Abstract

We investigate the response of an acousto-optic bistable device to time-varying acoustic inputs. The device modeled by a two-hump one-dimensional autonomous nonlinear map in which the (implied) is justifiably period is determined by the feedback time of the device. Our newly added time-varying input has a map period much greater than the feedback time and for simplicity is taken in the form of a periodic square pulse. We use numerical simulation and a matrix method to predict the general behavior of the output intensity at specific instants of time. Background knowledge, viz., general comments on the nature of one-two-hump one-dimensional maps and their distinction, is also presented in a unified fashion to aid in and the understanding of the dynamics of the device. We find that novel changes of the output period can occur for significant feedback amplitudes, and that these changes can be sensitively controlled.

© 1992 Optical Society of America

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  1. E. Ott, C. Grebogi, J. A. Yorke, “Experimental control of chaos” in Chaos, D. K. Campbell, ed. (American Institute of Physics, New York, 1990), pp. 153–172.
  2. W. L. Ditto, S. N. Rauseo, M. L. Spano, “Experimental control of chaos,” Phys. Rev. Lett. 65, 3211–3214 (1990).
    [CrossRef] [PubMed]
  3. J. Singer, Y.-Z. Wang, H. H. Bau, “Controlling a chaotic system,” Phys. Rev. Lett. 66, 1123–1125 (1991).
    [CrossRef] [PubMed]
  4. A. J. Irwin, S. J. Fraser, R. Kapral, “Stochastically induced coherence in bistable systems,” Phys. Rev. Lett. 64, 2343–2346 (1990).
    [CrossRef] [PubMed]
  5. A. Hubler, E. Luscher, “Resonant stimulation and control of nonlinear oscillators,” Naturwissenschaften 76, 67–69 (1989).
    [CrossRef]
  6. P. P. Banerjee, A. Ghafoor, “Design of a binary parallel optical processor,” Appl. Opt. 27, 4766–4770 (1988).
    [CrossRef] [PubMed]
  7. H. M. Gibbs, Optical Bistability (Academic, New York, 1985).
  8. A. Korpel, Acousto-Optics (Dekker, New York, 1989).
  9. J. Chrostowski, C. Delisle, “Bistable optical switching based on Bragg diffraction,” Opt. Commun. 41, 71–74 (1982).
    [CrossRef]
  10. J. Chrostowski, R. Vallee, C. Delisle, “Self-pulsing and chaos in acoustooptic bistability,” Can. J. Phys. 61, 1143–1148 (1983).
    [CrossRef]
  11. H. Jerominek, C. Delisle, J. Y. D. Pomerleau, R. Tremblay, “Bistability, optical regenerative oscillations and chaos in an integrated acoustoopic device,” Can. J. Phys. 63, 227–233 (1985).
    [CrossRef]
  12. F. A. Hopf, D. L. Kaplan, H. M. Gibbs, R. L. Shoemaker, “Bifurcations to chaos in optical bistability,” Phys. Rev. A 25, 2172–2182 (1982).
    [CrossRef]
  13. M. W. Derstine, H. M. Gibbs, F. A. Hopf, D. L. Kaplan, “Alternate paths to chaos in optical bistability,” Phys. Rev. A 27, 3200–3208 (1983).
    [CrossRef]
  14. H.-J. Zhang, J.-H. Dai, P.-Y. Wang, F.-L. Zhang, G. Xu, S.-P. Yang, “Chaos in liquid-crystal optical bistability,” in Directions in Chaos (2), H. B. Lin, ed. (World Scientific, Singapore, 1988), pp. 46–89.
  15. F. T. Arecchi, R. Meucci, G. Puccioni, J. Tredicce, “Experimental evidence of subharmonic bifurcations, multistability, and turbulence in a Q-switched gas laser,” Phys. Rev. Lett. 49, 1217–1220 (1982).
    [CrossRef]
  16. L. A. Lugiato, L. M. Narducci, D. K. Bandy, C. A. Pennise, “Breathing, spiking and chaos in a laser with injected signal,” Opt. Commun. 46, 64–68 (1983).
    [CrossRef]
  17. E. N. Lorenz, “Deterministic nonperiodic flow,” J. Atmos. Sci. 20, 130–141 (1963).
    [CrossRef]
  18. J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (Springer-Verlag, New York, 1983).
  19. M. LeBerre, E. Ressayre, A. Tallet, H. M. Gibbs, “High-dimension chaotic attractors of a nonlinear ring cavity,” Phys. Rev. Lett. 56, 274–277 (1986).
    [CrossRef]
  20. J. Belair, L. Glass, “Universality and self-similarity in the bifurcations of circle maps,” Physica 16D, 143–154 (1985).
  21. P. Collet, J. P. Eckmann, Iterated Maps on the Interval as Dynamical Systems (Birkhauser, Boston, 1980).
  22. N. Metropolis, M. L. Stein, P. R. Stein, “On finite limit sets for transformation on the unit interval,” J. Comb. Theory 15, 25–44 (1973).
    [CrossRef]
  23. P. Mandel, R. Kapral, “Subharmonic and chaotic bifurcation structure in optical bistability,” Opt. Commun. 47, 151–156 (1983).
    [CrossRef]
  24. Starting values of x that are sufficiently close to any subbasin boundary, such as the position of a vertical line in Fig. 4, each of which acts as an unstable fixed point, will iterate for a long time according to the linearized map in the neighborhood of that point before leaving that neighborhood.
  25. In order to show that the hundredth iterate represents a period-four orbit rather than, for instance, four fixed-point attractors, we can graphically compound the map with its hundredth iterate, e.g., f101 = f0100° f, by the usual method [see H. Kaplan, Am. J. Phys. 55, 1023 (1987)].
    [CrossRef]
  26. P. Cvitanovic, “Invariant measurement of strange sets in terms of limit cycles,” Phys. Rev. Lett. 61, 2729–2732 (1988).
    [CrossRef] [PubMed]

