Abstract

The observed motion of stellar-image centroids is shown to have a chaotic attractor with a correlation dimension of ~6. The existence of a chaotic attractor in star wander, or equivalently in wave-front tilts, indicates that the atmospheric processes that cause image degradation may be more accurately described as chaotic, not so random as is usually assumed. This new result has important implications for the accurate modeling of atmospheric processes, the operation of adaptive optics systems, and the processing of stellar images.

© 1991 Optical Society of America

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References

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  1. J.-P. Eckmann, D. Ruelle, Rev. Mod. Phys. 57, 617 (1985).
    [Crossref]
  2. R. H. Cromwell, V. R. Haemerle, N. J. Woolf, Proc. Soc. Photo-Opt. Instrum. Eng. 1236, 520 (1990).
  3. N. H. Packard, J. P. Crutchfield, J. D. Farmer, R. S. Shaw, Phys. Rev. Lett. 45, 712 (1980).
    [Crossref]
  4. F. Takens, in Dynamical Systems and Turbulence, D. Rank, L. S. Young, eds. (Springer-Verlag, Berlin, 1981), pp. 366–381.
    [Crossref]
  5. P. Grassberger, J. Procaccia, Phys. Rev. Lett. 50, 346 (1980).
    [Crossref]
  6. P. Atten, B. Malraison, in Le Chaos, P. Berge, ed. (Eyrolles, Paris, 1988), pp. 283–325.
  7. D. Ruelle, Proc. R. Soc. London Ser. A 427, 241 (1990).
    [Crossref]
  8. J. D. Farmer, J. J. Sidorowich, Phys. Rev. Lett. 59, 845 (1987).
    [Crossref] [PubMed]
  9. A. Lapedes, R. Farber, in Neural Information Processing Systems, D. Anderson, ed. (American Institute of Physics, New York, 1988), pp. 442–456.
  10. J. Primot, G. Rousset, J. C. Fontanella, in Very Large Telescopes and Their Instrumentation, M.-H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1988), pp. 682–683.
  11. H. Leung, S. Haykin, Appl. Phys. Lett. 56, 593 (1990).
    [Crossref]
  12. H. Leung, S. Haykin, Proc. Soc. Photo-Opt. Instrum. Eng. 1152, 18 (1989).

1990 (3)

R. H. Cromwell, V. R. Haemerle, N. J. Woolf, Proc. Soc. Photo-Opt. Instrum. Eng. 1236, 520 (1990).

D. Ruelle, Proc. R. Soc. London Ser. A 427, 241 (1990).
[Crossref]

H. Leung, S. Haykin, Appl. Phys. Lett. 56, 593 (1990).
[Crossref]

1989 (1)

H. Leung, S. Haykin, Proc. Soc. Photo-Opt. Instrum. Eng. 1152, 18 (1989).

1987 (1)

J. D. Farmer, J. J. Sidorowich, Phys. Rev. Lett. 59, 845 (1987).
[Crossref] [PubMed]

1985 (1)

J.-P. Eckmann, D. Ruelle, Rev. Mod. Phys. 57, 617 (1985).
[Crossref]

1980 (2)

N. H. Packard, J. P. Crutchfield, J. D. Farmer, R. S. Shaw, Phys. Rev. Lett. 45, 712 (1980).
[Crossref]

P. Grassberger, J. Procaccia, Phys. Rev. Lett. 50, 346 (1980).
[Crossref]

Atten, P.

P. Atten, B. Malraison, in Le Chaos, P. Berge, ed. (Eyrolles, Paris, 1988), pp. 283–325.

Cromwell, R. H.

R. H. Cromwell, V. R. Haemerle, N. J. Woolf, Proc. Soc. Photo-Opt. Instrum. Eng. 1236, 520 (1990).

Crutchfield, J. P.

N. H. Packard, J. P. Crutchfield, J. D. Farmer, R. S. Shaw, Phys. Rev. Lett. 45, 712 (1980).
[Crossref]

Eckmann, J.-P.

J.-P. Eckmann, D. Ruelle, Rev. Mod. Phys. 57, 617 (1985).
[Crossref]

Farber, R.

A. Lapedes, R. Farber, in Neural Information Processing Systems, D. Anderson, ed. (American Institute of Physics, New York, 1988), pp. 442–456.

Farmer, J. D.

J. D. Farmer, J. J. Sidorowich, Phys. Rev. Lett. 59, 845 (1987).
[Crossref] [PubMed]

N. H. Packard, J. P. Crutchfield, J. D. Farmer, R. S. Shaw, Phys. Rev. Lett. 45, 712 (1980).
[Crossref]

Fontanella, J. C.

J. Primot, G. Rousset, J. C. Fontanella, in Very Large Telescopes and Their Instrumentation, M.-H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1988), pp. 682–683.

Grassberger, P.

P. Grassberger, J. Procaccia, Phys. Rev. Lett. 50, 346 (1980).
[Crossref]

Haemerle, V. R.

R. H. Cromwell, V. R. Haemerle, N. J. Woolf, Proc. Soc. Photo-Opt. Instrum. Eng. 1236, 520 (1990).

Haykin, S.

H. Leung, S. Haykin, Appl. Phys. Lett. 56, 593 (1990).
[Crossref]

H. Leung, S. Haykin, Proc. Soc. Photo-Opt. Instrum. Eng. 1152, 18 (1989).

Lapedes, A.

