Abstract

A CO2 laser with feedback shows different dynamic regimes depending on the dominant role of one or two of three coexisting unstable stationary points. These regimes have been characterized by statistical distributions of return times to a Poincaré section at constant intensity. In particular, in the regime of Shil’nikov chaos the iteration maps of return times display a statistical spread owing to a transient fluctuation enhancement phenomenon peculiar to macroscopic systems, which is absent in low-dimensional chaotic dynamics.

© 1988 Optical Society of America

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  1. F. T. Arecchi, R. Meucci, and W. Gadomski, Phys. Rev. Lett. 58, 2205 (1987).
    [CrossRef] [PubMed]
  2. L. P. Shil’nikov, Dokl. Akad. Nauk SSSR 160, 558 (1965);L. P. Shil’nikov, Mat. Sb. 77, 119, 461 (1968);Mat. Sb. 81, 92, 123 (1970).
  3. P. Glendinning and C. Sparrow, J. Stat. Phys. 35, 645 (1984);P. Gaspard, R. Kapral, and G. Nicolis, J. Stat. Phys. 35, 697 (1984).
    [CrossRef]
  4. F. Argoul, A. Arneodo, and P. Richetti, Phys. Lett. A 120, 269 (1987).
    [CrossRef]
  5. F. T. Arecchi, A. Lapucci, R. Meucci, J. A. Roversi, and P. Coullet, submitted to Phys. Rev. Lett.
  6. F. T. Arecchi, V. Degiorgio, and B. Querzola, Phys. Rev. Lett. 19, 168 (1967).
    [CrossRef]
  7. F. Haake, Phys. Rev. Lett. 41, 1685 (1978).
    [CrossRef]
  8. F. T. Arecchi and A. Politi, Phys. Rev. Lett. 45, 1215 (1980);F. T. Arecchi, A. Politi, and L. Ulivi, Nuovo Cimento 71B, 119 (1982).
    [CrossRef]
  9. F. T. Arecchi, W. Gadomski, and R. Meucci, Phys. Rev. A 34, 1617 (1986).
    [CrossRef] [PubMed]
  10. F. T. Arecchi, in Instabilities and Chaos in Quantum Optics, F. T. Arecchi and R. G. Harrison, eds., Vol. 34 of Springer Series in Synergetics (Springer-Verlag, Berlin, 1987), p. 9.
    [CrossRef]
  11. F. T. Arecchi, R. Meucci, G. P. Puccioni, and J. R. Tredicce, Phys. Rev. Lett. 49, 1217 (1982).
    [CrossRef]
  12. T. Midavaine, D. Dangoisse, and P. Glorieux, Phys. Rev. Lett. 55, 1189 (1985).
    [CrossRef]
  13. F. T. Arecchi, G. L. Lippi, G. P. Puccioni, and J. R. Tredicce, Opt. Commun. 51, 308 (1984).
    [CrossRef]
  14. G. L. Lippi, J. R. Tredicce, N. B. Abraham, and F. T. Arecchi, Opt. Commun. 53, 129 (1985).
    [CrossRef]
  15. A. Arneodo, P. H. Coullet, E. A. Spiegel, and C. Tresser, Physica 14D, 327 (1985).
  16. J. P. Crutchfield, D. Farmer, and B. A. Hubermann, Phys. Rep. 92, 45 (1982).
    [CrossRef]

1987 (2)

F. T. Arecchi, R. Meucci, and W. Gadomski, Phys. Rev. Lett. 58, 2205 (1987).
[CrossRef] [PubMed]

F. Argoul, A. Arneodo, and P. Richetti, Phys. Lett. A 120, 269 (1987).
[CrossRef]

1986 (1)

F. T. Arecchi, W. Gadomski, and R. Meucci, Phys. Rev. A 34, 1617 (1986).
[CrossRef] [PubMed]

1985 (3)

T. Midavaine, D. Dangoisse, and P. Glorieux, Phys. Rev. Lett. 55, 1189 (1985).
[CrossRef]

G. L. Lippi, J. R. Tredicce, N. B. Abraham, and F. T. Arecchi, Opt. Commun. 53, 129 (1985).
[CrossRef]

A. Arneodo, P. H. Coullet, E. A. Spiegel, and C. Tresser, Physica 14D, 327 (1985).

1984 (2)