1991 (1)

J. Singer, Y.-Z. Wang, H. H. Bau, “Controlling a chaotic system,” Phys. Rev. Lett. 66, 1123–1125 (1991).
[CrossRef] [PubMed]

1990 (2)

A. J. Irwin, S. J. Fraser, R. Kapral, “Stochastically induced coherence in bistable systems,” Phys. Rev. Lett. 64, 2343–2346 (1990).
[CrossRef] [PubMed]

W. L. Ditto, S. N. Rauseo, M. L. Spano, “Experimental control of chaos,” Phys. Rev. Lett. 65, 3211–3214 (1990).
[CrossRef] [PubMed]

1989 (1)

A. Hubler, E. Luscher, “Resonant stimulation and control of nonlinear oscillators,” Naturwissenschaften 76, 67–69 (1989).
[CrossRef]

1988 (2)

P. P. Banerjee, A. Ghafoor, “Design of a binary parallel optical processor,” Appl. Opt. 27, 4766–4770 (1988).
[CrossRef] [PubMed]

P. Cvitanovic, “Invariant measurement of strange sets in terms of limit cycles,” Phys. Rev. Lett. 61, 2729–2732 (1988).
[CrossRef] [PubMed]

1987 (1)

In order to show that the hundredth iterate represents a period-four orbit rather than, for instance, four fixed-point attractors, we can graphically compound the map with its hundredth iterate, e.g., f101 = f0100° f, by the usual method [see H. Kaplan, Am. J. Phys. 55, 1023 (1987)].
[CrossRef]

1986 (1)

M. LeBerre, E. Ressayre, A. Tallet, H. M. Gibbs, “High-dimension chaotic attractors of a nonlinear ring cavity,” Phys. Rev. Lett. 56, 274–277 (1986).
[CrossRef]

1985 (2)

J. Belair, L. Glass, “Universality and self-similarity in the bifurcations of circle maps,” Physica 16D, 143–154 (1985).

H. Jerominek, C. Delisle, J. Y. D. Pomerleau, R. Tremblay, “Bistability, optical regenerative oscillations and chaos in an integrated acoustoopic device,” Can. J. Phys. 63, 227–233 (1985).
[CrossRef]

1983 (4)

J. Chrostowski, R. Vallee, C. Delisle, “Self-pulsing and chaos in acoustooptic bistability,” Can. J. Phys. 61, 1143–1148 (1983).
[CrossRef]

P. Mandel, R. Kapral, “Subharmonic and chaotic bifurcation structure in optical bistability,” Opt. Commun. 47, 151–156 (1983).
[CrossRef]

M. W. Derstine, H. M. Gibbs, F. A. Hopf, D. L. Kaplan, “Alternate paths to chaos in optical bistability,” Phys. Rev. A 27, 3200–3208 (1983).
[CrossRef]

