Abstract

Checks of homoclinic chaos made with nonlinear analysis techniques have been performed on the signals coming from a CO2 laser containing CH3I as a saturable absorber. The one-dimensional return maps of the regimes appearing inside the alternating periodic chaotic sequence are typical of homoclinic chaos. Numerical simulations give results in good agreement with the experimental observations. In the case of a fast absorber, a homoclinic tangency to a cycle created in a suberitical Hopf bifurcation is seen to be responsible for the homo-clinic behavior observed in the model.

© 1991 Optical Society of America

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  1. I. Burak, P. L. Houston, D. G. Sutton, and J. I. Steinfeld, “Mechanism of passive Q-switching in CO2lasers,” IEEE J. Quantum Electron. QE-7, 73–82 (1971).
    [Crossref]
  2. J. Dupré, F. Meyer, and C. Meyer, “Influence des phénomènes de relaxation sur la forme des impulsions fournies par un laser CO2déclenché par un absorbant saturable,” Rev. Phys. Appl. 10, 285–293 (1975).
    [Crossref]
  3. E. Arimondo, F. Casagrande, L. Lugiato, and P. Glorieux, “Repetitive passive Q-switching and bistability in lasers with saturable absorber,” Appl. Phys. B 30, 57–77 (1983).
    [Crossref]
  4. M. L. Asquini and F. Casagrande, “Passive Q-switching in lasers with saturable absorbers: improved treatment of a four level method,” Nuovo Cimento 2D, 917–931 (1983).
    [Crossref]
  5. A. Jacques and P. Glorieux, “Observation of bistability in a CO2laser exhibiting passive Q-switching,” Opt. Commun. 40, 455–460 (1982).
    [Crossref]
  6. E. Arimondo and E. Menchi, “Analysis of Q-switch in a CO2laser with saturable absorber,” Appl. Phys. B 37, 55–61 (1985).
    [Crossref]
  7. M. Tachikawa, K. Tanii, M. Kajita, and T. Shimizu, “Undamped undulation superposed on the passive Q-switching pulse of a CO2laser,” Appl. Phys. B 39, 83–90 (1986).
    [Crossref]
  8. K. Tanii, M. Tachikawa, M. Kajita, and T. Shimizu, “Sinusoidal self-modulation in the output of a CO2laser with an intracavity saturable absorber,” J. Opt. Soc. Am. B 5, 24–28 (1988).
    [Crossref]
  9. E. Arimondo, P. Bootz, P. Glorieux, and E. Menchi, “Pulse shape and phase diagram in the passive Q-switching of CO2lasers,” J. Opt. Soc. Am. B 2, 193–201 (1985).
    [Crossref]
  10. M. Tachikawa, K. Tanii, and T. Shimizu, “Comprehensive interpretation of passive Q-switching and optical bistability in a CO2laser with an intracavity saturable absorber,” J. Opt. Soc. Am. B 4, 387–395 (1987).
    [Crossref]
  11. B. Zambon, F. de Tomasi, D. Hennequin, and E. Arimondo, “Investigations of models for the laser with a saturable absorber: theoretical and experimental studies of the stationary regimes,” Phys. Rev. A 40, 3782–3795 (1989).
    [Crossref] [PubMed]
  12. D. Dangoisse, A. Bekkali, F. Papoff, and P. Glorieux, “Shil’nikov dynamics in a passive Q-switching laser,” Europhys. Lett. 6, 335–340 (1988).
    [Crossref]
  13. D. Hennequin, F. de Tomasi, B. Zambon, and E. Arimondo, “Homoclinic orbits and cycles in the instabilities of a laser with saturable absorber,” Phys. Rev. A 37, 2243–2246 (1988).
    [Crossref] [PubMed]
  14. M. Tachikawa, K. Tanii, and T. Shimizu, “Laser instability and chaotic pulsation in a CO2laser with intracavity saturable absorber,” J. Opt. Soc. Am. B 5, 1077–1081 (1988).
    [Crossref]
  15. F. de Tomasi, D. Hennequin, B. Zambon, and E. Arimondo, “Instabilities and chaos in an infrared laser with saturable absorber: experiments and vibrorotational model,” J. Opt. Soc. Am. B 6, 45–57 (1989).
    [Crossref]
  16. M. Tachikawa, F. L. Hong, K. Tanii, and T. Shimizu, “Deterministic chaos in passive Q-switching pulsation of a CO2laser with saturable absorber,” Phys. Rev. Lett. 60, 2266–2268 (1988).
    [Crossref] [PubMed]
  17. P. Glendinning and C. Sparrow, “Local and global behavior near homoclinic orbits,” J. Stat. Phys. 35, 645–696 (1984).
    [Crossref]
  18. P. Gaspard, R. Kapral, and G. Nicolis, “Bifurcation phenomena near homoclinic systems: a two-parameter analysis,” J. Stat. Phys. 35, 697–727 (1984).
    [Crossref]
  19. L. P. Shil’nikov, “A case of the existence of a countable number of periodic motions,” Sov. Math. Dokl. 6, 163–166 (1965); “A contribution to the problem of the structure of an extended neighborhood of a rough equilibrium state of saddle focus type,” Math. USSR Sbornik 10, 91–102 (1970).
  20. D. Hennequin, F. de Tomasi, L. Fronzoni, B. Zambon, and E. Arimondo, “Influence of noise on the quasi-homoclinic behavior of a laser with saturable absorber,” Opt. Commun. 70, 253–258 (1989).
    [Crossref]
  21. D. Hennequin, M. Lefranc, A. Bekkali, D. Dangoisse, and P. Glorieux, “Characterization of Shil’nikov chaos in a CO2laser containing a saturable absorber” in Measures of Complexity and Chaos, N. B. Abraham, A. M. Albano, A. Passa mente, and P. E. Rapp, eds. (Plenum, New York, 1989), pp. 299–302.
    [Crossref]
  22. F. Papoff, A. Fioretti, E. Arimondo, and N. B. Abraham, “Time return maps and distributions for the laser with a sat urable absorber,” in Measures of Complexity and Chaos, N. B. Abraham, A. M. Albano, A. Passamente, and P. E. Rapp, eds. (Plenum, New York, 1989), pp. 309–312.
    [Crossref]
  23. F. T. Arecchi, A. Lapucci, R. Meucci, J. A. Roversi, and P. H. Coullet, “Experimental characterization of Shil’nikov chaos by statistics of return times,” Europhys. Lett. 6, 677–682 (1988); F. T. Arecchi, W. Gadomski, A. Lapucci, H. Mancini, R. Meucci, and J. A. Roversi, “Laser with feed back: an optical implementation of competing instabilities, Shil’nikov chaos, and transient fluctuation enhancement,” J. Opt. Soc. Am. B 5, 1153–1159 (1988).
    [Crossref]
  24. F. Argoul, A. Arneodo, and P. Richetti, “Experimental evidence for homoclinic chaos in the Belousov–Zhabotinskii reaction,” Phys. Lett. A 120, 269–275 (1987); F. Argoul, A. Arneodo, and P. Richetti, “Dynamique symbolique dans la réaction de Belousov–Zhabotinskii: une illustration expérimentale de la théorie de Shil’nikov des orbites homoclines,” J. Chim. Phys. 84, 1367–1385 (1987).
    [Crossref]
  25. E. Arimondo and P. Glorieux, “Saturated absorption experiments on a dressed molecule. Application to the spectroscopy of the ν6band of CH3I,” Phys. Rev. A 19, 1067–1083 (1979).
    [Crossref]
  26. See, e.g., J. S. Turner, J. C. Roux, W. D. McCormick, and H. L. Swinney, “Alternating periodic and chaotic regimes in chemical reaction–experiments and theory,” Phys. Lett 85A, 9–12 (1981).
  27. A. Arneodo, P. Coullet, E. A. Spiegel, and C. Tresser, “Asymptotic chaos,” Physica D 14, 327–347 (1985).
    [Crossref]
  28. H. T. Powell and G. J. Wolga, “Repetitive passive Q-switching of single-frequency lasers,” IEEE J. Quantum Electron QE-7, 213–219 (1971).
    [Crossref]
  29. T. Erneux and P. Mandel, “Bifurcation phenomena in a with saturable absorber I and II,” Z. Phys. B 44, 353–363, 365–374 (1981); P. Mandel and T. Erneux, “Stationary, harmonic and pulsed operations of an optically bistable with saturable absorber. I and II,” Phys. Rev. A 30, 1893–1901, 1902–1909 (1984); T. Erneux, P. Mandel, and J. Magnan, “Quasi-periodicity in lasers with saturable absorbers,” Phys. Rev. A 29, 2690–2699 (1984); D. E. Chyba, N. B. Abraham, and A. M. Albano, “Semiclassical analysis of a detuned ring laser with a saturable absorber. New results for steady states,” Phys. Rev. A 35, 2936–2950 (1987).
    [Crossref] [PubMed]
  30. C. Tresser, “About some theorems by L. P. Shil’nikov,” Ann. Inst. Henri Poincaré 40, 441–461 (1984).
  31. P. Fredrickson, J. L. Kaplan, E. D. Yorke, and J. A. Yorke, “The Lyapunov dimension of strange attractors,” J. Diff. 49, 185–207 (1983).
    [Crossref]
  32. F. Ledrappier, “Some relations between dimension Lyapunov exponent,” Commun. Math. Phys. 81, 229–238 (1981).
    [Crossref]
  33. P. Gaspard and X. J. Wang, “Homoclinic orbits and mixed mode oscillations in far from equilibrium systems,” J. Phys. 48, 151–199 (1987).
  34. C. Sparrow, The Lorenz Equations, Bifurcations, Chaos Strange Attractors (Springer-Verlag, Berlin, 1982), pp. 211–220, App. E.

1989 (3)

D. Hennequin, F. de Tomasi, L. Fronzoni, B. Zambon, and E. Arimondo, “Influence of noise on the quasi-homoclinic behavior of a laser with saturable absorber,” Opt. Commun. 70, 253–258 (1989).
[Crossref]

B. Zambon, F. de Tomasi, D. Hennequin, and E. Arimondo, “Investigations of models for the laser with a saturable absorber: theoretical and experimental studies of the stationary regimes,” Phys. Rev. A 40, 3782–3795 (1989).
[Crossref] [PubMed]

F. de Tomasi, D. Hennequin, B. Zambon, and E. Arimondo, “Instabilities and chaos in an infrared laser with saturable absorber: experiments and vibrorotational model,” J. Opt. Soc. Am. B 6, 45–57 (1989).
[Crossref]

1988 (6)

K. Tanii, M. Tachikawa, M. Kajita, and T. Shimizu, “Sinusoidal self-modulation in the output of a CO2laser with an intracavity saturable absorber,” J. Opt. Soc. Am. B 5, 24–28 (1988).
[Crossref]

M. Tachikawa, K. Tanii, and T. Shimizu, “Laser instability and chaotic pulsation in a CO2laser with intracavity saturable absorber,” J. Opt. Soc. Am. B 5, 1077–1081 (1988).
[Crossref]

D. Dangoisse, A. Bekkali, F. Papoff, and P. Glorieux, “Shil’nikov dynamics in a passive Q-switching laser,” Europhys. Lett. 6, 335–340 (1988).
[Crossref]

D. Hennequin, F. de Tomasi, B. Zambon, and E. Arimondo, “Homoclinic orbits and cycles in the instabilities of a laser with saturable absorber,” Phys. Rev. A 37, 2243–2246 (1988).
[Crossref] [PubMed]

F. T. Arecchi, A. Lapucci, R. Meucci, J. A. Roversi, and P. H. Coullet, “Experimental characterization of Shil’nikov chaos by statistics of return times,” Europhys. Lett. 6, 677–682 (1988); F. T. Arecchi, W. Gadomski, A. Lapucci, H. Mancini, R. Meucci, and J. A. Roversi, “Laser with feed back: an optical implementation of competing instabilities, Shil’nikov chaos, and transient fluctuation enhancement,” J. Opt. Soc. Am. B 5, 1153–1159 (1988).
[Crossref]

M. Tachikawa, F. L. Hong, K. Tanii, and T. Shimizu, “Deterministic chaos in passive Q-switching pulsation of a CO2laser with saturable absorber,” Phys. Rev. Lett. 60, 2266–2268 (1988).
[Crossref] [PubMed]

1987 (3)

F. Argoul, A. Arneodo, and P. Richetti, “Experimental evidence for homoclinic chaos in the Belousov–Zhabotinskii reaction,” Phys. Lett. A 120, 269–275 (1987); F. Argoul, A. Arneodo, and P. Richetti, “Dynamique symbolique dans la réaction de Belousov–Zhabotinskii: une illustration expérimentale de la théorie de Shil’nikov des orbites homoclines,” J. Chim. Phys. 84, 1367–1385 (1987).
[Crossref]

P. Gaspard and X. J. Wang, “Homoclinic orbits and mixed mode oscillations in far from equilibrium systems,” J. Phys. 48, 151–199 (1987).

M. Tachikawa, K. Tanii, and T. Shimizu, “Comprehensive interpretation of passive Q-switching and optical bistability in a CO2laser with an intracavity saturable absorber,” J. Opt. Soc. Am. B 4, 387–395 (1987).
[Crossref]

1986 (1)

M. Tachikawa, K. Tanii, M. Kajita, and T. Shimizu, “Undamped undulation superposed on the passive Q-switching pulse of a CO2laser,” Appl. Phys. B 39, 83–90 (1986).
[Crossref]

1985 (3)

E. Arimondo, P. Bootz, P. Glorieux, and E. Menchi, “Pulse shape and phase diagram in the passive Q-switching of CO2lasers,” J. Opt. Soc. Am. B 2, 193–201 (1985).
[Crossref]

E. Arimondo and E. Menchi, “Analysis of Q-switch in a CO2laser with saturable absorber,” Appl. Phys. B 37, 55–61 (1985).
[Crossref]

A. Arneodo, P. Coullet, E. A. Spiegel, and C. Tresser, “Asymptotic chaos,” Physica D 14, 327–347 (1985).
[Crossref]

1984 (3)

C. Tresser, “About some theorems by L. P. Shil’nikov,” Ann. Inst. Henri Poincaré 40, 441–461 (1984).

P. Glendinning and C. Sparrow, “Local and global behavior near homoclinic orbits,” J. Stat. Phys. 35, 645–696 (1984).
[Crossref]

P. Gaspard, R. Kapral, and G. Nicolis, “Bifurcation phenomena near homoclinic systems: a two-parameter analysis,” J. Stat. Phys. 35, 697–727 (1984).
[Crossref]

1983 (3)

P. Fredrickson, J. L. Kaplan, E. D. Yorke, and J. A. Yorke, “The Lyapunov dimension of strange attractors,” J. Diff. 49, 185–207 (1983).
[Crossref]

E. Arimondo, F. Casagrande, L. Lugiato, and P. Glorieux, “Repetitive passive Q-switching and bistability in lasers with saturable absorber,” Appl. Phys. B 30, 57–77 (1983).
[Crossref]

M. L. Asquini and F. Casagrande, “Passive Q-switching in lasers with saturable absorbers: improved treatment of a four level method,” Nuovo Cimento 2D, 917–931 (1983).
[Crossref]

1982 (1)

A. Jacques and P. Glorieux, “Observation of bistability in a CO2laser exhibiting passive Q-switching,” Opt. Commun. 40, 455–460 (1982).
[Crossref]

1981 (3)

F. Ledrappier, “Some relations between dimension Lyapunov exponent,” Commun. Math. Phys. 81, 229–238 (1981).
[Crossref]

See, e.g., J. S. Turner, J. C. Roux, W. D. McCormick, and H. L. Swinney, “Alternating periodic and chaotic regimes in chemical reaction–experiments and theory,” Phys. Lett 85A, 9–12 (1981).