F. T. Arecchi, G. L. Lippi, G. P. Puccioni, and J. R. Tredicce, Opt. Commun. 51, 308 (1984).
[CrossRef]

P. Glendinning and C. Sparrow, J. Stat. Phys. 35, 645 (1984);P. Gaspard, R. Kapral, and G. Nicolis, J. Stat. Phys. 35, 697 (1984).
[CrossRef]

1982 (2)

F. T. Arecchi, R. Meucci, G. P. Puccioni, and J. R. Tredicce, Phys. Rev. Lett. 49, 1217 (1982).
[CrossRef]

J. P. Crutchfield, D. Farmer, and B. A. Hubermann, Phys. Rep. 92, 45 (1982).
[CrossRef]

1980 (1)

F. T. Arecchi and A. Politi, Phys. Rev. Lett. 45, 1215 (1980);F. T. Arecchi, A. Politi, and L. Ulivi, Nuovo Cimento 71B, 119 (1982).
[CrossRef]

1978 (1)

F. Haake, Phys. Rev. Lett. 41, 1685 (1978).
[CrossRef]

1967 (1)

F. T. Arecchi, V. Degiorgio, and B. Querzola, Phys. Rev. Lett. 19, 168 (1967).
[CrossRef]

1965 (1)

L. P. Shil’nikov, Dokl. Akad. Nauk SSSR 160, 558 (1965);L. P. Shil’nikov, Mat. Sb. 77, 119, 461 (1968);Mat. Sb. 81, 92, 123 (1970).

Abraham, N. B.

G. L. Lippi, J. R. Tredicce, N. B. Abraham, and F. T. Arecchi, Opt. Commun. 53, 129 (1985).
[CrossRef]

Arecchi, F. T.

F. T. Arecchi, R. Meucci, and W. Gadomski, Phys. Rev. Lett. 58, 2205 (1987).
[CrossRef] [PubMed]

F. T. Arecchi, W. Gadomski, and R. Meucci, Phys. Rev. A 34, 1617 (1986).
[CrossRef] [PubMed]

G. L. Lippi, J. R. Tredicce, N. B. Abraham, and F. T. Arecchi, Opt. Commun. 53, 129 (1985).
[CrossRef]

F. T. Arecchi, G. L. Lippi, G. P. Puccioni, and J. R. Tredicce, Opt. Commun. 51, 308 (1984).
[CrossRef]

F. T. Arecchi, R. Meucci, G. P. Puccioni, and J. R. Tredicce, Phys. Rev. Lett. 49, 1217 (1982).
[CrossRef]

F. T. Arecchi and A. Politi, Phys. Rev. Lett. 45, 1215 (1980);F. T. Arecchi, A. Politi, and L. Ulivi, Nuovo Cimento 71B, 119 (1982).
[CrossRef]

F. T. Arecchi, V. Degiorgio, and B. Querzola, Phys. Rev. Lett. 19, 168 (1967).
[CrossRef]

F. T. Arecchi, A. Lapucci, R. Meucci, J. A. Roversi, and P. Coullet, submitted to Phys. Rev. Lett.

F. T. Arecchi, in Instabilities and Chaos in Quantum Optics, F. T. Arecchi and R. G. Harrison, eds., Vol. 34 of Springer Series in Synergetics (Springer-Verlag, Berlin, 1987), p. 9.
[CrossRef]

Argoul, F.

F. Argoul, A. Arneodo, and P. Richetti, Phys. Lett. A 120, 269 (1987).
[CrossRef]

Arneodo, A.

F. Argoul, A. Arneodo, and P. Richetti, Phys. Lett. A 120, 269 (1987).
[CrossRef]

A. Arneodo, P. H. Coullet, E. A. Spiegel, and C. Tresser, Physica 14D, 327 (1985).

Coullet, P.

F. T. Arecchi, A. Lapucci, R. Meucci, J. A. Roversi, and P. Coullet, submitted to Phys. Rev. Lett.

Coullet, P. H.

A. Arneodo, P. H. Coullet, E. A. Spiegel, and C. Tresser, Physica 14D, 327 (1985).

Crutchfield, J. P.