L. A. Lugiato, L. M. Narducci, D. K. Bandy, C. A. Pennise, “Breathing, spiking and chaos in a laser with injected signal,” Opt. Commun. 46, 64–68 (1983).
[CrossRef]

1982 (3)

F. T. Arecchi, R. Meucci, G. Puccioni, J. Tredicce, “Experimental evidence of subharmonic bifurcations, multistability, and turbulence in a Q-switched gas laser,” Phys. Rev. Lett. 49, 1217–1220 (1982).
[CrossRef]

F. A. Hopf, D. L. Kaplan, H. M. Gibbs, R. L. Shoemaker, “Bifurcations to chaos in optical bistability,” Phys. Rev. A 25, 2172–2182 (1982).
[CrossRef]

J. Chrostowski, C. Delisle, “Bistable optical switching based on Bragg diffraction,” Opt. Commun. 41, 71–74 (1982).
[CrossRef]

1973 (1)

N. Metropolis, M. L. Stein, P. R. Stein, “On finite limit sets for transformation on the unit interval,” J. Comb. Theory 15, 25–44 (1973).
[CrossRef]

1963 (1)

E. N. Lorenz, “Deterministic nonperiodic flow,” J. Atmos. Sci. 20, 130–141 (1963).
[CrossRef]

Arecchi, F. T.

F. T. Arecchi, R. Meucci, G. Puccioni, J. Tredicce, “Experimental evidence of subharmonic bifurcations, multistability, and turbulence in a Q-switched gas laser,” Phys. Rev. Lett. 49, 1217–1220 (1982).
[CrossRef]

Bandy, D. K.

L. A. Lugiato, L. M. Narducci, D. K. Bandy, C. A. Pennise, “Breathing, spiking and chaos in a laser with injected signal,” Opt. Commun. 46, 64–68 (1983).
[CrossRef]

Banerjee, P. P.

Bau, H. H.

J. Singer, Y.-Z. Wang, H. H. Bau, “Controlling a chaotic system,” Phys. Rev. Lett. 66, 1123–1125 (1991).
[CrossRef] [PubMed]

Belair, J.

J. Belair, L. Glass, “Universality and self-similarity in the bifurcations of circle maps,” Physica 16D, 143–154 (1985).

Chrostowski, J.

J. Chrostowski, R. Vallee, C. Delisle, “Self-pulsing and chaos in acoustooptic bistability,” Can. J. Phys. 61, 1143–1148 (1983).
[CrossRef]

J. Chrostowski, C. Delisle, “Bistable optical switching based on Bragg diffraction,” Opt. Commun. 41, 71–74 (1982).
[CrossRef]

Collet, P.

P. Collet, J. P. Eckmann, Iterated Maps on the Interval as Dynamical Systems (Birkhauser, Boston, 1980).

Cvitanovic, P.

P. Cvitanovic, “Invariant measurement of strange sets in terms of limit cycles,” Phys. Rev. Lett. 61, 2729–2732 (1988).
[CrossRef] [PubMed]

Dai, J.-H.

H.-J. Zhang, J.-H. Dai, P.-Y. Wang, F.-L. Zhang, G. Xu, S.-P. Yang, “Chaos in liquid-crystal optical bistability,” in Directions in Chaos (2), H. B. Lin, ed. (World Scientific, Singapore, 1988), pp. 46–89.

Delisle, C.

H. Jerominek, C. Delisle, J. Y. D. Pomerleau, R. Tremblay, “Bistability, optical regenerative oscillations and chaos in an integrated acoustoopic device,” Can. J. Phys. 63, 227–233 (1985).
[CrossRef]

J. Chrostowski, R. Vallee, C. Delisle, “Self-pulsing and chaos in acoustooptic bistability,” Can. J. Phys. 61, 1143–1148 (1983).
[CrossRef]

J. Chrostowski, C. Delisle, “Bistable optical switching based on Bragg diffraction,” Opt. Commun. 41, 71–74 (1982).
[CrossRef]

Derstine, M. W.

M. W. Derstine, H. M. Gibbs, F. A. Hopf, D. L. Kaplan, “Alternate paths to chaos in optical bistability,” Phys. Rev. A 27, 3200–3208 (1983).
[CrossRef]

Ditto, W. L.

W. L. Ditto, S. N. Rauseo, M. L. Spano, “Experimental control of chaos,” Phys. Rev. Lett. 65, 3211–3214 (1990).
[CrossRef] [PubMed]

Eckmann, J. P.