A. Lapedes, R. Farber, in Neural Information Processing Systems, D. Anderson, ed. (American Institute of Physics, New York, 1988), pp. 442–456.

Leung, H.

H. Leung, S. Haykin, Appl. Phys. Lett. 56, 593 (1990).
[Crossref]

H. Leung, S. Haykin, Proc. Soc. Photo-Opt. Instrum. Eng. 1152, 18 (1989).

Malraison, B.

P. Atten, B. Malraison, in Le Chaos, P. Berge, ed. (Eyrolles, Paris, 1988), pp. 283–325.

Packard, N. H.

N. H. Packard, J. P. Crutchfield, J. D. Farmer, R. S. Shaw, Phys. Rev. Lett. 45, 712 (1980).
[Crossref]

Primot, J.

J. Primot, G. Rousset, J. C. Fontanella, in Very Large Telescopes and Their Instrumentation, M.-H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1988), pp. 682–683.

Procaccia, J.

P. Grassberger, J. Procaccia, Phys. Rev. Lett. 50, 346 (1980).
[Crossref]

Rousset, G.

J. Primot, G. Rousset, J. C. Fontanella, in Very Large Telescopes and Their Instrumentation, M.-H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1988), pp. 682–683.

Ruelle, D.

D. Ruelle, Proc. R. Soc. London Ser. A 427, 241 (1990).
[Crossref]

J.-P. Eckmann, D. Ruelle, Rev. Mod. Phys. 57, 617 (1985).
[Crossref]

Shaw, R. S.

N. H. Packard, J. P. Crutchfield, J. D. Farmer, R. S. Shaw, Phys. Rev. Lett. 45, 712 (1980).
[Crossref]

Sidorowich, J. J.

J. D. Farmer, J. J. Sidorowich, Phys. Rev. Lett. 59, 845 (1987).
[Crossref] [PubMed]

Takens, F.

F. Takens, in Dynamical Systems and Turbulence, D. Rank, L. S. Young, eds. (Springer-Verlag, Berlin, 1981), pp. 366–381.
[Crossref]

Woolf, N. J.

R. H. Cromwell, V. R. Haemerle, N. J. Woolf, Proc. Soc. Photo-Opt. Instrum. Eng. 1236, 520 (1990).

Appl. Phys. Lett. (1)

H. Leung, S. Haykin, Appl. Phys. Lett. 56, 593 (1990).
[Crossref]

Phys. Rev. Lett. (3)

N. H. Packard, J. P. Crutchfield, J. D. Farmer, R. S. Shaw, Phys. Rev. Lett. 45, 712 (1980).
[Crossref]

P. Grassberger, J. Procaccia, Phys. Rev. Lett. 50, 346 (1980).
[Crossref]

J. D. Farmer, J. J. Sidorowich, Phys. Rev. Lett. 59, 845 (1987).
[Crossref] [PubMed]

Proc. R. Soc. London Ser. A (1)

D. Ruelle, Proc. R. Soc. London Ser. A 427, 241 (1990).
[Crossref]

Proc. Soc. Photo-Opt. Instrum. Eng. (2)

H. Leung, S. Haykin, Proc. Soc. Photo-Opt. Instrum. Eng. 1152, 18 (1989).

R. H. Cromwell, V. R. Haemerle, N. J. Woolf, Proc. Soc. Photo-Opt. Instrum. Eng. 1236, 520 (1990).

Rev. Mod. Phys. (1)

J.-P. Eckmann, D. Ruelle, Rev. Mod. Phys. 57, 617 (1985).
[Crossref]

Other (4)

F. Takens, in Dynamical Systems and Turbulence, D. Rank, L. S. Young, eds. (Springer-Verlag, Berlin, 1981), pp. 366–381.
[Crossref]

A. Lapedes, R. Farber, in Neural Information Processing Systems, D. Anderson, ed. (American Institute of Physics, New York, 1988), pp. 442–456.

J. Primot, G. Rousset, J. C. Fontanella, in Very Large Telescopes and Their Instrumentation, M.-H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1988), pp. 682–683.

P. Atten, B. Malraison, in Le Chaos, P. Berge, ed. (Eyrolles, Paris, 1988), pp. 283–325.

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Figures (3)

Fig. 1
Fig. 1

90-sec, 2701-point time series of the x coordinate of the stellar image centroids for one of the data sets.

Fig. 2
Fig. 2

Correlation integral of the star-wander data. From left to right, the curves correspond to embedding dimensions, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 26, and 32.

Fig. 3
Fig. 3

Attractor dimensions as a function of embedding dimension for two data sets are plotted in the two lower curves. The theoretical relationship for random noise (the straight line) and the result of an analysis of the 2701-point random number sequence are shown for reference.

Equations (3)

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x ( t 0 ) , , x ( t N - 1 ) , y ( t 0 ) , , y ( t N - 1 ) , x ( t 0 + τ ) , , x ( t N - 1 + τ ) , y ( t 0 + τ ) , , y ( t N - 1 + τ ) , x [ t 0 + ( n 2 - 1 ) τ ] , , x [ t N - 1 + ( n 2 - 1 ) τ ] , y [ t 0 + ( n 2 - 1 ) τ ] , , y [ t N - 1 + ( n 2 - 1 ) τ ] .
{ x ( t i ) , y ( t i ) , , x [ t i + ( n 2 - 1 ) τ ] , y [ t i + ( n 2 - 1 ) τ ] } .
C ( r ) = lim N 1 N 2 θ ( r - z i - z j ) .

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