J. P. Crutchfield, D. Farmer, and B. A. Hubermann, Phys. Rep. 92, 45 (1982).
[CrossRef]

Dangoisse, D.

T. Midavaine, D. Dangoisse, and P. Glorieux, Phys. Rev. Lett. 55, 1189 (1985).
[CrossRef]

Degiorgio, V.

F. T. Arecchi, V. Degiorgio, and B. Querzola, Phys. Rev. Lett. 19, 168 (1967).
[CrossRef]

Farmer, D.

J. P. Crutchfield, D. Farmer, and B. A. Hubermann, Phys. Rep. 92, 45 (1982).
[CrossRef]

Gadomski, W.

F. T. Arecchi, R. Meucci, and W. Gadomski, Phys. Rev. Lett. 58, 2205 (1987).
[CrossRef] [PubMed]

F. T. Arecchi, W. Gadomski, and R. Meucci, Phys. Rev. A 34, 1617 (1986).
[CrossRef] [PubMed]

Glendinning, P.

P. Glendinning and C. Sparrow, J. Stat. Phys. 35, 645 (1984);P. Gaspard, R. Kapral, and G. Nicolis, J. Stat. Phys. 35, 697 (1984).
[CrossRef]

Glorieux, P.

T. Midavaine, D. Dangoisse, and P. Glorieux, Phys. Rev. Lett. 55, 1189 (1985).
[CrossRef]

Haake, F.

F. Haake, Phys. Rev. Lett. 41, 1685 (1978).
[CrossRef]

Hubermann, B. A.

J. P. Crutchfield, D. Farmer, and B. A. Hubermann, Phys. Rep. 92, 45 (1982).
[CrossRef]

Lapucci, A.

F. T. Arecchi, A. Lapucci, R. Meucci, J. A. Roversi, and P. Coullet, submitted to Phys. Rev. Lett.

Lippi, G. L.

G. L. Lippi, J. R. Tredicce, N. B. Abraham, and F. T. Arecchi, Opt. Commun. 53, 129 (1985).
[CrossRef]

F. T. Arecchi, G. L. Lippi, G. P. Puccioni, and J. R. Tredicce, Opt. Commun. 51, 308 (1984).
[CrossRef]

Meucci, R.

F. T. Arecchi, R. Meucci, and W. Gadomski, Phys. Rev. Lett. 58, 2205 (1987).
[CrossRef] [PubMed]

F. T. Arecchi, W. Gadomski, and R. Meucci, Phys. Rev. A 34, 1617 (1986).
[CrossRef] [PubMed]

F. T. Arecchi, R. Meucci, G. P. Puccioni, and J. R. Tredicce, Phys. Rev. Lett. 49, 1217 (1982).
[CrossRef]

F. T. Arecchi, A. Lapucci, R. Meucci, J. A. Roversi, and P. Coullet, submitted to Phys. Rev. Lett.

Midavaine, T.

T. Midavaine, D. Dangoisse, and P. Glorieux, Phys. Rev. Lett. 55, 1189 (1985).
[CrossRef]

Politi, A.

F. T. Arecchi and A. Politi, Phys. Rev. Lett. 45, 1215 (1980);F. T. Arecchi, A. Politi, and L. Ulivi, Nuovo Cimento 71B, 119 (1982).
[CrossRef]

Puccioni, G. P.

F. T. Arecchi, G. L. Lippi, G. P. Puccioni, and J. R. Tredicce, Opt. Commun. 51, 308 (1984).
[CrossRef]

F. T. Arecchi, R. Meucci, G. P. Puccioni, and J. R. Tredicce, Phys. Rev. Lett. 49, 1217 (1982).
[CrossRef]

Querzola, B.

F. T. Arecchi, V. Degiorgio, and B. Querzola, Phys. Rev. Lett. 19, 168 (1967).
[CrossRef]

Richetti, P.

F. Argoul, A. Arneodo, and P. Richetti, Phys. Lett. A 120, 269 (1987).
[CrossRef]

Roversi, J. A.

F. T. Arecchi, A. Lapucci, R. Meucci, J. A. Roversi, and P. Coullet, submitted to Phys. Rev. Lett.

Shil’nikov, L. P.