T. Erneux and P. Mandel, “Bifurcation phenomena in a with saturable absorber I and II,” Z. Phys. B 44, 353–363, 365–374 (1981); P. Mandel and T. Erneux, “Stationary, harmonic and pulsed operations of an optically bistable with saturable absorber. I and II,” Phys. Rev. A 30, 1893–1901, 1902–1909 (1984); T. Erneux, P. Mandel, and J. Magnan, “Quasi-periodicity in lasers with saturable absorbers,” Phys. Rev. A 29, 2690–2699 (1984); D. E. Chyba, N. B. Abraham, and A. M. Albano, “Semiclassical analysis of a detuned ring laser with a saturable absorber. New results for steady states,” Phys. Rev. A 35, 2936–2950 (1987).
[Crossref] [PubMed]

1979 (1)

E. Arimondo and P. Glorieux, “Saturated absorption experiments on a dressed molecule. Application to the spectroscopy of the ν6band of CH3I,” Phys. Rev. A 19, 1067–1083 (1979).
[Crossref]

1975 (1)

J. Dupré, F. Meyer, and C. Meyer, “Influence des phénomènes de relaxation sur la forme des impulsions fournies par un laser CO2déclenché par un absorbant saturable,” Rev. Phys. Appl. 10, 285–293 (1975).
[Crossref]

1971 (2)

H. T. Powell and G. J. Wolga, “Repetitive passive Q-switching of single-frequency lasers,” IEEE J. Quantum Electron QE-7, 213–219 (1971).
[Crossref]

I. Burak, P. L. Houston, D. G. Sutton, and J. I. Steinfeld, “Mechanism of passive Q-switching in CO2lasers,” IEEE J. Quantum Electron. QE-7, 73–82 (1971).
[Crossref]

1965 (1)

L. P. Shil’nikov, “A case of the existence of a countable number of periodic motions,” Sov. Math. Dokl. 6, 163–166 (1965); “A contribution to the problem of the structure of an extended neighborhood of a rough equilibrium state of saddle focus type,” Math. USSR Sbornik 10, 91–102 (1970).

Abraham, N. B.

F. Papoff, A. Fioretti, E. Arimondo, and N. B. Abraham, “Time return maps and distributions for the laser with a sat urable absorber,” in Measures of Complexity and Chaos, N. B. Abraham, A. M. Albano, A. Passamente, and P. E. Rapp, eds. (Plenum, New York, 1989), pp. 309–312.
[Crossref]

Arecchi, F. T.

F. T. Arecchi, A. Lapucci, R. Meucci, J. A. Roversi, and P. H. Coullet, “Experimental characterization of Shil’nikov chaos by statistics of return times,” Europhys. Lett. 6, 677–682 (1988); F. T. Arecchi, W. Gadomski, A. Lapucci, H. Mancini, R. Meucci, and J. A. Roversi, “Laser with feed back: an optical implementation of competing instabilities, Shil’nikov chaos, and transient fluctuation enhancement,” J. Opt. Soc. Am. B 5, 1153–1159 (1988).
[Crossref]

Argoul, F.

F. Argoul, A. Arneodo, and P. Richetti, “Experimental evidence for homoclinic chaos in the Belousov–Zhabotinskii reaction,” Phys. Lett. A 120, 269–275 (1987); F. Argoul, A. Arneodo, and P. Richetti, “Dynamique symbolique dans la réaction de Belousov–Zhabotinskii: une illustration expérimentale de la théorie de Shil’nikov des orbites homoclines,” J. Chim. Phys. 84, 1367–1385 (1987).
[Crossref]

Arimondo, E.

B. Zambon, F. de Tomasi, D. Hennequin, and E. Arimondo, “Investigations of models for the laser with a saturable absorber: theoretical and experimental studies of the stationary regimes,” Phys. Rev. A 40, 3782–3795 (1989).
[Crossref] [PubMed]

F. de Tomasi, D. Hennequin, B. Zambon, and E. Arimondo, “Instabilities and chaos in an infrared laser with saturable absorber: experiments and vibrorotational model,” J. Opt. Soc. Am. B 6, 45–57 (1989).
[Crossref]

D. Hennequin, F. de Tomasi, L. Fronzoni, B. Zambon, and E. Arimondo, “Influence of noise on the quasi-homoclinic behavior of a laser with saturable absorber,” Opt. Commun. 70, 253–258 (1989).
[Crossref]

D. Hennequin, F. de Tomasi, B. Zambon, and E. Arimondo, “Homoclinic orbits and cycles in the instabilities of a laser with saturable absorber,” Phys. Rev. A 37, 2243–2246 (1988).
[Crossref] [PubMed]

E. Arimondo, P. Bootz, P. Glorieux, and E. Menchi, “Pulse shape and phase diagram in the passive Q-switching of CO2lasers,” J. Opt. Soc. Am. B 2, 193–201 (1985).
[Crossref]

E. Arimondo and E. Menchi, “Analysis of Q-switch in a CO2laser with saturable absorber,” Appl. Phys. B 37, 55–61 (1985).
[Crossref]

E. Arimondo, F. Casagrande, L. Lugiato, and P. Glorieux, “Repetitive passive Q-switching and bistability in lasers with saturable absorber,” Appl. Phys. B 30, 57–77 (1983).
[Crossref]

E. Arimondo and P. Glorieux, “Saturated absorption experiments on a dressed molecule. Application to the spectroscopy of the ν6band of CH3I,” Phys. Rev. A 19, 1067–1083 (1979).
[Crossref]

F. Papoff, A. Fioretti, E. Arimondo, and N. B. Abraham, “Time return maps and distributions for the laser with a sat urable absorber,” in Measures of Complexity and Chaos, N. B. Abraham, A. M. Albano, A. Passamente, and P. E. Rapp, eds. (Plenum, New York, 1989), pp. 309–312.
[Crossref]

Arneodo, A.

F. Argoul, A. Arneodo, and P. Richetti, “Experimental evidence for homoclinic chaos in the Belousov–Zhabotinskii reaction,” Phys. Lett. A 120, 269–275 (1987); F. Argoul, A. Arneodo, and P. Richetti, “Dynamique symbolique dans la réaction de Belousov–Zhabotinskii: une illustration expérimentale de la théorie de Shil’nikov des orbites homoclines,” J. Chim. Phys. 84, 1367–1385 (1987).
[Crossref]

A. Arneodo, P. Coullet, E. A. Spiegel, and C. Tresser, “Asymptotic chaos,” Physica D 14, 327–347 (1985).
[Crossref]

Asquini, M. L.

M. L. Asquini and F. Casagrande, “Passive Q-switching in lasers with saturable absorbers: improved treatment of a four level method,” Nuovo Cimento 2D, 917–931 (1983).
[Crossref]

Bekkali, A.

D. Dangoisse, A. Bekkali, F. Papoff, and P. Glorieux, “Shil’nikov dynamics in a passive Q-switching laser,” Europhys. Lett. 6, 335–340 (1988).
[Crossref]

D. Hennequin, M. Lefranc, A. Bekkali, D. Dangoisse, and P. Glorieux, “Characterization of Shil’nikov chaos in a CO2laser containing a saturable absorber” in Measures of Complexity and Chaos, N. B. Abraham, A. M. Albano, A. Passa mente, and P. E. Rapp, eds. (Plenum, New York, 1989), pp. 299–302.
[Crossref]

Bootz, P.

Burak, I.

I. Burak, P. L. Houston, D. G. Sutton, and J. I. Steinfeld, “Mechanism of passive Q-switching in CO2lasers,” IEEE J. Quantum Electron. QE-7, 73–82 (1971).
[Crossref]

Casagrande, F.

E. Arimondo, F. Casagrande, L. Lugiato, and P. Glorieux, “Repetitive passive Q-switching and bistability in lasers with saturable absorber,” Appl. Phys. B 30, 57–77 (1983).
[Crossref]

M. L. Asquini and F. Casagrande, “Passive Q-switching in lasers with saturable absorbers: improved treatment of a four level method,” Nuovo Cimento 2D, 917–931 (1983).
[Crossref]

Coullet, P.

A. Arneodo, P. Coullet, E. A. Spiegel, and C. Tresser, “Asymptotic chaos,” Physica D 14, 327–347 (1985).
[Crossref]

Coullet, P. H.

F. T. Arecchi, A. Lapucci, R. Meucci, J. A. Roversi, and P. H. Coullet, “Experimental characterization of Shil’nikov chaos by statistics of return times,” Europhys. Lett. 6, 677–682 (1988); F. T. Arecchi, W. Gadomski, A. Lapucci, H. Mancini, R. Meucci, and J. A. Roversi, “Laser with feed back: an optical implementation of competing instabilities, Shil’nikov chaos, and transient fluctuation enhancement,” J. Opt. Soc. Am. B 5, 1153–1159 (1988).
[Crossref]

Dangoisse, D.

D. Dangoisse, A. Bekkali, F. Papoff, and P. Glorieux, “Shil’nikov dynamics in a passive Q-switching laser,” Europhys. Lett. 6, 335–340 (1988).
[Crossref]

D. Hennequin, M. Lefranc, A. Bekkali, D. Dangoisse, and P. Glorieux, “Characterization of Shil’nikov chaos in a CO2laser containing a saturable absorber” in Measures of Complexity and Chaos, N. B. Abraham, A. M. Albano, A. Passa mente, and P. E. Rapp, eds. (Plenum, New York, 1989), pp. 299–302.
[Crossref]

de Tomasi, F.

D. Hennequin, F. de Tomasi, L. Fronzoni, B. Zambon, and E. Arimondo, “Influence of noise on the quasi-homoclinic behavior of a laser with saturable absorber,” Opt. Commun. 70, 253–258 (1989).
[Crossref]

F. de Tomasi, D. Hennequin, B. Zambon, and E. Arimondo, “Instabilities and chaos in an infrared laser with saturable absorber: experiments and vibrorotational model,” J. Opt. Soc. Am. B 6, 45–57 (1989).
[Crossref]

B. Zambon, F. de Tomasi, D. Hennequin, and E. Arimondo, “Investigations of models for the laser with a saturable absorber: theoretical and experimental studies of the stationary regimes,” Phys. Rev. A 40, 3782–3795 (1989).
[Crossref] [PubMed]

D. Hennequin, F. de Tomasi, B. Zambon, and E. Arimondo, “Homoclinic orbits and cycles in the instabilities of a laser with saturable absorber,” Phys. Rev. A 37, 2243–2246 (1988).
[Crossref] [PubMed]

Dupré, J.

J. Dupré, F. Meyer, and C. Meyer, “Influence des phénomènes de relaxation sur la forme des impulsions fournies par un laser CO2déclenché par un absorbant saturable,” Rev. Phys. Appl. 10, 285–293 (1975).
[Crossref]

Erneux, T.

T. Erneux and P. Mandel, “Bifurcation phenomena in a with saturable absorber I and II,” Z. Phys. B 44, 353–363, 365–374 (1981); P. Mandel and T. Erneux, “Stationary, harmonic and pulsed operations of an optically bistable with saturable absorber. I and II,” Phys. Rev. A 30, 1893–1901, 1902–1909 (1984); T. Erneux, P. Mandel, and J. Magnan, “Quasi-periodicity in lasers with saturable absorbers,” Phys. Rev. A 29, 2690–2699 (1984); D. E. Chyba, N. B. Abraham, and A. M. Albano, “Semiclassical analysis of a detuned ring laser with a saturable absorber. New results for steady states,” Phys. Rev. A 35, 2936–2950 (1987).
[Crossref] [PubMed]

Fioretti, A.

F. Papoff, A. Fioretti, E. Arimondo, and N. B. Abraham, “Time return maps and distributions for the laser with a sat urable absorber,” in Measures of Complexity and Chaos, N. B. Abraham, A. M. Albano, A. Passamente, and P. E. Rapp, eds. (Plenum, New York, 1989), pp. 309–312.
[Crossref]

Fredrickson, P.

P. Fredrickson, J. L. Kaplan, E. D. Yorke, and J. A. Yorke, “The Lyapunov dimension of strange attractors,” J. Diff. 49, 185–207 (1983).
[Crossref]

Fronzoni, L.

D. Hennequin, F. de Tomasi, L. Fronzoni, B. Zambon, and E. Arimondo, “Influence of noise on the quasi-homoclinic behavior of a laser with saturable absorber,” Opt. Commun. 70, 253–258 (1989).
[Crossref]

Gaspard, P.

P. Gaspard and X. J. Wang, “Homoclinic orbits and mixed mode oscillations in far from equilibrium systems,” J. Phys. 48, 151–199 (1987).

P. Gaspard, R. Kapral, and G. Nicolis, “Bifurcation phenomena near homoclinic systems: a two-parameter analysis,” J. Stat. Phys. 35, 697–727 (1984).
[Crossref]

Glendinning, P.

P. Glendinning and C. Sparrow, “Local and global behavior near homoclinic orbits,” J. Stat. Phys. 35, 645–696 (1984).
[Crossref]

Glorieux, P.

D. Dangoisse, A. Bekkali, F. Papoff, and P. Glorieux, “Shil’nikov dynamics in a passive Q-switching laser,” Europhys. Lett. 6, 335–340 (1988).
[Crossref]

E. Arimondo, P. Bootz, P. Glorieux, and E. Menchi, “Pulse shape and phase diagram in the passive Q-switching of CO2lasers,” J. Opt. Soc. Am. B 2, 193–201 (1985).
[Crossref]

E. Arimondo, F. Casagrande, L. Lugiato, and P. Glorieux, “Repetitive passive Q-switching and bistability in lasers with saturable absorber,” Appl. Phys. B 30, 57–77 (1983).
[Crossref]

A. Jacques and P. Glorieux, “Observation of bistability in a CO2laser exhibiting passive Q-switching,” Opt. Commun. 40, 455–460 (1982).
[Crossref]

E. Arimondo and P. Glorieux, “Saturated absorption experiments on a dressed molecule. Application to the spectroscopy of the ν6band of CH3I,” Phys. Rev. A 19, 1067–1083 (1979).
[Crossref]

D. Hennequin, M. Lefranc, A. Bekkali, D. Dangoisse, and P. Glorieux, “Characterization of Shil’nikov chaos in a CO2laser containing a saturable absorber” in Measures of Complexity and Chaos, N. B. Abraham, A. M. Albano, A. Passa mente, and P. E. Rapp, eds. (Plenum, New York, 1989), pp. 299–302.
[Crossref]

Hennequin, D.

D. Hennequin, F. de Tomasi, L. Fronzoni, B. Zambon, and E. Arimondo, “Influence of noise on the quasi-homoclinic behavior of a laser with saturable absorber,” Opt. Commun. 70, 253–258 (1989).
[Crossref]

F. de Tomasi, D. Hennequin, B. Zambon, and E. Arimondo, “Instabilities and chaos in an infrared laser with saturable absorber: experiments and vibrorotational model,” J. Opt. Soc. Am. B 6, 45–57 (1989).
[Crossref]

B. Zambon, F. de Tomasi, D. Hennequin, and E. Arimondo, “Investigations of models for the laser with a saturable absorber: theoretical and experimental studies of the stationary regimes,” Phys. Rev. A 40, 3782–3795 (1989).
[Crossref] [PubMed]

D. Hennequin, F. de Tomasi, B. Zambon, and E. Arimondo, “Homoclinic orbits and cycles in the instabilities of a laser with saturable absorber,” Phys. Rev. A 37, 2243–2246 (1988).
[Crossref] [PubMed]

D. Hennequin, M. Lefranc, A. Bekkali, D. Dangoisse, and P. Glorieux, “Characterization of Shil’nikov chaos in a CO2laser containing a saturable absorber” in Measures of Complexity and Chaos, N. B. Abraham, A. M. Albano, A. Passa mente, and P. E. Rapp, eds. (Plenum, New York, 1989), pp. 299–302.
[Crossref]

Hong, F. L.

M. Tachikawa, F. L. Hong, K. Tanii, and T. Shimizu, “Deterministic chaos in passive Q-switching pulsation of a CO2laser with saturable absorber,” Phys. Rev. Lett. 60, 2266–2268 (1988).
[Crossref] [PubMed]

Houston, P. L.

I. Burak, P. L. Houston, D. G. Sutton, and J. I. Steinfeld, “Mechanism of passive Q-switching in CO2lasers,” IEEE J. Quantum Electron. QE-7, 73–82 (1971).
[Crossref]

Jacques, A.

A. Jacques and P. Glorieux, “Observation of bistability in a CO2laser exhibiting passive Q-switching,” Opt. Commun. 40, 455–460 (1982).
[Crossref]

Kajita, M.

K. Tanii, M. Tachikawa, M. Kajita, and T. Shimizu, “Sinusoidal self-modulation in the output of a CO2laser with an intracavity saturable absorber,” J. Opt. Soc. Am. B 5, 24–28 (1988).
[Crossref]

M. Tachikawa, K. Tanii, M. Kajita, and T. Shimizu, “Undamped undulation superposed on the passive Q-switching pulse of a CO2laser,” Appl. Phys. B 39, 83–90 (1986).
[Crossref]

Kaplan, J. L.

P. Fredrickson, J. L. Kaplan, E. D. Yorke, and J. A. Yorke, “The Lyapunov dimension of strange attractors,” J. Diff. 49, 185–207 (1983).
[Crossref]

Kapral, R.

P. Gaspard, R. Kapral, and G. Nicolis, “Bifurcation phenomena near homoclinic systems: a two-parameter analysis,” J. Stat. Phys. 35, 697–727 (1984).
[Crossref]

Lapucci, A.

F. T. Arecchi, A. Lapucci, R. Meucci, J. A. Roversi, and P. H. Coullet, “Experimental characterization of Shil’nikov chaos by statistics of return times,” Europhys. Lett. 6, 677–682 (1988); F. T. Arecchi, W. Gadomski, A. Lapucci, H. Mancini, R. Meucci, and J. A. Roversi, “Laser with feed back: an optical implementation of competing instabilities, Shil’nikov chaos, and transient fluctuation enhancement,” J. Opt. Soc. Am. B 5, 1153–1159 (1988).
[Crossref]

Ledrappier, F.

F. Ledrappier, “Some relations between dimension Lyapunov exponent,” Commun. Math. Phys. 81, 229–238 (1981).
[Crossref]

Lefranc, M.

D. Hennequin, M. Lefranc, A. Bekkali, D. Dangoisse, and P. Glorieux, “Characterization of Shil’nikov chaos in a CO2laser containing a saturable absorber” in Measures of Complexity and Chaos, N. B. Abraham, A. M. Albano, A. Passa mente, and P. E. Rapp, eds. (Plenum, New York, 1989), pp. 299–302.
[Crossref]

Lugiato, L.

E. Arimondo, F. Casagrande, L. Lugiato, and P. Glorieux, “Repetitive passive Q-switching and bistability in lasers with saturable absorber,” Appl. Phys. B 30, 57–77 (1983).
[Crossref]

Mandel, P.

T. Erneux and P. Mandel, “Bifurcation phenomena in a with saturable absorber I and II,” Z. Phys. B 44, 353–363, 365–374 (1981); P. Mandel and T. Erneux, “Stationary, harmonic and pulsed operations of an optically bistable with saturable absorber. I and II,” Phys. Rev. A 30, 1893–1901, 1902–1909 (1984); T. Erneux, P. Mandel, and J. Magnan, “Quasi-periodicity in lasers with saturable absorbers,” Phys. Rev. A 29, 2690–2699 (1984); D. E. Chyba, N. B. Abraham, and A. M. Albano, “Semiclassical analysis of a detuned ring laser with a saturable absorber. New results for steady states,” Phys. Rev. A 35, 2936–2950 (1987).
[Crossref] [PubMed]

McCormick, W. D.

See, e.g., J. S. Turner, J. C. Roux, W. D. McCormick, and H. L. Swinney, “Alternating periodic and chaotic regimes in chemical reaction–experiments and theory,” Phys. Lett 85A, 9–12 (1981).

Menchi, E.

E. Arimondo and E. Menchi, “Analysis of Q-switch in a CO2laser with saturable absorber,” Appl. Phys. B 37, 55–61 (1985).
[Crossref]

E. Arimondo, P. Bootz, P. Glorieux, and E. Menchi, “Pulse shape and phase diagram in the passive Q-switching of CO2lasers,” J. Opt. Soc. Am. B 2, 193–201 (1985).
[Crossref]

Meucci, R.

F. T. Arecchi, A. Lapucci, R. Meucci, J. A. Roversi, and P. H. Coullet, “Experimental characterization of Shil’nikov chaos by statistics of return times,” Europhys. Lett. 6, 677–682 (1988); F. T. Arecchi, W. Gadomski, A. Lapucci, H. Mancini, R. Meucci, and J. A. Roversi, “Laser with feed back: an optical implementation of competing instabilities, Shil’nikov chaos, and transient fluctuation enhancement,” J. Opt. Soc. Am. B 5, 1153–1159 (1988).
[Crossref]

Meyer, C.

J. Dupré, F. Meyer, and C. Meyer, “Influence des phénomènes de relaxation sur la forme des impulsions fournies par un laser CO2déclenché par un absorbant saturable,” Rev. Phys. Appl. 10, 285–293 (1975).
[Crossref]

Meyer, F.

J. Dupré, F. Meyer, and C. Meyer, “Influence des phénomènes de relaxation sur la forme des impulsions fournies par un laser CO2déclenché par un absorbant saturable,” Rev. Phys. Appl. 10, 285–293 (1975).
[Crossref]

Nicolis, G.

P. Gaspard, R. Kapral, and G. Nicolis, “Bifurcation phenomena near homoclinic systems: a two-parameter analysis,” J. Stat. Phys. 35, 697–727 (1984).
[Crossref]

Papoff, F.

D. Dangoisse, A. Bekkali, F. Papoff, and P. Glorieux, “Shil’nikov dynamics in a passive Q-switching laser,” Europhys. Lett. 6, 335–340 (1988).
[Crossref]

F. Papoff, A. Fioretti, E. Arimondo, and N. B. Abraham, “Time return maps and distributions for the laser with a sat urable absorber,” in Measures of Complexity and Chaos, N. B. Abraham, A. M. Albano, A. Passamente, and P. E. Rapp, eds. (Plenum, New York, 1989), pp. 309–312.
[Crossref]

Powell, H. T.

H. T. Powell and G. J. Wolga, “Repetitive passive Q-switching of single-frequency lasers,” IEEE J. Quantum Electron QE-7, 213–219 (1971).
[Crossref]

Richetti, P.

F. Argoul, A. Arneodo, and P. Richetti, “Experimental evidence for homoclinic chaos in the Belousov–Zhabotinskii reaction,” Phys. Lett. A 120, 269–275 (1987); F. Argoul, A. Arneodo, and P. Richetti, “Dynamique symbolique dans la réaction de Belousov–Zhabotinskii: une illustration expérimentale de la théorie de Shil’nikov des orbites homoclines,” J. Chim. Phys. 84, 1367–1385 (1987).
[Crossref]

Roux, J. C.

See, e.g., J. S. Turner, J. C. Roux, W. D. McCormick, and H. L. Swinney, “Alternating periodic and chaotic regimes in chemical reaction–experiments and theory,” Phys. Lett 85A, 9–12 (1981).

Roversi, J. A.

F. T. Arecchi, A. Lapucci, R. Meucci, J. A. Roversi, and P. H. Coullet, “Experimental characterization of Shil’nikov chaos by statistics of return times,” Europhys. Lett. 6, 677–682 (1988); F. T. Arecchi, W. Gadomski, A. Lapucci, H. Mancini, R. Meucci, and J. A. Roversi, “Laser with feed back: an optical implementation of competing instabilities, Shil’nikov chaos, and transient fluctuation enhancement,” J. Opt. Soc. Am. B 5, 1153–1159 (1988).
[Crossref]

Shil’nikov, L. P.

L. P. Shil’nikov, “A case of the existence of a countable number of periodic motions,” Sov. Math. Dokl. 6, 163–166 (1965); “A contribution to the problem of the structure of an extended neighborhood of a rough equilibrium state of saddle focus type,” Math. USSR Sbornik 10, 91–102 (1970).

Shimizu, T.

Sparrow, C.

P. Glendinning and C. Sparrow, “Local and global behavior near homoclinic orbits,” J. Stat. Phys. 35, 645–696 (1984).
[Crossref]

C. Sparrow, The Lorenz Equations, Bifurcations, Chaos Strange Attractors (Springer-Verlag, Berlin, 1982), pp. 211–220, App. E.

Spiegel, E. A.

A. Arneodo, P. Coullet, E. A. Spiegel, and C. Tresser, “Asymptotic chaos,” Physica D 14, 327–347 (1985).
[Crossref]

Steinfeld, J. I.

I. Burak, P. L. Houston, D. G. Sutton, and J. I. Steinfeld, “Mechanism of passive Q-switching in CO2lasers,” IEEE J. Quantum Electron. QE-7, 73–82 (1971).
[Crossref]

Sutton, D. G.

I. Burak, P. L. Houston, D. G. Sutton, and J. I. Steinfeld, “Mechanism of passive Q-switching in CO2lasers,” IEEE J. Quantum Electron. QE-7, 73–82 (1971).
[Crossref]

Swinney, H. L.

See, e.g., J. S. Turner, J. C. Roux, W. D. McCormick, and H. L. Swinney, “Alternating periodic and chaotic regimes in chemical reaction–experiments and theory,” Phys. Lett 85A, 9–12 (1981).

Tachikawa, M.

Tanii, K.

Tresser, C.

A. Arneodo, P. Coullet, E. A. Spiegel, and C. Tresser, “Asymptotic chaos,” Physica D 14, 327–347 (1985).
[Crossref]

C. Tresser, “About some theorems by L. P. Shil’nikov,” Ann. Inst. Henri Poincaré 40, 441–461 (1984).

Turner, J. S.

See, e.g., J. S. Turner, J. C. Roux, W. D. McCormick, and H. L. Swinney, “Alternating periodic and chaotic regimes in chemical reaction–experiments and theory,” Phys. Lett 85A, 9–12 (1981).

Wang, X. J.

P. Gaspard and X. J. Wang, “Homoclinic orbits and mixed mode oscillations in far from equilibrium systems,” J. Phys. 48, 151–199 (1987).

Wolga, G. J.

H. T. Powell and G. J. Wolga, “Repetitive passive Q-switching of single-frequency lasers,” IEEE J. Quantum Electron QE-7, 213–219 (1971).
[Crossref]

Yorke, E. D.

P. Fredrickson, J. L. Kaplan, E. D. Yorke, and J. A. Yorke, “The Lyapunov dimension of strange attractors,” J. Diff. 49, 185–207 (1983).
[Crossref]

Yorke, J. A.

P. Fredrickson, J. L. Kaplan, E. D. Yorke, and J. A. Yorke, “The Lyapunov dimension of strange attractors,” J. Diff. 49, 185–207 (1983).
[Crossref]

Zambon, B.

B. Zambon, F. de Tomasi, D. Hennequin, and E. Arimondo, “Investigations of models for the laser with a saturable absorber: theoretical and experimental studies of the stationary regimes,” Phys. Rev. A 40, 3782–3795 (1989).
[Crossref] [PubMed]

F. de Tomasi, D. Hennequin, B. Zambon, and E. Arimondo, “Instabilities and chaos in an infrared laser with saturable absorber: experiments and vibrorotational model,” J. Opt. Soc. Am. B 6, 45–57 (1989).
[Crossref]

D. Hennequin, F. de Tomasi, L. Fronzoni, B. Zambon, and E. Arimondo, “Influence of noise on the quasi-homoclinic behavior of a laser with saturable absorber,” Opt. Commun. 70, 253–258 (1989).
[Crossref]

D. Hennequin, F. de Tomasi, B. Zambon, and E. Arimondo, “Homoclinic orbits and cycles in the instabilities of a laser with saturable absorber,” Phys. Rev. A 37, 2243–2246 (1988).
[Crossref] [PubMed]

Ann. Inst. Henri Poincaré (1)

C. Tresser, “About some theorems by L. P. Shil’nikov,” Ann. Inst. Henri Poincaré 40, 441–461 (1984).

Appl. Phys. B (3)

E. Arimondo, F. Casagrande, L. Lugiato, and P. Glorieux, “Repetitive passive Q-switching and bistability in lasers with saturable absorber,” Appl. Phys. B 30, 57–77 (1983).
[Crossref]

E. Arimondo and E. Menchi, “Analysis of Q-switch in a CO2laser with saturable absorber,” Appl. Phys. B 37, 55–61 (1985).
[Crossref]

M. Tachikawa, K. Tanii, M. Kajita, and T. Shimizu, “Undamped undulation superposed on the passive Q-switching pulse of a CO2laser,” Appl. Phys. B 39, 83–90 (1986).
[Crossref]

Commun. Math. Phys. (1)

F. Ledrappier, “Some relations between dimension Lyapunov exponent,” Commun. Math. Phys. 81, 229–238 (1981).
[Crossref]

Europhys. Lett. (2)

F. T. Arecchi, A. Lapucci, R. Meucci, J. A. Roversi, and P. H. Coullet, “Experimental characterization of Shil’nikov chaos by statistics of return times,” Europhys. Lett. 6, 677–682 (1988); F. T. Arecchi, W. Gadomski, A. Lapucci, H. Mancini, R. Meucci, and J. A. Roversi, “Laser with feed back: an optical implementation of competing instabilities, Shil’nikov chaos, and transient fluctuation enhancement,” J. Opt. Soc. Am. B 5, 1153–1159 (1988).
[Crossref]

D. Dangoisse, A. Bekkali, F. Papoff, and P. Glorieux, “Shil’nikov dynamics in a passive Q-switching laser,” Europhys. Lett. 6, 335–340 (1988).
[Crossref]

IEEE J. Quantum Electron (1)

H. T. Powell and G. J. Wolga, “Repetitive passive Q-switching of single-frequency lasers,” IEEE J. Quantum Electron QE-7, 213–219 (1971).
[Crossref]

IEEE J. Quantum Electron. (1)

I. Burak, P. L. Houston, D. G. Sutton, and J. I. Steinfeld, “Mechanism of passive Q-switching in CO2lasers,” IEEE J. Quantum Electron. QE-7, 73–82 (1971).
[Crossref]

J. Diff. (1)

P. Fredrickson, J. L. Kaplan, E. D. Yorke, and J. A. Yorke, “The Lyapunov dimension of strange attractors,” J. Diff. 49, 185–207 (1983).
[Crossref]

J. Opt. Soc. Am. B (5)

J. Phys. (1)

P. Gaspard and X. J. Wang, “Homoclinic orbits and mixed mode oscillations in far from equilibrium systems,” J. Phys. 48, 151–199 (1987).

J. Stat. Phys. (2)

P. Glendinning and C. Sparrow, “Local and global behavior near homoclinic orbits,” J. Stat. Phys. 35, 645–696 (1984).
[Crossref]

P. Gaspard, R. Kapral, and G. Nicolis, “Bifurcation phenomena near homoclinic systems: a two-parameter analysis,” J. Stat. Phys. 35, 697–727 (1984).
[Crossref]

Nuovo Cimento (1)

M. L. Asquini and F. Casagrande, “Passive Q-switching in lasers with saturable absorbers: improved treatment of a four level method,” Nuovo Cimento 2D, 917–931 (1983).
[Crossref]

Opt. Commun. (2)

A. Jacques and P. Glorieux, “Observation of bistability in a CO2laser exhibiting passive Q-switching,” Opt. Commun. 40, 455–460 (1982).
[Crossref]

D. Hennequin, F. de Tomasi, L. Fronzoni, B. Zambon, and E. Arimondo, “Influence of noise on the quasi-homoclinic behavior of a laser with saturable absorber,” Opt. Commun. 70, 253–258 (1989).
[Crossref]

Phys. Lett (1)

See, e.g., J. S. Turner, J. C. Roux, W. D. McCormick, and H. L. Swinney, “Alternating periodic and chaotic regimes in chemical reaction–experiments and theory,” Phys. Lett 85A, 9–12 (1981).

Phys. Lett. A (1)

F. Argoul, A. Arneodo, and P. Richetti, “Experimental evidence for homoclinic chaos in the Belousov–Zhabotinskii reaction,” Phys. Lett. A 120, 269–275 (1987); F. Argoul, A. Arneodo, and P. Richetti, “Dynamique symbolique dans la réaction de Belousov–Zhabotinskii: une illustration expérimentale de la théorie de Shil’nikov des orbites homoclines,” J. Chim. Phys. 84, 1367–1385 (1987).
[Crossref]

Phys. Rev. A (3)

E. Arimondo and P. Glorieux, “Saturated absorption experiments on a dressed molecule. Application to the spectroscopy of the ν6band of CH3I,” Phys. Rev. A 19, 1067–1083 (1979).
[Crossref]

D. Hennequin, F. de Tomasi, B. Zambon, and E. Arimondo, “Homoclinic orbits and cycles in the instabilities of a laser with saturable absorber,” Phys. Rev. A 37, 2243–2246 (1988).
[Crossref] [PubMed]

B. Zambon, F. de Tomasi, D. Hennequin, and E. Arimondo, “Investigations of models for the laser with a saturable absorber: theoretical and experimental studies of the stationary regimes,” Phys. Rev. A 40, 3782–3795 (1989).
[Crossref] [PubMed]

Phys. Rev. Lett. (1)

M. Tachikawa, F. L. Hong, K. Tanii, and T. Shimizu, “Deterministic chaos in passive Q-switching pulsation of a CO2laser with saturable absorber,” Phys. Rev. Lett. 60, 2266–2268 (1988).
[Crossref] [PubMed]

Physica D (1)

A. Arneodo, P. Coullet, E. A. Spiegel, and C. Tresser, “Asymptotic chaos,” Physica D 14, 327–347 (1985).
[Crossref]

Rev. Phys. Appl. (1)

J. Dupré, F. Meyer, and C. Meyer, “Influence des phénomènes de relaxation sur la forme des impulsions fournies par un laser CO2déclenché par un absorbant saturable,” Rev. Phys. Appl. 10, 285–293 (1975).
[Crossref]

Sov. Math. Dokl. (1)

L. P. Shil’nikov, “A case of the existence of a countable number of periodic motions,” Sov. Math. Dokl. 6, 163–166 (1965); “A contribution to the problem of the structure of an extended neighborhood of a rough equilibrium state of saddle focus type,” Math. USSR Sbornik 10, 91–102 (1970).

Z. Phys. B (1)

T. Erneux and P. Mandel, “Bifurcation phenomena in a with saturable absorber I and II,” Z. Phys. B 44, 353–363, 365–374 (1981); P. Mandel and T. Erneux, “Stationary, harmonic and pulsed operations of an optically bistable with saturable absorber. I and II,” Phys. Rev. A 30, 1893–1901, 1902–1909 (1984); T. Erneux, P. Mandel, and J. Magnan, “Quasi-periodicity in lasers with saturable absorbers,” Phys. Rev. A 29, 2690–2699 (1984); D. E. Chyba, N. B. Abraham, and A. M. Albano, “Semiclassical analysis of a detuned ring laser with a saturable absorber. New results for steady states,” Phys. Rev. A 35, 2936–2950 (1987).
[Crossref] [PubMed]

Other (3)

C. Sparrow, The Lorenz Equations, Bifurcations, Chaos Strange Attractors (Springer-Verlag, Berlin, 1982), pp. 211–220, App. E.

D. Hennequin, M. Lefranc, A. Bekkali, D. Dangoisse, and P. Glorieux, “Characterization of Shil’nikov chaos in a CO2laser containing a saturable absorber” in Measures of Complexity and Chaos, N. B. Abraham, A. M. Albano, A. Passa mente, and P. E. Rapp, eds. (Plenum, New York, 1989), pp. 299–302.
[Crossref]

F. Papoff, A. Fioretti, E. Arimondo, and N. B. Abraham, “Time return maps and distributions for the laser with a sat urable absorber,” in Measures of Complexity and Chaos, N. B. Abraham, A. M. Albano, A. Passamente, and P. E. Rapp, eds. (Plenum, New York, 1989), pp. 309–312.
[Crossref]

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Figures (13)

Fig. 1
Fig. 1

Analysis of the C(2) experimental regime: (a) temporal sequence; (b) three-dimensional reconstruction of the attractor in the phase space I(t), I(t + τ), I(t + 2τ), where τ = 3.2 μsec; (c) Poincaré section performed in the reinjection loop of the attractor in a plane indicated by the dashed line in (a); (d) first return map of the Poincaré section of(c). Each point of coordinates (In, In+1) is represented by the number p of undulations that separate the nth and the (n + 1)th crossings through the Poincaré section. The symbols O, I, +, and × correspond, respectively, to p = 0, 1, 2, and 3. The dashed curves separate regions corresponding to a given p, which is indicated for each region at the top of each the figure. The experimental conditions are given in the text.

Fig. 2
Fig. 2

1-D maps of C(1) chaos: (a) close to the P(1) regime, (b) close to the P(2) regime. The symbols are the same as in Fig. 1(d).

Fig. 3
Fig. 3

Schematic representation of the energy levels used in the theoretical model of the LSA. The quantities with overbars refer to the absorber medium, those without overbars to the active medium.

Fig. 4
Fig. 4

(a) Evolution of the eigenvalues of the I0 (dashed lines) and the I+ (solid lines) eigenvalues as functions of the A parameter. AH = 1.943 is the Hopf bifurcation point. (b) Values of ρ/λ (solid line) and (λ21) (ρ/λ) (dashed line) versus A. Their absolute values are both smaller than 1.

Fig. 5
Fig. 5

Bifurcation diagram of the model of the LSA for parameters of Table 1. The letters indicate the following regimes: a, P(0);b, P(1); c, P(2); d, P(3); e, T; f, I+.

Fig. 6
Fig. 6

Analysis of the C(1) regime of Fig. 5: (a) temporal sequence (time in reduced units); (b) Poincaré section performed at I = 0.21, İ > 0, in the reinjection loop of the attractor; (c) first return map of the Poincaré section of (b). The numbers 0 and 1 indicate the number of turns around I+ associated with each branch. Parameters used for calculation are given in Table 1 with A = 1.773.

Fig. 7
Fig. 7

Analysis of the C(2) of the diagram of Fig. 5: (a) Temporal sequence (time in reduced units). (b) Poincaré section performed at I = 0.21, İ > 0, in the reinjection loop of the attractor. (c) First return map of the Poincaré section of (b). 0–2 indicate the number of turns associated with each branch. (d) Projection of the attractor on the hyperplane (I, U, W). The parameters used for calculation are given in Table 1 with A = 1.860.

Fig. 8
Fig. 8

Plot of the largest transverse Floquet multiplier associated with the P(3) periodic orbit as a function of A.

Fig. 9
Fig. 9

Bifurcation diagram of the model of the LSA for the parameters of Table 1, except for ɛ ¯ = 6. Periodic regimes P(n) with n ≤ 8 are present.

Fig. 10
Fig. 10

(a) Temporal signal obtained for A = 1.997 with the reduced model ( ɛ ¯ infinite). A pulse with 41 undulations can be seen. (b) First return map of the Poincaré section plane I = 0.2, İ > 0, with 42 branches. The branches converge geometrically with an asymptotic rate equal to 0.882. Except for ɛ ¯, the parameters may be found in Table 1, as they can for Figs. 1113.

Fig. 11
Fig. 11

Bifurcation diagram obtained with infinite ɛ ¯. The sudden transition from chaos to I+, which can be seen at the right, corresponds to a homocliniclike bifurcation.

Fig. 12
Fig. 12

Temporal signal obtained for A = 1.888 with the reduced model ( ɛ ¯ infinite). It is similar to signals of the four-variable model computed for equivalent pump parameters and may be compared with Fig. 7(a).

Fig. 13
Fig. 13

Intersection with a section plane transverse to P(∞) of the stable (Ws) and unstable (Wu) manifolds of P(∞) in the neighborhood of the unstable cycle. The point where the two manifolds cross is the intersection of P(∞) with the section plane:(a) For A = 1.991, Ws and Wu are disconnected. Region I is the attraction basin of I+, and region II is the part of phase space where motion on the attractor takes place. The two regions are separated by Ws. (b) For A = 1.997, Wu and Ws are tangent to each other.

Tables (1)

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Table 1 Values of the Parameters Used in the Numerical Simulations

Equations (16)

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ϕ ˙ = ϕ ( ζ A ( M 1 - M 2 ) - ζ ¯ A ¯ M ¯ - 2 κ ) ,
M ˙ 1 = - A ( M 1 - M 2 ) ϕ + P M 0 - ( γ 10 + γ 12 ) M 1 ,
M ˙ 2 = A ( M 1 - M 2 ) ϕ - γ 20 M 2 + γ 12 M 1 ,
M ˙ 0 = γ 10 M 1 + γ 20 M 2 - P M 0 ,
M ¯ ˙ = - 2 A ¯ ϕ M ¯ - γ ¯ ( M ¯ - M ¯ * ) ,
I ˙ = I ( ζ A M - ζ ¯ A ¯ M ¯ - 2 κ ) ,
M ˙ = - ( I + 1 ) γ 2 M + P M 0 + γ 1 ( N - M 0 ) ,
M ˙ 0 = γ 2 N - γ 1 M - ( γ 2 + P ) M 0 ,
M ¯ ˙ = - 2 A ¯ ϕ M ¯ - γ ¯ ( M ¯ - M ¯ * ) ,
a = A ¯ A γ 2 γ ¯ ,             b = ( γ 1 γ 2 ) 2 , ɛ = γ 2 2 κ ,             ɛ ¯ = γ ¯ 2 κ , A = P ( b + 1 ) A ζ N 2 κ γ 2 ,             A ¯ = M ¯ * ζ ¯ A ¯ 2 κ , U = M ζ A 2 κ ,             U ¯ = M ¯ ζ ¯ A ¯ 2 κ ,
W = A 2 κ ζ γ 2 [ P M 0 + γ 1 ( N - M 0 ) ] ,
I ˙ = I ( U - U ¯ - 1 ) ,
U ˙ = ɛ [ W - U ( 1 + I ) ] ,
W ˙ = ɛ ( A + b U - W ) ,
U ¯ ˙ = ɛ ¯ [ A ¯ - U ¯ ( 1 + a I ) ] .
I 0 = 0 ,             U 0 = W 0 = A ( 1 - b ) - 1 ,             U ¯ 0 = A ¯ , I + = a ( A + b - 1 ) - ( A ¯ + 1 ) + { [ a ( A + b - 1 ) - ( A ¯ + 1 ) ] 2 + 4 a ( A - A th ) } 1 / 2 2 a , U + = A ( 1 + I + - b ) - 1 , W + = A ( 1 + I + ) ( 1 + I + - b ) - 1 , U ¯ + = A ¯ ( 1 + a I + ) - 1 .

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