P. Collet, J. P. Eckmann, Iterated Maps on the Interval as Dynamical Systems (Birkhauser, Boston, 1980).

Fraser, S. J.

A. J. Irwin, S. J. Fraser, R. Kapral, “Stochastically induced coherence in bistable systems,” Phys. Rev. Lett. 64, 2343–2346 (1990).
[CrossRef] [PubMed]

Ghafoor, A.

Gibbs, H. M.

M. LeBerre, E. Ressayre, A. Tallet, H. M. Gibbs, “High-dimension chaotic attractors of a nonlinear ring cavity,” Phys. Rev. Lett. 56, 274–277 (1986).
[CrossRef]

M. W. Derstine, H. M. Gibbs, F. A. Hopf, D. L. Kaplan, “Alternate paths to chaos in optical bistability,” Phys. Rev. A 27, 3200–3208 (1983).
[CrossRef]

F. A. Hopf, D. L. Kaplan, H. M. Gibbs, R. L. Shoemaker, “Bifurcations to chaos in optical bistability,” Phys. Rev. A 25, 2172–2182 (1982).
[CrossRef]

H. M. Gibbs, Optical Bistability (Academic, New York, 1985).

Glass, L.

J. Belair, L. Glass, “Universality and self-similarity in the bifurcations of circle maps,” Physica 16D, 143–154 (1985).

Grebogi, C.

E. Ott, C. Grebogi, J. A. Yorke, “Experimental control of chaos” in Chaos, D. K. Campbell, ed. (American Institute of Physics, New York, 1990), pp. 153–172.

Guckenheimer, J.

J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (Springer-Verlag, New York, 1983).

Holmes, P.

J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (Springer-Verlag, New York, 1983).

Hopf, F. A.

M. W. Derstine, H. M. Gibbs, F. A. Hopf, D. L. Kaplan, “Alternate paths to chaos in optical bistability,” Phys. Rev. A 27, 3200–3208 (1983).
[CrossRef]

F. A. Hopf, D. L. Kaplan, H. M. Gibbs, R. L. Shoemaker, “Bifurcations to chaos in optical bistability,” Phys. Rev. A 25, 2172–2182 (1982).
[CrossRef]

Hubler, A.

A. Hubler, E. Luscher, “Resonant stimulation and control of nonlinear oscillators,” Naturwissenschaften 76, 67–69 (1989).
[CrossRef]

Irwin, A. J.

A. J. Irwin, S. J. Fraser, R. Kapral, “Stochastically induced coherence in bistable systems,” Phys. Rev. Lett. 64, 2343–2346 (1990).
[CrossRef] [PubMed]

Jerominek, H.

H. Jerominek, C. Delisle, J. Y. D. Pomerleau, R. Tremblay, “Bistability, optical regenerative oscillations and chaos in an integrated acoustoopic device,” Can. J. Phys. 63, 227–233 (1985).
[CrossRef]

Kaplan, D. L.

M. W. Derstine, H. M. Gibbs, F. A. Hopf, D. L. Kaplan, “Alternate paths to chaos in optical bistability,” Phys. Rev. A 27, 3200–3208 (1983).
[CrossRef]

F. A. Hopf, D. L. Kaplan, H. M. Gibbs, R. L. Shoemaker, “Bifurcations to chaos in optical bistability,” Phys. Rev. A 25, 2172–2182 (1982).
[CrossRef]

Kaplan, H.

In order to show that the hundredth iterate represents a period-four orbit rather than, for instance, four fixed-point attractors, we can graphically compound the map with its hundredth iterate, e.g., f101 = f0100° f, by the usual method [see H. Kaplan, Am. J. Phys. 55, 1023 (1987)].
[CrossRef]

Kapral, R.

A. J. Irwin, S. J. Fraser, R. Kapral, “Stochastically induced coherence in bistable systems,” Phys. Rev. Lett. 64, 2343–2346 (1990).
[CrossRef] [PubMed]

P. Mandel, R. Kapral, “Subharmonic and chaotic bifurcation structure in optical bistability,” Opt. Commun. 47, 151–156 (1983).
[CrossRef]

Korpel, A.

A. Korpel, Acousto-Optics (Dekker, New York, 1989).

LeBerre, M.

M. LeBerre, E. Ressayre, A. Tallet, H. M. Gibbs, “High-dimension chaotic attractors of a nonlinear ring cavity,” Phys. Rev. Lett. 56, 274–277 (1986).
[CrossRef]

Lorenz, E. N.

E. N. Lorenz, “Deterministic nonperiodic flow,” J. Atmos. Sci. 20, 130–141 (1963).
[CrossRef]

Lugiato, L. A.

L. A. Lugiato, L. M. Narducci, D. K. Bandy, C. A. Pennise, “Breathing, spiking and chaos in a laser with injected signal,” Opt. Commun. 46, 64–68 (1983).
[CrossRef]

Luscher, E.

A. Hubler, E. Luscher, “Resonant stimulation and control of nonlinear oscillators,” Naturwissenschaften 76, 67–69 (1989).
[CrossRef]

Mandel, P.

P. Mandel, R. Kapral, “Subharmonic and chaotic bifurcation structure in optical bistability,” Opt. Commun. 47, 151–156 (1983).
[CrossRef]

Metropolis, N.

N. Metropolis, M. L. Stein, P. R. Stein, “On finite limit sets for transformation on the unit interval,” J. Comb. Theory 15, 25–44 (1973).
[CrossRef]

Meucci, R.

F. T. Arecchi, R. Meucci, G. Puccioni, J. Tredicce, “Experimental evidence of subharmonic bifurcations, multistability, and turbulence in a Q-switched gas laser,” Phys. Rev. Lett. 49, 1217–1220 (1982).
[CrossRef]

Narducci, L. M.

L. A. Lugiato, L. M. Narducci, D. K. Bandy, C. A. Pennise, “Breathing, spiking and chaos in a laser with injected signal,” Opt. Commun. 46, 64–68 (1983).
[CrossRef]

Ott, E.

E. Ott, C. Grebogi, J. A. Yorke, “Experimental control of chaos” in Chaos, D. K. Campbell, ed. (American Institute of Physics, New York, 1990), pp. 153–172.

Pennise, C. A.

L. A. Lugiato, L. M. Narducci, D. K. Bandy, C. A. Pennise, “Breathing, spiking and chaos in a laser with injected signal,” Opt. Commun. 46, 64–68 (1983).
[CrossRef]

Pomerleau, J. Y. D.

H. Jerominek, C. Delisle, J. Y. D. Pomerleau, R. Tremblay, “Bistability, optical regenerative oscillations and chaos in an integrated acoustoopic device,” Can. J. Phys. 63, 227–233 (1985).
[CrossRef]

Puccioni, G.

F. T. Arecchi, R. Meucci, G. Puccioni, J. Tredicce, “Experimental evidence of subharmonic bifurcations, multistability, and turbulence in a Q-switched gas laser,” Phys. Rev. Lett. 49, 1217–1220 (1982).
[CrossRef]

Rauseo, S. N.

W. L. Ditto, S. N. Rauseo, M. L. Spano, “Experimental control of chaos,” Phys. Rev. Lett. 65, 3211–3214 (1990).
[CrossRef] [PubMed]

Ressayre, E.

M. LeBerre, E. Ressayre, A. Tallet, H. M. Gibbs, “High-dimension chaotic attractors of a nonlinear ring cavity,” Phys. Rev. Lett. 56, 274–277 (1986).
[CrossRef]

Shoemaker, R. L.

F. A. Hopf, D. L. Kaplan, H. M. Gibbs, R. L. Shoemaker, “Bifurcations to chaos in optical bistability,” Phys. Rev. A 25, 2172–2182 (1982).
[CrossRef]

Singer, J.

J. Singer, Y.-Z. Wang, H. H. Bau, “Controlling a chaotic system,” Phys. Rev. Lett. 66, 1123–1125 (1991).
[CrossRef] [PubMed]

Spano, M. L.

W. L. Ditto, S. N. Rauseo, M. L. Spano, “Experimental control of chaos,” Phys. Rev. Lett. 65, 3211–3214 (1990).
[CrossRef] [PubMed]

Stein, M. L.

N. Metropolis, M. L. Stein, P. R. Stein, “On finite limit sets for transformation on the unit interval,” J. Comb. Theory 15, 25–44 (1973).
[CrossRef]

Stein, P. R.

N. Metropolis, M. L. Stein, P. R. Stein, “On finite limit sets for transformation on the unit interval,” J. Comb. Theory 15, 25–44 (1973).
[CrossRef]

Tallet, A.

M. LeBerre, E. Ressayre, A. Tallet, H. M. Gibbs, “High-dimension chaotic attractors of a nonlinear ring cavity,” Phys. Rev. Lett. 56, 274–277 (1986).
[CrossRef]

Tredicce, J.

F. T. Arecchi, R. Meucci, G. Puccioni, J. Tredicce, “Experimental evidence of subharmonic bifurcations, multistability, and turbulence in a Q-switched gas laser,” Phys. Rev. Lett. 49, 1217–1220 (1982).
[CrossRef]

Tremblay, R.

H. Jerominek, C. Delisle, J. Y. D. Pomerleau, R. Tremblay, “Bistability, optical regenerative oscillations and chaos in an integrated acoustoopic device,” Can. J. Phys. 63, 227–233 (1985).
[CrossRef]

Vallee, R.

J. Chrostowski, R. Vallee, C. Delisle, “Self-pulsing and chaos in acoustooptic bistability,” Can. J. Phys. 61, 1143–1148 (1983).
[CrossRef]

Wang, P.-Y.

H.-J. Zhang, J.-H. Dai, P.-Y. Wang, F.-L. Zhang, G. Xu, S.-P. Yang, “Chaos in liquid-crystal optical bistability,” in Directions in Chaos (2), H. B. Lin, ed. (World Scientific, Singapore, 1988), pp. 46–89.

Wang, Y.-Z.

J. Singer, Y.-Z. Wang, H. H. Bau, “Controlling a chaotic system,” Phys. Rev. Lett. 66, 1123–1125 (1991).
[CrossRef] [PubMed]

Xu, G.

H.-J. Zhang, J.-H. Dai, P.-Y. Wang, F.-L. Zhang, G. Xu, S.-P. Yang, “Chaos in liquid-crystal optical bistability,” in Directions in Chaos (2), H. B. Lin, ed. (World Scientific, Singapore, 1988), pp. 46–89.

Yang, S.-P.

H.-J. Zhang, J.-H. Dai, P.-Y. Wang, F.-L. Zhang, G. Xu, S.-P. Yang, “Chaos in liquid-crystal optical bistability,” in Directions in Chaos (2), H. B. Lin, ed. (World Scientific, Singapore, 1988), pp. 46–89.

Yorke, J. A.

E. Ott, C. Grebogi, J. A. Yorke, “Experimental control of chaos” in Chaos, D. K. Campbell, ed. (American Institute of Physics, New York, 1990), pp. 153–172.

Zhang, F.-L.

H.-J. Zhang, J.-H. Dai, P.-Y. Wang, F.-L. Zhang, G. Xu, S.-P. Yang, “Chaos in liquid-crystal optical bistability,” in Directions in Chaos (2), H. B. Lin, ed. (World Scientific, Singapore, 1988), pp. 46–89.

Zhang, H.-J.

H.-J. Zhang, J.-H. Dai, P.-Y. Wang, F.-L. Zhang, G. Xu, S.-P. Yang, “Chaos in liquid-crystal optical bistability,” in Directions in Chaos (2), H. B. Lin, ed. (World Scientific, Singapore, 1988), pp. 46–89.

Am. J. Phys. (1)

In order to show that the hundredth iterate represents a period-four orbit rather than, for instance, four fixed-point attractors, we can graphically compound the map with its hundredth iterate, e.g., f101 = f0100° f, by the usual method [see H. Kaplan, Am. J. Phys. 55, 1023 (1987)].
[CrossRef]

Appl. Opt. (1)

Can. J. Phys. (2)

J. Chrostowski, R. Vallee, C. Delisle, “Self-pulsing and chaos in acoustooptic bistability,” Can. J. Phys. 61, 1143–1148 (1983).
[CrossRef]

H. Jerominek, C. Delisle, J. Y. D. Pomerleau, R. Tremblay, “Bistability, optical regenerative oscillations and chaos in an integrated acoustoopic device,” Can. J. Phys. 63, 227–233 (1985).
[CrossRef]

J. Atmos. Sci. (1)

E. N. Lorenz, “Deterministic nonperiodic flow,” J. Atmos. Sci. 20, 130–141 (1963).
[CrossRef]

J. Comb. Theory (1)

N. Metropolis, M. L. Stein, P. R. Stein, “On finite limit sets for transformation on the unit interval,” J. Comb. Theory 15, 25–44 (1973).
[CrossRef]

Naturwissenschaften (1)

A. Hubler, E. Luscher, “Resonant stimulation and control of nonlinear oscillators,” Naturwissenschaften 76, 67–69 (1989).
[CrossRef]

Opt. Commun. (3)

J. Chrostowski, C. Delisle, “Bistable optical switching based on Bragg diffraction,” Opt. Commun. 41, 71–74 (1982).
[CrossRef]

P. Mandel, R. Kapral, “Subharmonic and chaotic bifurcation structure in optical bistability,” Opt. Commun. 47, 151–156 (1983).
[CrossRef]

L. A. Lugiato, L. M. Narducci, D. K. Bandy, C. A. Pennise, “Breathing, spiking and chaos in a laser with injected signal,” Opt. Commun. 46, 64–68 (1983).
[CrossRef]

Phys. Rev. A (2)

F. A. Hopf, D. L. Kaplan, H. M. Gibbs, R. L. Shoemaker, “Bifurcations to chaos in optical bistability,” Phys. Rev. A 25, 2172–2182 (1982).
[CrossRef]

M. W. Derstine, H. M. Gibbs, F. A. Hopf, D. L. Kaplan, “Alternate paths to chaos in optical bistability,” Phys. Rev. A 27, 3200–3208 (1983).
[CrossRef]

Phys. Rev. Lett. (6)

W. L. Ditto, S. N. Rauseo, M. L. Spano, “Experimental control of chaos,” Phys. Rev. Lett. 65, 3211–3214 (1990).
[CrossRef] [PubMed]

J. Singer, Y.-Z. Wang, H. H. Bau, “Controlling a chaotic system,” Phys. Rev. Lett. 66, 1123–1125 (1991).
[CrossRef] [PubMed]

A. J. Irwin, S. J. Fraser, R. Kapral, “Stochastically induced coherence in bistable systems,” Phys. Rev. Lett. 64, 2343–2346 (1990).
[CrossRef] [PubMed]

F. T. Arecchi, R. Meucci, G. Puccioni, J. Tredicce, “Experimental evidence of subharmonic bifurcations, multistability, and turbulence in a Q-switched gas laser,” Phys. Rev. Lett. 49, 1217–1220 (1982).
[CrossRef]

M. LeBerre, E. Ressayre, A. Tallet, H. M. Gibbs, “High-dimension chaotic attractors of a nonlinear ring cavity,” Phys. Rev. Lett. 56, 274–277 (1986).
[CrossRef]

P. Cvitanovic, “Invariant measurement of strange sets in terms of limit cycles,” Phys. Rev. Lett. 61, 2729–2732 (1988).
[CrossRef] [PubMed]

Physica (1)

J. Belair, L. Glass, “Universality and self-similarity in the bifurcations of circle maps,” Physica 16D, 143–154 (1985).

Other (7)

P. Collet, J. P. Eckmann, Iterated Maps on the Interval as Dynamical Systems (Birkhauser, Boston, 1980).

J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (Springer-Verlag, New York, 1983).

H. M. Gibbs, Optical Bistability (Academic, New York, 1985).

A. Korpel, Acousto-Optics (Dekker, New York, 1989).

E. Ott, C. Grebogi, J. A. Yorke, “Experimental control of chaos” in Chaos, D. K. Campbell, ed. (American Institute of Physics, New York, 1990), pp. 153–172.

H.-J. Zhang, J.-H. Dai, P.-Y. Wang, F.-L. Zhang, G. Xu, S.-P. Yang, “Chaos in liquid-crystal optical bistability,” in Directions in Chaos (2), H. B. Lin, ed. (World Scientific, Singapore, 1988), pp. 46–89.

Starting values of x that are sufficiently close to any subbasin boundary, such as the position of a vertical line in Fig. 4, each of which acts as an unstable fixed point, will iterate for a long time according to the linearized map in the neighborhood of that point before leaving that neighborhood.

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Figures (7)

Fig. 1
Fig. 1

Schematic for (a) AO Bragg diffraction, and (b) Bragg diffraction with positive feedback.

Fig. 2
Fig. 2

First-order diffracted intensity as a function of sound pressure showing an idealized sin2 curve.

Fig. 3
Fig. 3

Behavior of g(m)x near one of its (degenerate) fixed points when the control parameter corresponds to (a) a tangent bifurcation, and (b) a period-doubling bifurcation.

Fig. 4
Fig. 4

The 3rd and 201st iterate of the quadratic map for λ = 3.8284, …, the tangent bifurcation value for a stable period-three orbit. Apart from unavoidable errors, which are intrinsic to finite sampling of the abscissa, almost all the points should be at three vertical levels, which are indicated by the stable orbit points of period three. Vertical lines indicate the positions of unstable periodic orbit points and their preimages.

Fig. 5
Fig. 5

(a) Skeleton curves for periods one, two, and three for b < 0.5. (b) The solid curves show period-doubling bifurcation lines, the lower one between period-one and period-two attractors, and the upper pair between period-two and period-four attractors. The dashed curve shows a tangent bifurcation for the period-two basin. The dotted curves are period-two skeleton lines. (c) Each point in the shaded area belongs to a map with at least one period-two basin. The shaded diamond indicates period-two bistability (after Ref. 20).

Fig. 6
Fig. 6

Plots of (a) f 0 ( 1 ) , f 0 ( 2 ) , f 0 ( 4 ) , f and f 0 ( 100 ) for ϕ = 0.6 and (b) f 1 ( 1 ) , f 1 ( 2 ) , f 1 ( 4 ), and f 1 ( 100 ) for ϕ = 0.7 and B = 0.6.

Fig. 7
Fig. 7

Plots of (a) the composite map F and (b) its tenth iterate.

Equations (30)

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d ψ 0 / d ξ = - j ( α ^ / 2 ) ψ 1 , d ψ 1 / d ξ = - j ( α ^ / 2 ) ψ 0 ,
ψ 1 = - j ψ inc sin ( α ^ / 2 ) ,
α ^ = α ^ 0 + β ¯ ψ 1 2 ,
y n + 1 = sin 2 [ ( α ^ 0 + β ˜ y n ) / 2 ] ,
B = β ˜ / 2 π ,             ϕ = α ^ 0 / 2 π ,             u n + 1 = ( β ˜ / 2 π ) y n + 1 ,
u n + 1 = B sin 2 π ( ϕ + u n ) f ( u n ) ,
x n + 1 = τ + b sin 2 π x n .
u n = x n - τ + b ,             B = 2 b ,             ϕ 0 = τ - b + 1 / 4.
x n + 1 = λ x n ( 1 - x n ) g ( x n ) ,
g ( m ) ( x ) = x ,             d [ g ( m ) ] / d x = 1 ,             d 2 [ g ( m ) ] / d x 2 0 ,
g ( m / 2 ) ( x ) = x = g ( m ) ( x ) ,             d ( g ( m / 2 ) ) / d x = - 1 , d [ g ( m ) ] / d x = 1 ,             d 2 [ g ( m ) ] / d x 2 = 0.
u n + 1 = B sin 2 π [ ϕ ( n ) + u n ] ,
ϕ ( n ) = { ϕ 0 , 0 n n 0 ϕ 1 , n 0 + 1 n n 0 + n 1 ,
F = F 1 F 0
F 0 f 0 ( n 0 ) ,             F 1 f 1 ( n 1 ) ,
u n + 1 = B sin 2 π ( ϕ 0 + u n ) = f 0 ( u n ) ,
u n + 1 = B sin 2 π ( ϕ 0 + u n ) = f 1 ( u n ) .
n = [ ( k + 2 ) / 2 ] n 0 + [ ( k + 1 ) / 2 ] n 1 ,             k = 1 , 3 , 5 , ,
ϕ ( n ) = ϕ n ,             n = 0 , 1 , N - 1 ,
M x = y .
M ¯ = P M P T = [ A 1 0 0 0 A 2 0 0 0 0 · · · · · · · · · 0 0 A k 0 B 1 B 2 B k C ] ,
[ 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 . . . . . . . . . . 0 0 . . . . . . 0 1 0 ] ,
A i m x ¯ i = I m x ¯ i = x ¯ i .
F ( m ) ( x ¯ i ) = x ¯ i ;             x ¯ i = ( x i 1 , x i 2 , , x i m ) ,
M = [ 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 ] .
M ¯ = P M P T = [ 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 ] ,             P = [ 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 ] .
C = [ 0 1 0 0 ] ,
M = [ 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 1 ] .
M n 0 + N , n 1 + N = T N - 1 M n 0 , n 1 T N ,
T N = T 1 N , T 1 = [ 0 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 ] .

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