L. P. Shil’nikov, Dokl. Akad. Nauk SSSR 160, 558 (1965);L. P. Shil’nikov, Mat. Sb. 77, 119, 461 (1968);Mat. Sb. 81, 92, 123 (1970).

Sparrow, C.

P. Glendinning and C. Sparrow, J. Stat. Phys. 35, 645 (1984);P. Gaspard, R. Kapral, and G. Nicolis, J. Stat. Phys. 35, 697 (1984).
[CrossRef]

Spiegel, E. A.

A. Arneodo, P. H. Coullet, E. A. Spiegel, and C. Tresser, Physica 14D, 327 (1985).

Tredicce, J. R.

G. L. Lippi, J. R. Tredicce, N. B. Abraham, and F. T. Arecchi, Opt. Commun. 53, 129 (1985).
[CrossRef]

F. T. Arecchi, G. L. Lippi, G. P. Puccioni, and J. R. Tredicce, Opt. Commun. 51, 308 (1984).
[CrossRef]

F. T. Arecchi, R. Meucci, G. P. Puccioni, and J. R. Tredicce, Phys. Rev. Lett. 49, 1217 (1982).
[CrossRef]

Tresser, C.

A. Arneodo, P. H. Coullet, E. A. Spiegel, and C. Tresser, Physica 14D, 327 (1985).

Dokl. Akad. Nauk SSSR (1)

L. P. Shil’nikov, Dokl. Akad. Nauk SSSR 160, 558 (1965);L. P. Shil’nikov, Mat. Sb. 77, 119, 461 (1968);Mat. Sb. 81, 92, 123 (1970).

J. Stat. Phys. (1)

P. Glendinning and C. Sparrow, J. Stat. Phys. 35, 645 (1984);P. Gaspard, R. Kapral, and G. Nicolis, J. Stat. Phys. 35, 697 (1984).
[CrossRef]

Opt. Commun. (2)

F. T. Arecchi, G. L. Lippi, G. P. Puccioni, and J. R. Tredicce, Opt. Commun. 51, 308 (1984).
[CrossRef]

G. L. Lippi, J. R. Tredicce, N. B. Abraham, and F. T. Arecchi, Opt. Commun. 53, 129 (1985).
[CrossRef]

Phys. Lett. A (1)

F. Argoul, A. Arneodo, and P. Richetti, Phys. Lett. A 120, 269 (1987).
[CrossRef]

Phys. Rep. (1)

J. P. Crutchfield, D. Farmer, and B. A. Hubermann, Phys. Rep. 92, 45 (1982).
[CrossRef]

Phys. Rev. A (1)

F. T. Arecchi, W. Gadomski, and R. Meucci, Phys. Rev. A 34, 1617 (1986).
[CrossRef] [PubMed]

Phys. Rev. Lett. (6)

F. T. Arecchi, V. Degiorgio, and B. Querzola, Phys. Rev. Lett. 19, 168 (1967).
[CrossRef]

F. Haake, Phys. Rev. Lett. 41, 1685 (1978).
[CrossRef]

F. T. Arecchi and A. Politi, Phys. Rev. Lett. 45, 1215 (1980);F. T. Arecchi, A. Politi, and L. Ulivi, Nuovo Cimento 71B, 119 (1982).
[CrossRef]

F. T. Arecchi, R. Meucci, and W. Gadomski, Phys. Rev. Lett. 58, 2205 (1987).
[CrossRef] [PubMed]

F. T. Arecchi, R. Meucci, G. P. Puccioni, and J. R. Tredicce, Phys. Rev. Lett. 49, 1217 (1982).
[CrossRef]

T. Midavaine, D. Dangoisse, and P. Glorieux, Phys. Rev. Lett. 55, 1189 (1985).
[CrossRef]

Physica (1)

A. Arneodo, P. H. Coullet, E. A. Spiegel, and C. Tresser, Physica 14D, 327 (1985).

Other (2)

F. T. Arecchi, in Instabilities and Chaos in Quantum Optics, F. T. Arecchi and R. G. Harrison, eds., Vol. 34 of Springer Series in Synergetics (Springer-Verlag, Berlin, 1987), p. 9.
[CrossRef]

F. T. Arecchi, A. Lapucci, R. Meucci, J. A. Roversi, and P. Coullet, submitted to Phys. Rev. Lett.

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Figures (11)

Fig. 1
Fig. 1

Experimental setup. M, total reflecting mirror mounted on a piezoelectric (P.Z.T.) drive; E.O. MOD, electro-optic modulator; B.S., ZnSe beam splitter; G, grating; D, HgCdTe detector; B, bias voltage; H.V., high-voltage amplifier; A, amplifier.

