Abstract

A new class of instabilities for a plane-wave intracavity field in an optical ring resonator is identified. Dynamical systems techniques are explained and applied to the map. A bifurcation diagram is given that organizes the important information, and global pictures are developed that describe the evolution of the attractor and its basin boundary. Anomalous behavior observed in earlier numerical studies is explained.

© 1985 Optical Society of America

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References

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  1. A comprehensive review of chaos in optical bistability can be found in C. M. Bowden, H. M. Gibbs, S. L. McCall, eds. Optical Bistability II (Plenum, New York, 1984).
    [CrossRef]
  2. H. Haken, Phys. Lett. 53A, 77 (1975).
  3. K. Ikeda, Opt. Commun. 30, 257 (1979);K. Ikeda, H. Daido, O. Akimoto, Phys. Rev. Lett. 45, 709 (1980).
    [CrossRef]
  4. F. T. Arrechi, R. Meucci, G. Puccioni, J. Tredicce, Phys. Rev. Lett. 49, 1217 (1982);R. S. Gioggia, N. B. Abraham, Phys. Rev. Lett. 51, 650 (1983);K. Otsuka, H. Kawaguchi, Phys. Rev. A 29, 2953 (1984).
    [CrossRef]
  5. R. G. Harrison, W. J. Firth, C. A. Emshary, I. A. Al-Saidi, Phys. Rev. Lett. 51, 562 (1983);R. G. Harrison, W. J. Firth, I. A. Al-Saidi, Phys. Rev. Lett. 53, 258 (1984).
    [CrossRef]
  6. M. LeBerre, E. Ressayre, A. Tallet, K. Tai, F. A. Hopf, H. M. Gibbs, J. V. Moloney, “Optical bistability and instabilities via diffraction-free-encoding and a single feedback mirror,” presented at International Conference on Quantum Electronics, Los Angeles, California, 1984.The first reported observation of period-2 oscillations in a cw experiment was described.
  7. J. V. Moloney, F. A. Hopf, H. M. Gibbs, Phys. Rev. A 25, 3442 (1982);J. V. Moloney, Phys. Rev. Lett. (to be published).
    [CrossRef]
  8. W. J. Firth, E. M. Wright, Phys. Lett. 92A, 211 (1982).
  9. R. R. Snapp, H. J. Carmichael, W. C. Schieve, Opt. Commun. 40, 68 (1981);Phys. Rev. A 26, 3408 (1982).
    [CrossRef]
  10. J. V. Moloney, Opt. Commun. 48, 435 (1984).
    [CrossRef]
  11. C. Grebogi, E. Ott, J. A. Yorke, Phys. Rev. Lett. 48, 1507 (1982);Phys. D 7, 181 (1983).
    [CrossRef]
  12. J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields (Springer-Verlag, New York, 1983).
  13. W. J. Firth, Opt. Commun. 39, 343 (1981);E. Abraham, W. J. Firth, J. Carr, Phys. Lett. 69A, 47 (1982);E. Abraham, W. J. Firth, Opt. Acta, 30, 1541 (1983).
    [CrossRef]
  14. J. Carr, J. C. Eilbeck, “One dimensional approximations for a quadratic Ikeda map,” Los Alamos National Laboratory Tech. Notes (1984).
  15. P. Holmes, D. Whitley, Phys. D, 7, 111 (1983), and references therein.
    [CrossRef]
  16. S. Smale, Bull Am. Math. Soc. 73, 747 (1967).
    [CrossRef]
  17. Y. Ueda, Ann. N. Y. Acad. Sci. 357, 422 (1980).
    [CrossRef]
  18. J. D. Cresser, P. Meystre, in Optical Bistability, C. M. Bowden, M. Ciftan, H. R. Robl, eds. (Plenum, New York, 1981), p. 265 and references therein.
    [CrossRef]
  19. N. K. Gavrilov, L. P. Silnikov, Mat. Sb. 88, 467 (1972);Mat. Sb. 90, 139 (1973).
    [CrossRef]
  20. S. E. Newhouse, Progress in Mathematics (Birkhauser, Boston, Mass., 1980), Vol. 8, pp. 1–114.
  21. J. A. Yorke, Institute for Physical Science and Technology, Department of Mathematics, University of Maryland, College Park, Maryland 20742 (personal communication).
  22. K. T. Alligood, J. A. Yorke, “Period doubling cascades of attractors: a prerequisite for horseshoes,” Department of Mathematics, Michigan State University, East Lansing, Michigan 48824 (personal communication).
  23. J. V. Moloney, Phys. Rev. Lett. 51, 556 (1984).
    [CrossRef]

1984 (2)

J. V. Moloney, Opt. Commun. 48, 435 (1984).
[CrossRef]

J. V. Moloney, Phys. Rev. Lett. 51, 556 (1984).
[CrossRef]

1983 (2)

P. Holmes, D. Whitley, Phys. D, 7, 111 (1983), and references therein.
[CrossRef]

R. G. Harrison, W. J. Firth, C. A. Emshary, I. A. Al-Saidi, Phys. Rev. Lett. 51, 562 (1983);R. G. Harrison, W. J. Firth, I. A. Al-Saidi, Phys. Rev. Lett. 53, 258 (1984).
[CrossRef]

1982 (4)

J. V. Moloney, F. A. Hopf, H. M. Gibbs, Phys. Rev. A 25, 3442 (1982);J. V. Moloney, Phys. Rev. Lett. (to be published).
[CrossRef]

W. J. Firth, E. M. Wright, Phys. Lett. 92A, 211 (1982).

C. Grebogi, E. Ott, J. A. Yorke, Phys. Rev. Lett. 48, 1507 (1982);Phys. D 7, 181 (1983).
[CrossRef]

F. T. Arrechi, R. Meucci, G. Puccioni, J. Tredicce, Phys. Rev. Lett. 49, 1217 (1982);R. S. Gioggia, N. B. Abraham, Phys. Rev. Lett. 51, 650 (1983);K. Otsuka, H. Kawaguchi, Phys. Rev. A 29, 2953 (1984).
[CrossRef]

1981 (2)

W. J. Firth, Opt. Commun. 39, 343 (1981);E. Abraham, W. J. Firth, J. Carr, Phys. Lett. 69A, 47 (1982);E. Abraham, W. J. Firth, Opt. Acta, 30, 1541 (1983).
[CrossRef]

R. R. Snapp, H. J. Carmichael, W. C. Schieve, Opt. Commun. 40, 68 (1981);Phys. Rev. A 26, 3408 (1982).
[CrossRef]

1980 (1)

Y. Ueda, Ann. N. Y. Acad. Sci. 357, 422 (1980).
[CrossRef]

1979 (1)

K. Ikeda, Opt. Commun. 30, 257 (1979);K. Ikeda, H. Daido, O. Akimoto, Phys. Rev. Lett. 45, 709 (1980).
[CrossRef]

1975 (1)

H. Haken, Phys. Lett. 53A, 77 (1975).

1972 (1)

N. K. Gavrilov, L. P. Silnikov, Mat. Sb. 88, 467 (1972);Mat. Sb. 90, 139 (1973).
[CrossRef]

1967 (1)

S. Smale, Bull Am. Math. Soc. 73, 747 (1967).
[CrossRef]

Alligood, K. T.

K. T. Alligood, J. A. Yorke, “Period doubling cascades of attractors: a prerequisite for horseshoes,” Department of Mathematics, Michigan State University, East Lansing, Michigan 48824 (personal communication).

Al-Saidi, I. A.

R. G. Harrison, W. J. Firth, C. A. Emshary, I. A. Al-Saidi, Phys. Rev. Lett. 51, 562 (1983);R. G. Harrison, W. J. Firth, I. A. Al-Saidi, Phys. Rev. Lett. 53, 258 (1984).
[CrossRef]

Arrechi, F. T.

F. T. Arrechi, R. Meucci, G. Puccioni, J. Tredicce, Phys. Rev. Lett. 49, 1217 (1982);R. S. Gioggia, N. B. Abraham, Phys. Rev. Lett. 51, 650 (1983);K. Otsuka, H. Kawaguchi, Phys. Rev. A 29, 2953 (1984).
[CrossRef]

Carmichael, H. J.

R. R. Snapp, H. J. Carmichael, W. C. Schieve, Opt. Commun. 40, 68 (1981);Phys. Rev. A 26, 3408 (1982).
[CrossRef]

Carr, J.

J. Carr, J. C. Eilbeck, “One dimensional approximations for a quadratic Ikeda map,” Los Alamos National Laboratory Tech. Notes (1984).

Cresser, J. D.

J. D. Cresser, P. Meystre, in Optical Bistability, C. M. Bowden, M. Ciftan, H. R. Robl, eds. (Plenum, New York, 1981), p. 265 and references therein.
[CrossRef]

Eilbeck, J. C.

J. Carr, J. C. Eilbeck, “One dimensional approximations for a quadratic Ikeda map,” Los Alamos National Laboratory Tech. Notes (1984).

Emshary, C. A.

R. G. Harrison, W. J. Firth, C. A. Emshary, I. A. Al-Saidi, Phys. Rev. Lett. 51, 562 (1983);R. G. Harrison, W. J. Firth, I. A. Al-Saidi, Phys. Rev. Lett. 53, 258 (1984).
[CrossRef]

Firth, W. J.

R. G. Harrison, W. J. Firth, C. A. Emshary, I. A. Al-Saidi, Phys. Rev. Lett. 51, 562 (1983);R. G. Harrison, W. J. Firth, I. A. Al-Saidi, Phys. Rev. Lett. 53, 258 (1984).
[CrossRef]

W. J. Firth, E. M. Wright, Phys. Lett. 92A, 211 (1982).

W. J. Firth, Opt. Commun. 39, 343 (1981);E. Abraham, W. J. Firth, J. Carr, Phys. Lett. 69A, 47 (1982);E. Abraham, W. J. Firth, Opt. Acta, 30, 1541 (1983).
[CrossRef]

Gavrilov, N. K.

N. K. Gavrilov, L. P. Silnikov, Mat. Sb. 88, 467 (1972);Mat. Sb. 90, 139 (1973).
[CrossRef]

Gibbs, H. M.

J. V. Moloney, F. A. Hopf, H. M. Gibbs, Phys. Rev. A 25, 3442 (1982);J. V. Moloney, Phys. Rev. Lett. (to be published).
[CrossRef]

M. LeBerre, E. Ressayre, A. Tallet, K. Tai, F. A. Hopf, H. M. Gibbs, J. V. Moloney, “Optical bistability and instabilities via diffraction-free-encoding and a single feedback mirror,” presented at International Conference on Quantum Electronics, Los Angeles, California, 1984.The first reported observation of period-2 oscillations in a cw experiment was described.

Grebogi, C.

C. Grebogi, E. Ott, J. A. Yorke, Phys. Rev. Lett. 48, 1507 (1982);Phys. D 7, 181 (1983).
[CrossRef]

Guckenheimer, J.

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

Haken, H.

H. Haken, Phys. Lett. 53A, 77 (1975).

Harrison, R. G.

R. G. Harrison, W. J. Firth, C. A. Emshary, I. A. Al-Saidi, Phys. Rev. Lett. 51, 562 (1983);R. G. Harrison, W. J. Firth, I. A. Al-Saidi, Phys. Rev. Lett. 53, 258 (1984).
[CrossRef]

Holmes, P.

P. Holmes, D. Whitley, Phys. D, 7, 111 (1983), and references therein.
[CrossRef]

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

Hopf, F. A.

J. V. Moloney, F. A. Hopf, H. M. Gibbs, Phys. Rev. A 25, 3442 (1982);J. V. Moloney, Phys. Rev. Lett. (to be published).
[CrossRef]

M. LeBerre, E. Ressayre, A. Tallet, K. Tai, F. A. Hopf, H. M. Gibbs, J. V. Moloney, “Optical bistability and instabilities via diffraction-free-encoding and a single feedback mirror,” presented at International Conference on Quantum Electronics, Los Angeles, California, 1984.The first reported observation of period-2 oscillations in a cw experiment was described.

Ikeda, K.

K. Ikeda, Opt. Commun. 30, 257 (1979);K. Ikeda, H. Daido, O. Akimoto, Phys. Rev. Lett. 45, 709 (1980).
[CrossRef]

LeBerre, M.

M. LeBerre, E. Ressayre, A. Tallet, K. Tai, F. A. Hopf, H. M. Gibbs, J. V. Moloney, “Optical bistability and instabilities via diffraction-free-encoding and a single feedback mirror,” presented at International Conference on Quantum Electronics, Los Angeles, California, 1984.The first reported observation of period-2 oscillations in a cw experiment was described.

Meucci, R.

F. T. Arrechi, R. Meucci, G. Puccioni, J. Tredicce, Phys. Rev. Lett. 49, 1217 (1982);R. S. Gioggia, N. B. Abraham, Phys. Rev. Lett. 51, 650 (1983);K. Otsuka, H. Kawaguchi, Phys. Rev. A 29, 2953 (1984).
[CrossRef]

Meystre, P.

J. D. Cresser, P. Meystre, in Optical Bistability, C. M. Bowden, M. Ciftan, H. R. Robl, eds. (Plenum, New York, 1981), p. 265 and references therein.
[CrossRef]

Moloney, J. V.

J. V. Moloney, Phys. Rev. Lett. 51, 556 (1984).
[CrossRef]

J. V. Moloney, Opt. Commun. 48, 435 (1984).
[CrossRef]

J. V. Moloney, F. A. Hopf, H. M. Gibbs, Phys. Rev. A 25, 3442 (1982);J. V. Moloney, Phys. Rev. Lett. (to be published).
[CrossRef]

M. LeBerre, E. Ressayre, A. Tallet, K. Tai, F. A. Hopf, H. M. Gibbs, J. V. Moloney, “Optical bistability and instabilities via diffraction-free-encoding and a single feedback mirror,” presented at International Conference on Quantum Electronics, Los Angeles, California, 1984.The first reported observation of period-2 oscillations in a cw experiment was described.

Newhouse, S. E.

S. E. Newhouse, Progress in Mathematics (Birkhauser, Boston, Mass., 1980), Vol. 8, pp. 1–114.

Ott, E.

C. Grebogi, E. Ott, J. A. Yorke, Phys. Rev. Lett. 48, 1507 (1982);Phys. D 7, 181 (1983).
[CrossRef]

Puccioni, G.

F. T. Arrechi, R. Meucci, G. Puccioni, J. Tredicce, Phys. Rev. Lett. 49, 1217 (1982);R. S. Gioggia, N. B. Abraham, Phys. Rev. Lett. 51, 650 (1983);K. Otsuka, H. Kawaguchi, Phys. Rev. A 29, 2953 (1984).
[CrossRef]

Ressayre, E.

M. LeBerre, E. Ressayre, A. Tallet, K. Tai, F. A. Hopf, H. M. Gibbs, J. V. Moloney, “Optical bistability and instabilities via diffraction-free-encoding and a single feedback mirror,” presented at International Conference on Quantum Electronics, Los Angeles, California, 1984.The first reported observation of period-2 oscillations in a cw experiment was described.

Schieve, W. C.

R. R. Snapp, H. J. Carmichael, W. C. Schieve, Opt. Commun. 40, 68 (1981);Phys. Rev. A 26, 3408 (1982).
[CrossRef]

Silnikov, L. P.

N. K. Gavrilov, L. P. Silnikov, Mat. Sb. 88, 467 (1972);Mat. Sb. 90, 139 (1973).
[CrossRef]

Smale, S.

S. Smale, Bull Am. Math. Soc. 73, 747 (1967).
[CrossRef]

Snapp, R. R.

R. R. Snapp, H. J. Carmichael, W. C. Schieve, Opt. Commun. 40, 68 (1981);Phys. Rev. A 26, 3408 (1982).
[CrossRef]

Tai, K.

M. LeBerre, E. Ressayre, A. Tallet, K. Tai, F. A. Hopf, H. M. Gibbs, J. V. Moloney, “Optical bistability and instabilities via diffraction-free-encoding and a single feedback mirror,” presented at International Conference on Quantum Electronics, Los Angeles, California, 1984.The first reported observation of period-2 oscillations in a cw experiment was described.

Tallet, A.

M. LeBerre, E. Ressayre, A. Tallet, K. Tai, F. A. Hopf, H. M. Gibbs, J. V. Moloney, “Optical bistability and instabilities via diffraction-free-encoding and a single feedback mirror,” presented at International Conference on Quantum Electronics, Los Angeles, California, 1984.The first reported observation of period-2 oscillations in a cw experiment was described.

Tredicce, J.

F. T. Arrechi, R. Meucci, G. Puccioni, J. Tredicce, Phys. Rev. Lett. 49, 1217 (1982);R. S. Gioggia, N. B. Abraham, Phys. Rev. Lett. 51, 650 (1983);K. Otsuka, H. Kawaguchi, Phys. Rev. A 29, 2953 (1984).
[CrossRef]

Ueda, Y.

Y. Ueda, Ann. N. Y. Acad. Sci. 357, 422 (1980).
[CrossRef]

Whitley, D.

P. Holmes, D. Whitley, Phys. D, 7, 111 (1983), and references therein.
[CrossRef]

Wright, E. M.

W. J. Firth, E. M. Wright, Phys. Lett. 92A, 211 (1982).

Yorke, J. A.

C. Grebogi, E. Ott, J. A. Yorke, Phys. Rev. Lett. 48, 1507 (1982);Phys. D 7, 181 (1983).
[CrossRef]

K. T. Alligood, J. A. Yorke, “Period doubling cascades of attractors: a prerequisite for horseshoes,” Department of Mathematics, Michigan State University, East Lansing, Michigan 48824 (personal communication).

J. A. Yorke, Institute for Physical Science and Technology, Department of Mathematics, University of Maryland, College Park, Maryland 20742 (personal communication).

Ann. N. Y. Acad. Sci. (1)

Y. Ueda, Ann. N. Y. Acad. Sci. 357, 422 (1980).
[CrossRef]

Bull Am. Math. Soc. (1)

S. Smale, Bull Am. Math. Soc. 73, 747 (1967).
[CrossRef]

Mat. Sb. (1)

N. K. Gavrilov, L. P. Silnikov, Mat. Sb. 88, 467 (1972);Mat. Sb. 90, 139 (1973).
[CrossRef]

Opt. Commun. (4)

W. J. Firth, Opt. Commun. 39, 343 (1981);E. Abraham, W. J. Firth, J. Carr, Phys. Lett. 69A, 47 (1982);E. Abraham, W. J. Firth, Opt. Acta, 30, 1541 (1983).
[CrossRef]

R. R. Snapp, H. J. Carmichael, W. C. Schieve, Opt. Commun. 40, 68 (1981);Phys. Rev. A 26, 3408 (1982).
[CrossRef]

J. V. Moloney, Opt. Commun. 48, 435 (1984).
[CrossRef]

K. Ikeda, Opt. Commun. 30, 257 (1979);K. Ikeda, H. Daido, O. Akimoto, Phys. Rev. Lett. 45, 709 (1980).
[CrossRef]

Phys. D (1)

P. Holmes, D. Whitley, Phys. D, 7, 111 (1983), and references therein.
[CrossRef]

Phys. Lett. (2)

H. Haken, Phys. Lett. 53A, 77 (1975).

W. J. Firth, E. M. Wright, Phys. Lett. 92A, 211 (1982).

Phys. Rev. A (1)

J. V. Moloney, F. A. Hopf, H. M. Gibbs, Phys. Rev. A 25, 3442 (1982);J. V. Moloney, Phys. Rev. Lett. (to be published).
[CrossRef]

Phys. Rev. Lett. (4)

F. T. Arrechi, R. Meucci, G. Puccioni, J. Tredicce, Phys. Rev. Lett. 49, 1217 (1982);R. S. Gioggia, N. B. Abraham, Phys. Rev. Lett. 51, 650 (1983);K. Otsuka, H. Kawaguchi, Phys. Rev. A 29, 2953 (1984).
[CrossRef]

R. G. Harrison, W. J. Firth, C. A. Emshary, I. A. Al-Saidi, Phys. Rev. Lett. 51, 562 (1983);R. G. Harrison, W. J. Firth, I. A. Al-Saidi, Phys. Rev. Lett. 53, 258 (1984).
[CrossRef]

C. Grebogi, E. Ott, J. A. Yorke, Phys. Rev. Lett. 48, 1507 (1982);Phys. D 7, 181 (1983).
[CrossRef]

J. V. Moloney, Phys. Rev. Lett. 51, 556 (1984).
[CrossRef]

Other (8)

J. D. Cresser, P. Meystre, in Optical Bistability, C. M. Bowden, M. Ciftan, H. R. Robl, eds. (Plenum, New York, 1981), p. 265 and references therein.
[CrossRef]

S. E. Newhouse, Progress in Mathematics (Birkhauser, Boston, Mass., 1980), Vol. 8, pp. 1–114.

J. A. Yorke, Institute for Physical Science and Technology, Department of Mathematics, University of Maryland, College Park, Maryland 20742 (personal communication).

K. T. Alligood, J. A. Yorke, “Period doubling cascades of attractors: a prerequisite for horseshoes,” Department of Mathematics, Michigan State University, East Lansing, Michigan 48824 (personal communication).

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

J. Carr, J. C. Eilbeck, “One dimensional approximations for a quadratic Ikeda map,” Los Alamos National Laboratory Tech. Notes (1984).

M. LeBerre, E. Ressayre, A. Tallet, K. Tai, F. A. Hopf, H. M. Gibbs, J. V. Moloney, “Optical bistability and instabilities via diffraction-free-encoding and a single feedback mirror,” presented at International Conference on Quantum Electronics, Los Angeles, California, 1984.The first reported observation of period-2 oscillations in a cw experiment was described.

A comprehensive review of chaos in optical bistability can be found in C. M. Bowden, H. M. Gibbs, S. L. McCall, eds. Optical Bistability II (Plenum, New York, 1984).
[CrossRef]

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Figures (16)

Fig. 1
Fig. 1

Graphical construction for fixed-point determination. The upper (U), middle (M), and lower (L) fixed points are indicated. The middle-branch fixed point is always unstable (see text).

Fig. 2
Fig. 2

Two-dimensional surface showing the modulus of the three-fixed-point solution plotted against the (p, a) parameter plane. A slice through this surface at fixed p gives the usual hysteresis loop.

Fig. 3
Fig. 3

The phase portraits of the lower-branch fixed point (L) after it has period doubled. The unstable manifold (Wu) is a continuous curve along which points move from L to the period-2 points under iteration of the map [Eq. (1)]. Points move along the stable manifold (Ws) toward the unstable saddle point L. Notice that under the action of the map, points are alternately flipped back and forth about L as the eigenvalues are on the negative real axis [see Eq. (5)].

Fig. 4
Fig. 4

The stable (Ws) and unstable (Wu) manifolds of the lower-branch saddle point just touch, forming a homoclinic tangency. The long, fingerlike regions stretch out to touch the respective manifolds, and the area of each finger reduces (increases) by R2 for Wu (Ws) as it approaches the unstable saddle point. Under higher iterations of the map the manifolds oscillate wildly, forming complicated, interleaved fingers.

Fig. 5
Fig. 5

Bifurcation requires that an eigenvalue cross the unit circle |λ| = 1. (a) Eigenvalue pair at the saddle-node bifurcation point. Here one eigenvalue is at +1 and the other is at R2, maintaining the relationship λ1λ2 = R2 (see text). (b) Situation at the period-doubling (or flip) bifurcation point. Here one eigenvalue is at −1. (c) The only situation that can occur for Eq. (1) if the eigenvalues form a complex-conjugate pair. The eigenvalues are forced to lie on a circle of radius R < 1, and this fact rules out a Hopf bifurcation, for which a complex-conjugate pair would have to cross the unit circle.

Fig. 6
Fig. 6

(a) Usual hysteresis loop. As a increases past some value a saddle (M) node (U) pair is abruptly created in the phase plane. At the switch-up point a saddle (M) node (L) pair is mutually annihilated (reverse saddle node). (b) Sketch of the phase portraits at the parameter value indicated by the vertical dashed line in (a). The stable and unstable manifolds are indicated together with the lower- (L), middle- (M), and upper- (U) branch fixed points.

Fig 7
Fig 7

(a) Phase portraits (top) and associated eigenvalue behavior (bottom) as a saddle-node pair is created. The node changes to a spiral in (b) when the associated eigenvalue pair lies on the circle R (complex conjugates). As the period-doubling bifurcation point is approached (c), one direction at L (lower-branch fixed point) becomes strongly stable (one eigenvalue approaches −R2 while the other eigenvalue approaches −1). Immediately after bifurcation, L has become a saddle and the period-2 points are created (d).

Fig. 8
Fig. 8

Bifurcation diagram in (p, a) parameter space. Regions of one and three fixed points are indicated. The upper-branch fixed point (labeled 3) is always stable over the present parameter range, and the remaining boundaries in the figure relate to the lower-branch fixed point. Two separate period-doubling cascades to chaos are evident (compare with Fig. 1 of Ref. 10). The meaning of the various boundaries is discussed in the text.

Fig. 9
Fig. 9

Attractor basin boundary separating basins of attraction of the stable lower- (L) and upper- (U) branch fixed points. The stable manifold (Ws) defines the basin boundary, and the oscillations are an indication of an impending homoclinic tangency. The unstable middle- (M) branch fixed point is also shown.

Fig. 10
Fig. 10

Period-doubling cascade to chaos as the parameter p is varied (a = 0.8). (a) After bifurcation of the lower-branch fixed point to period 2 the manifolds (Ws) and (Wu) are already approaching homoclinic tangency (p = 4.5). (b) At p = 5.2, the period-2 orbit has bifurcation to period 4. At this stage the manifolds are well past tangency, and the stable manifold (Ws) is not shown for clarity. As one approaches the accumulation point of the period-doubling cascade (c) (period 16 at p = 5.26), the unstable manifold develops larger twists as a result of its transversal intersections with Ws. Finally, beyond the accumulation point the fully developed chaotic attractor is shown in (d) (p = 5.4). Notice the resemblance of this attractor to Wu in (c) even though the former was generated by iterating a single point 25,000 times. The Cantor-like structure is evident in this figure.

Fig. 11
Fig. 11

Period-6 attractor coexisting with a period-2 cycle. The latter has bifurcated from the lower-branch (L) fixed point (the saddle point associated with Ws and Wu in the figure). The period-6 attractor goes through its own period-doubling cascade to a six-piece chaotic attractor, independent of the period-2 orbit. Here the parameter p is held fixed at p = 5 and a is varied. (a) Period-6 attractor (heavy dots) coexisting with period 2 (asterisks) (a = 0.70). (b) Period-12 orbit (doubling of period 6) coexisting with period 2 (a = 0.72). (c) Six-piece chaotic attractor at the end of the cascade coexisting with period 2 (a = 0.73). (d) Six-piece chaotic attractor with the stable (Ws) and unstable (Wu) manifolds of the lower-branch (L) saddle point deleted.

Fig. 12
Fig. 12

(a) Unstable period-6 saddle points with their stable and unstable manifolds corresponding to Fig. 11(a). Notice the complicated basin structure of this six-piece attractor (defined by the stable manifolds of these saddle points). These basins of attraction are strongly interleaved with the period-2 attractor basins (see Fig. 11). (b) