in the EOE equations and the role of the inflection line,” Proc. R. Soc. A 445, 305–322 (1994).
[CrossRef]

1992 (1)

G. N. Gorelov and V. A. Sobolev, “Duck-trajectories in a thermal explosion problem,” Appl. Math. Lett. 5, 3–6 (1992).
[CrossRef]

1991 (2)

G. N. Gorelov and V. A. Sobolev, “Mathematical modeling of critical phenomena in thermal explosion theory,” Combust. Flame 87, 203–210 (1991).
[CrossRef]

M. Brøns and K. Bar-Eli, “Canard explosion and excitation in a model of the Belousov-Zhabotinsky reaction,” J. Phys. Chem. 95, 8706–8713 (1991).
[CrossRef]

1986 (1)

S. Baer and T. Erneux, “Singular Hopf bifurcation to relaxation oscillations,” SIAM J. Appl. Math. 46, 721–739 (1986).
[CrossRef]

1984 (1)

A. K. Zvonkin and M. A. Shubin, “Non-standard analysis and singular perturbations of ordinary differential equations,” Russ. Math. Surv. 39, 69–131 (1984).
[CrossRef]

1983 (2)

E. Benoit, “Systèmes lents-rapides dans R3 et leurs canards,” Astérisque 109–110, 159–191 (1983).

W. Eckhaus, “Relaxation oscillations including a standart chase on French ducks,” Lect. Notes Math. 985, 449–494(1983).
[CrossRef]

1981 (1)

E. Benoit, J. L. Calot, F. Diener, and M. Diener, “Chasse au canard,” Collectanea Mathematica 31–32, 37–119 (1981).

1976 (1)

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 Res. 14, 331–335 (1976).

Baer, S.

S. Baer and T. Erneux, “Singular Hopf bifurcation to relaxation oscillations,” SIAM J. Appl. Math. 46, 721–739 (1986).
[CrossRef]

Balle, S.

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, 074104 (2007).
[CrossRef] [PubMed]

F. Marino, G. Catalán, P. Sánchez, S. Balle, and O. Piro, “Thermo-optical “canard orbits” and excitable limit cycles,” Phys. Rev. Lett. 92, 073901 (2004).
[CrossRef] [PubMed]

S. Balle, Institut Mediterrani d’Estudis Avançats, CSIC-UIB, E-07071, Palma de Mallorca, Spain (personal communication, 2011).

Bar-Eli, K.

M. Brøns and K. Bar-Eli, “Asymptotic analysis of canards in the EOE equations and the role of the inflection line,” Proc. R. Soc. A 445, 305–322 (1994).
[CrossRef]

M. Brøns and K. Bar-Eli, “Canard explosion and excitation in a model of the Belousov-Zhabotinsky reaction,” J. Phys. Chem. 95, 8706–8713 (1991).
[CrossRef]

Benoit, E.

E. Benoit, “Systèmes lents-rapides dans R3 et leurs canards,” Astérisque 109–110, 159–191 (1983).

E. Benoit, J. L. Calot, F. Diener, and M. Diener, “Chasse au canard,” Collectanea Mathematica 31–32, 37–119 (1981).

Braaksma, B.

B. Braaksma, “Critical phenomena in dynamical systems of van der Pol type,” Ph.D. thesis (University of Utrecht, 1993).

Brøns, M.

M. Brøns, “Relaxation oscillations and canards in a nonlinear model of discontinuous plastic deformation in metals at very low temperatures,” Proc. R. Soc. A 461, 2289–2302 (2005).
[CrossRef]

M. Brøns and J. Sturis, “Explosion of limit cycles and chaotic waves in a simple nonlinear chemical system,” Phys. Rev. E 64, 026209 (2001).
[CrossRef]

M. Brøns and K. Bar-Eli, “Asymptotic analysis of canards in the EOE equations and the role of the inflection line,” Proc. R. Soc. A 445, 305–322 (1994).
[CrossRef]

M. Brøns and K. Bar-Eli, “Canard explosion and excitation in a model of the Belousov-Zhabotinsky reaction,” J. Phys. Chem. 95, 8706–8713 (1991).
[CrossRef]

Butuzov, V. F.

A. B. Vasil’eva, V. F. Butuzov, and L. V. Kalachev, The Boundary Function Method for Singular Perturbation Problems (SIAM, 1995), Vol.  14.
[CrossRef]

Calot, J. L.

E. Benoit, J. L. Calot, F. Diener, and M. Diener, “Chasse au canard,” Collectanea Mathematica 31–32, 37–119 (1981).

Catalán, G.