Fig. 2
Fig. 2

Plot of the normalized stationary intensity x* versus B for A = 6.667 and α = 9.0. Curves a, b, c, d, arid e refer to r = 0.0, 0.04, 0.08, 0.12, and 0.16, respectively. The horizontal dashed line corresponds to stationary solutions with z* = 0. Points 0, 1, and 2, indicated by arrows, are the stationary points for B = 0.838 and r = 0.16.

Fig. 3
Fig. 3

Schematic view of a trajectory in the phase space when the dynamics are affected by all three unstable stationary points. The values of A and α are the same as in Fig. 2, with r = 0.16 and B = 0.838. The normalized components of eigenvectors are given in Table 1.

Fig. 4
Fig. 4

Return-times measurement setup. COMP., comparator (threshold circuit); F.D., frequency divider; T.A.C.1, T.A.C.2, time-to-amplitude converters; M.A., multichannel analyzer; A/D, analog-to-digital converter; C, computer.

Fig. 5
Fig. 5

(a) Time plot of the laser intensity at the onset of the Hopf bifurcation (r = 0.330 and B = 0.270). (b) T.A.C. pulses, whose heights are proportional to the return times to a Poincaré section x(t) = constant [see the dashed line of Fig. 5(a)]. (c) Corresponding statistical distribution of return times.

Fig. 6
Fig. 6

(a) Time plot of the laser intensity. (b) T.A.C. pulses and (c) statistical distribution of return times for a local chaos around point 1 (r = 0.330 and B = 0.283).

Fig. 7
Fig. 7

(a) Time plot of the laser intensity. (b) T.A.C. pulses and (c) statistical distribution of return times in chaotic region of competing instabilities between points 1 and 2 (r = 0.380 and B = 0.285).

Fig. 8
Fig. 8

(a) Time plot of the laser intensity. (b) T.A.C. pulses and (c) statistical distribution of return times in a window of Shil’nikov chaos (r = 0.430 and B = 0.340).

Fig. 9
Fig. 9

Experimental iteration maps of the return times, (a) r = 0.487 and B = 0.350. (b) Maps corresponding to regular periodic situations, namely, 1, an electronic oscillator; 2, the laser in a regular periodic regime; 3, the laser just at the onset of the instability but still with a regular period.

Fig. 10
Fig. 10

Time plots of the intensity in the regime of Shil’nikov chaos, (a), (b) Refer to the same B value (B = 0.427) but two different gains r (0.487 and 0.696, respectively) of the feedback loop. (b) Shows long transients corresponding to a large number of small spirals around the saddle focus.

Fig. 11
Fig. 11

Numerical iteration maps for Shil’nikov chaos. Parameter values: ω/γ = 13.0, α/γ= 0.986, c = 0.01. (a) and (b), τ maps without and with noise δ∊ = 10−2, respectively, (c) Stable fixed point of the regular dynamics, broadened by a noise δ∊= 10−2.

Tables (1)

Tables Icon

Table 1 Eigenvalues and Eigenvector Components of the Fixed Points Shown in Fig. 2a

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

= K 0 x [ 1 + α sin 2 ( z ) y ] ,
y ˙ = γ ( y + x y A ) ,
ż = β ( z B + r x ) ,
B = r x * + arcsin ( { [ A / ( 1 + x * ) 1 ] / α } 1 / 2 ) .
Δ τ 0 / τ 0 14 % , Δ τ 2 / τ 2 80 % , Δ τ 0 / τ 0 40 % , Δ τ 2 / τ 2 250 % .
ζ n + 1 = ζ n λ / γ cos [ ω / γ ln ( ζ n ) ] + ,
τ n + 1 = ln [ exp ( λ / γ τ n ) cos ( ω / γ τ n ) + ] = ln [ φ ( τ ) + ] ,
τ / = [ φ ( τ ) + ] 1 .

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