F. Marino, G. Catalán, P. Sánchez, S. Balle, and O. Piro, “Thermo-optical “canard orbits” and excitable limit cycles,” Phys. Rev. Lett. 92, 073901 (2004).
[CrossRef] [PubMed]

Chan, A. W. L.

A. W. L. Chan, K. L. Lee, and C. Shu, “Self-starting photonic clock using semiconductor optical amplifier based Mach-Zehnder interferometer,” Electron. Lett. 40, 827–828 (2004).
[CrossRef]

Desroches, M.

M. Desroches, B. Krauskopf, and H. M. Osinga, “Numerical continuation of canard orbits in slow-fast dynamical systems,” Nonlinearity 23, 739–765 (2010).
[CrossRef]

Diener, F.

E. Benoit, J. L. Calot, F. Diener, and M. Diener, “Chasse au canard,” Collectanea Mathematica 31–32, 37–119 (1981).

Diener, M.

E. Benoit, J. L. Calot, F. Diener, and M. Diener, “Chasse au canard,” Collectanea Mathematica 31–32, 37–119 (1981).

M. Diener, Nessie et Les Canards (Publication IRMA, 1979).

Eckhaus, W.

W. Eckhaus, “Relaxation oscillations including a standart chase on French ducks,” Lect. Notes Math. 985, 449–494(1983).
[CrossRef]

Erneux, T.

S. Baer and T. Erneux, “Singular Hopf bifurcation to relaxation oscillations,” SIAM J. Appl. Math. 46, 721–739 (1986).
[CrossRef]

Gorelov, G. N.

G. N. Gorelov and V. A. Sobolev, “Duck-trajectories in a thermal explosion problem,” Appl. Math. Lett. 5, 3–6 (1992).
[CrossRef]

G. N. Gorelov and V. A. Sobolev, “Mathematical modeling of critical phenomena in thermal explosion theory,” Combust. Flame 87, 203–210 (1991).
[CrossRef]

Grasman, J.

J. Grasman, Asymptotic Methods for Relaxation Oscillations and Applications (Springer-Verlag, 1987).
[CrossRef]

Inaba, N.

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, 227–249 (2004).
[CrossRef]

Kalachev, L. V.

A. B. Vasil’eva, V. F. Butuzov, and L. V. Kalachev, The Boundary Function Method for Singular Perturbation Problems (SIAM, 1995), Vol.  14.
[CrossRef]

Kolesov, A. Y.

E. F. Mishchenko, Y. S. Kolesov, A. Y. Kolesov, and N. K. Rozov, Asymptotic Methods in Singularly Perturbed Systems(Plenum, 1995).

Kolesov, Y. S.

E. F. Mishchenko, Y. S. Kolesov, A. Y. Kolesov, and N. K. Rozov, Asymptotic Methods in Singularly Perturbed Systems(Plenum, 1995).

Krauskopf, B.

M. Desroches, B. Krauskopf, and H. M. Osinga, “Numerical continuation of canard orbits in slow-fast dynamical systems,” Nonlinearity 23, 739–765 (2010).
[CrossRef]

Kurths, J.

E. I. Volkov, E. Ullner, A. A. Zaikin, and J. Kurths, “Oscillatory amplification of stochastic resonance in excitable systems,” Phys. Rev. E 68, 026214 (2003).
[CrossRef]

Lee, K. L.

A. W. L. Chan, K. L. Lee, and C. Shu, “Self-starting photonic clock using semiconductor optical amplifier based Mach-Zehnder interferometer,” Electron. Lett. 40, 827–828 (2004).
[CrossRef]

Marino, F.

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, 074104 (2007).
[CrossRef] [PubMed]

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, 074104 (2007).
[CrossRef] [PubMed]

F. Marino, G. Catalán, P. Sánchez, S. Balle, and O. Piro, “Thermo-optical “canard orbits” and excitable limit cycles,” Phys. Rev. Lett. 92, 073901 (2004).
[CrossRef] [PubMed]

Mishchenko, E. F.

E. F. Mishchenko, Y. S. Kolesov, A. Y. Kolesov, and N. K. Rozov, Asymptotic Methods in Singularly Perturbed Systems(Plenum, 1995).

E. F. Mishchenko and N. K. Rozov, Differential Equations with Small Parameters and Relaxation Oscillations (Plenum, 1980).

Moehlis, J.

J. Moehlis, “Canards in a surface oxidation reaction,” J. Nonlinear Sci. 12, 319–345 (2002).
[CrossRef]

Murray, J. D.

J. D. Murray, Mathematical Biology (Springer-Verlag, 2003).

O’Malley, R. E.

R. E. O’Malley, Jr., Introduction to Singular Perturbations (Academic, 1974).

Osinga, H. M.

M. Desroches, B. Krauskopf, and H. M. Osinga, “Numerical continuation of canard orbits in slow-fast dynamical systems,” Nonlinearity 23, 739–765 (2010).
[CrossRef]

Piro, O.

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, 074104 (2007).
[CrossRef] [PubMed]

F. Marino, G. Catalán, P. Sánchez, S. Balle, and O. Piro, “Thermo-optical “canard orbits” and excitable limit cycles,” Phys. Rev. Lett. 92, 073901 (2004).
[CrossRef] [PubMed]

Rozov, N. K.

E. F. Mishchenko and N. K. Rozov, Differential Equations with Small Parameters and Relaxation Oscillations (Plenum, 1980).

E. F. Mishchenko, Y. S. Kolesov, A. Y. Kolesov, and N. K. Rozov, Asymptotic Methods in Singularly Perturbed Systems(Plenum, 1995).

Sánchez, P.

F. Marino, G. Catalán, P. Sánchez, S. Balle, and O. Piro, “Thermo-optical “canard orbits” and excitable limit cycles,” Phys. Rev. Lett. 92, 073901 (2004).
[CrossRef] [PubMed]

Sekikawa, M.

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, 227–249 (2004).
[CrossRef]

Shchepakina, E.

V. Sobolev and E. Shchepakina, “Duck trajectories in a problem of combustion theory,” Differ. Equ. 32, 1177–1186 (1996).

E. Shchepakina and V. Sobolev, “Black swans and canards in laser and combustion models,” in Singular Perturbations and Hysteresis, M.P.Mortell, R.E.O’Malley, A.Pokrovskii, and V.A.Sobolev, eds. (SIAM, 2005), pp. 207–255.
[CrossRef]

Shu, C.

A. W. L. Chan, K. L. Lee, and C. Shu, “Self-starting photonic clock using semiconductor optical amplifier based Mach-Zehnder interferometer,” Electron. Lett. 40, 827–828 (2004).
[CrossRef]

Shubin, M. A.

A. K. Zvonkin and M. A. Shubin, “Non-standard analysis and singular perturbations of ordinary differential equations,” Russ. Math. Surv. 39, 69–131 (1984).
[CrossRef]

Sobolev, V.

V. Sobolev and E. Shchepakina, “Duck trajectories in a problem of combustion theory,” Differ. Equ. 32, 1177–1186 (1996).

E. Shchepakina and V. Sobolev, “Black swans and canards in laser and combustion models,” in Singular Perturbations and Hysteresis, M.P.Mortell, R.E.O’Malley, A.Pokrovskii, and V.A.Sobolev, eds. (SIAM, 2005), pp. 207–255.
[CrossRef]

Sobolev, V. A.

G. N. Gorelov and V. A. Sobolev, “Duck-trajectories in a thermal explosion problem,” Appl. Math. Lett. 5, 3–6 (1992).
[CrossRef]

G. N. Gorelov and V. A. Sobolev, “Mathematical modeling of critical phenomena in thermal explosion theory,” Combust. Flame 87, 203–210 (1991).
[CrossRef]

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 Res. 14, 331–335 (1976).

Strygin, V. V.

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 Res. 14, 331–335 (1976).

Sturis, J.

M. Brøns and J. Sturis, “Explosion of limit cycles and chaotic waves in a simple nonlinear chemical system,” Phys. Rev. E 64, 026209 (2001).
[CrossRef]

Tsubouchi, T.

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, 227–249 (2004).
[CrossRef]

Ullner, E.

E. I. Volkov, E. Ullner, A. A. Zaikin, and J. Kurths, “Oscillatory amplification of stochastic resonance in excitable systems,” Phys. Rev. E 68, 026214 (2003).
[CrossRef]

Vasil’eva, A. B.

A. B. Vasil’eva, V. F. Butuzov, and L. V. Kalachev, The Boundary Function Method for Singular Perturbation Problems (SIAM, 1995), Vol.  14.
[CrossRef]

Volkov, E. I.

E. I. Volkov, E. Ullner, A. A. Zaikin, and J. Kurths, “Oscillatory amplification of stochastic resonance in excitable systems,” Phys. Rev. E 68, 026214 (2003).
[CrossRef]

Zaikin, A. A.

E. I. Volkov, E. Ullner, A. A. Zaikin, and J. Kurths, “Oscillatory amplification of stochastic resonance in excitable systems,” Phys. Rev. E 68, 026214 (2003).
[CrossRef]

Zvonkin, A. K.

A. K. Zvonkin and M. A. Shubin, “Non-standard analysis and singular perturbations of ordinary differential equations,” Russ. Math. Surv. 39, 69–131 (1984).
[CrossRef]

Appl. Math. Lett. (1)

G. N. Gorelov and V. A. Sobolev, “Duck-trajectories in a thermal explosion problem,” Appl. Math. Lett. 5, 3–6 (1992).
[CrossRef]

Astérisque (1)

E. Benoit, “Systèmes lents-rapides dans R3 et leurs canards,” Astérisque 109–110, 159–191 (1983).

Collectanea Mathematica (1)

E. Benoit, J. L. Calot, F. Diener, and M. Diener, “Chasse au canard,” Collectanea Mathematica 31–32, 37–119 (1981).

Combust. Flame (1)

G. N. Gorelov and V. A. Sobolev, “Mathematical modeling of critical phenomena in thermal explosion theory,” Combust. Flame 87, 203–210 (1991).
[CrossRef]

Cosmic Res. (1)

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 Res. 14, 331–335 (1976).

Differ. Equ. (1)

V. Sobolev and E. Shchepakina, “Duck trajectories in a problem of combustion theory,” Differ. Equ. 32, 1177–1186 (1996).

Electron. Lett. (1)

A. W. L. Chan, K. L. Lee, and C. Shu, “Self-starting photonic clock using semiconductor optical amplifier based Mach-Zehnder interferometer,” Electron. Lett. 40, 827–828 (2004).
[CrossRef]

J. Nonlinear Sci. (1)

J. Moehlis, “Canards in a surface oxidation reaction,” J. Nonlinear Sci. 12, 319–345 (2002).
[CrossRef]

J. Phys. Chem. (1)

M. Brøns and K. Bar-Eli, “Canard explosion and excitation in a model of the Belousov-Zhabotinsky reaction,” J. Phys. Chem. 95, 8706–8713 (1991).
[CrossRef]

Lect. Notes Math. (1)

W. Eckhaus, “Relaxation oscillations including a standart chase on French ducks,” Lect. Notes Math. 985, 449–494(1983).
[CrossRef]

Nonlinearity (1)

M. Desroches, B. Krauskopf, and H. M. Osinga, “Numerical continuation of canard orbits in slow-fast dynamical systems,” Nonlinearity 23, 739–765 (2010).
[CrossRef]

Phys. Rev. E (2)

M. Brøns and J. Sturis, “Explosion of limit cycles and chaotic waves in a simple nonlinear chemical system,” Phys. Rev. E 64, 026209 (2001).
[CrossRef]

E. I. Volkov, E. Ullner, A. A. Zaikin, and J. Kurths, “Oscillatory amplification of stochastic resonance in excitable systems,” Phys. Rev. E 68, 026214 (2003).
[CrossRef]

Phys. Rev. Lett. (2)

F. Marino, G. Catalán, P. Sánchez, S. Balle, and O. Piro, “Thermo-optical “canard orbits” and excitable limit cycles,” Phys. Rev. Lett. 92, 073901 (2004).
[CrossRef] [PubMed]

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, 074104 (2007).
[CrossRef] [PubMed]

Physica D (1)

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, 227–249 (2004).
[CrossRef]

Proc. R. Soc. A (2)

M. Brøns, “Relaxation oscillations and canards in a nonlinear model of discontinuous plastic deformation in metals at very low temperatures,” Proc. R. Soc. A 461, 2289–2302 (2005).
[CrossRef]

M. Brøns and K. Bar-Eli, “Asymptotic analysis of canards in the EOE equations and the role of the inflection line,” Proc. R. Soc. A 445, 305–322 (1994).
[CrossRef]

Russ. Math. Surv. (1)

A. K. Zvonkin and M. A. Shubin, “Non-standard analysis and singular perturbations of ordinary differential equations,” Russ. Math. Surv. 39, 69–131 (1984).
[CrossRef]

SIAM J. Appl. Math. (1)

S. Baer and T. Erneux, “Singular Hopf bifurcation to relaxation oscillations,” SIAM J. Appl. Math. 46, 721–739 (1986).
[CrossRef]

Other (11)

B. Braaksma, “Critical phenomena in dynamical systems of van der Pol type,” Ph.D. thesis (University of Utrecht, 1993).

E. F. Mishchenko, Y. S. Kolesov, A. Y. Kolesov, and N. K. Rozov, Asymptotic Methods in Singularly Perturbed Systems(Plenum, 1995).

M.P.Mortell, R.E.O’Malley, A.Pokrovskii, and V.A.Sobolev, eds. Singular Perturbations and Hysteresis (SIAM, 2005).
[CrossRef]

R. E. O’Malley, Jr., Introduction to Singular Perturbations (Academic, 1974).

A. B. Vasil’eva, V. F. Butuzov, and L. V. Kalachev, The Boundary Function Method for Singular Perturbation Problems (SIAM, 1995), Vol.  14.
[CrossRef]

E. F. Mishchenko and N. K. Rozov, Differential Equations with Small Parameters and Relaxation Oscillations (Plenum, 1980).

J. D. Murray, Mathematical Biology (Springer-Verlag, 2003).

J. Grasman, Asymptotic Methods for Relaxation Oscillations and Applications (Springer-Verlag, 1987).
[CrossRef]

M. Diener, Nessie et Les Canards (Publication IRMA, 1979).

E. Shchepakina and V. Sobolev, “Black swans and canards in laser and combustion models,” in Singular Perturbations and Hysteresis, M.P.Mortell, R.E.O’Malley, A.Pokrovskii, and V.A.Sobolev, eds. (SIAM, 2005), pp. 207–255.
[CrossRef]

S. Balle, Institut Mediterrani d’Estudis Avançats, CSIC-UIB, E-07071, Palma de Mallorca, Spain (personal communication, 2011).

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Figures (3)

Fig. 1
Fig. 1

(a) Canard (thick line) and the slow curve (thin line) of the van der Pol’s type system. (b) Slow curve (thick line) and the trajectories (thin lines) of the van der Pol’s type systems. Parts S 1 s and S 3 s of the slow curve are stable, S 2 u is unstable, and A 1 and A 2 are the jump points.

Fig. 2
Fig. 2

Slow curve of the system (11, 12); P i = 0.005 , R 1 = 0.49 , R 2 = 0.999 , α = 3 , and G 0 = 0.357 . Parts S 1 and S 3 are stable; S 2 is unstable.

Fig. 3
Fig. 3

Canard explosion of the system (11, 12): the slow curve (thin lines) and the trajectory (thick lines) for ε = 0.015 , P = 0.005 , R 1 = 0.49 , R 2 = 0.999 , λ = 0.005 , α = 3 , G 0 = 0.357 , ϕ e = 0.8899 , and (a)  μ = 1.9 , (b)  μ = 2 , (c)  μ = 2.002639 , (d)  μ = 2.0026395498095733099 , (e)  μ = 2.0026395498095733185 , (f)  μ = 2.0026395498097 , (g)  μ = 2.0027 , (h)  μ = 2.003 .

Equations (36)

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d x d t = f ( x , y , μ ) ,
ε d y d t = g ( x , y , μ ) ,
d x d τ = ε f ( x , y , μ ) ,
d y d τ = g ( x , y , μ ) .
y = h ( x , ε ) = h 0 ( x ) + ε h 1 ( x ) + ε 2 h 2 ( x ) + .
μ c = μ 0 + ε μ 1 + ε 2 μ 2 + .
ε d y d x = g ( x , y , μ ) f ( x , y , μ ) ,
ε d h d x f ( x , h ( x , ε ) , μ ) = g ( x , h ( x , ε ) , μ ) ,
d G d t = γ sp ( G 0 G P i Q ) ,
d ϕ d t = γ th ( ϕ ϕ e + λ G + μ P i Q ) ,
Q = ( 1 R 1 ) ( 1 + R 2 e G ) ( e G 1 ) 1 + R 1 R 2 e 2 G 2 R 1 R 2 e G cos ( ϕ α G ) ,
ε d G d τ = G 0 G P i Q ,
d ϕ d τ = ( ϕ ϕ e + λ G + μ P i Q ) ,
ϕ = ϕ 0 ( G ) + ε ϕ 1 ( G ) + ε 2 ϕ 2 ( G ) + , μ = μ c = μ 0 + ε μ 1 + ε 2 μ 2 + .
Q = Q 0 + ε Q 1 + ε 2 Q 2 + ,
Q 0 = Q 0 ( G , ϕ 0 ) = ( 1 R 1 ) ( 1 + R 2 e G ) ( e G 1 ) 1 + R 1 R 2 e 2 G 2 R 1 R 2 e G cos ( ϕ 0 α G ) ,
Q 1 = Q 1 ( G , ϕ 0 , ϕ 1 ) = ϕ 1 Q 0 2 2 R 1 R 2 e G sin ( ϕ 0 α G ) ( 1 R 1 ) ( 1 + R 2 e G ) ( e G 1 ) ,
( ϕ 0 + ε ϕ 1 + ε 2 ϕ 2 + ) ( G 0 G P i Q 0 ε P i Q 1 ε 2 P i Q 2 ) = ε [ ϕ 0 + ε ϕ 1 + ϕ e + λ G + ( μ 0 + ε μ 1 + ) P i ( Q 0 + ε Q 1 + ) ] .
g ( ϕ 0 , G ) = G 0 G P i Q 0 = 0 .
P i 2 R 1 R 2 e G ( 1 R 1 ) ( 1 + R 2 e G ) ( e G 1 ) [ R 1 R 2 e G cos ( ϕ 0 α G ) α sin ( ϕ 0 α G ) ] [ 1 + R 1 R 2 e 2 G 2 R 1 R 2 e G cos ( ϕ 0 α G ) ] 2 1 P i ( 1 R 1 ) ( 2 R 2 e G R 2 + 1 ) e G 1 + R 1 R 2 e 2 G 2 R 1 R 2 e G cos ( ϕ 0 α G ) = 0 .
ϕ 1 ( G 0 G P i Q 0 ) ϕ 0 P i Q 1 = ( ϕ 0 ϕ e + λ G + μ 0 P i Q 0 )
ϕ 0 P i Q 0 2 2 R 1 R 2 e G sin ( ϕ 0 α G ) ( 1 R 1 ) ( 1 + R 2 e G ) ( e G 1 ) ϕ 1 = ϕ 0 ϕ e + λ G + μ 0 P i Q 0 .
[ ϕ 0 ϕ e + λ G + μ 0 P i Q 0 ] | G = G j = 0
μ 0 = ( ϕ e ϕ 0 ( G j ) λ G j ) [ 1 + R 1 R 2 e 2 G j 2 R 1 R 2 e G j cos ( ϕ 0 ( G j ) α G j ) ] P i ( 1 R 1 ) ( 1 + R 2 e G j ) ( e G j 1 ) ,
ϕ 2 ( G 0 G P i Q 0 ) ϕ 1 P i Q 1 ϕ 0 P i Q 2 = ϕ 1 μ 0 P i Q 1 μ 1 P i Q 0
ϕ 1 P i Q 1 + ϕ 0 P i Q 2 = ϕ 1 + μ 0 P i Q 1 + μ 1 P i Q 0 .
[ P i Q 1 ( μ 0 ϕ 1 ) + ϕ 1 + μ 1 P i Q 0 ] | G = G j = 0
μ 1 = 1 Q 0 [ Q 1 ( ϕ 1 μ 0 ) ϕ 1 P i ] | G = G j
μ 1 = P i Q 1 ( G j , ϕ 0 ( G j ) , ϕ 1 ( G j ) ) ( ϕ 1 ( G j ) μ 0 ) ϕ 1 ( G j ) P i Q 0 ( G j , ϕ 0 ( G j ) ) .
G 0 c = ζ 0 + ε ζ 1 + ε 2 ζ 2 +
g ( ϕ 0 , G ) = ζ 0 G P i Q 0 = 0 ,
ϕ 0 = arccos 1 2 R 1 R 2 e G [ 1 + R 1 R 2 e 2 G P i ( 1 R 1 ) ( 1 + R 2 e G ) ( e G 1 ) ζ 0 G ] + α G .
ϕ 0 [ ζ 1 + P i Q 0 2 2 R 1 R 2 e G sin ( ϕ 0 α G ) ( 1 R 1 ) ( 1 + R 2 e G ) ( e G 1 ) ϕ 1 ] = ϕ 0 ϕ e + λ G + μ P i Q 0 ,
ϕ 1 ( ζ 1 P i Q 1 ) + ϕ 0 ( ζ 2 P i Q 2 ) = ϕ 1 μ P i Q 1
[ ϕ 0 ϕ e + λ G + μ ( ζ 0 G ) ] | G = G j = 0
[ ϕ 1 ( ζ 1 P i Q 1 ) + ϕ 1 + μ P i Q 1 ] | G = G j = 0 .

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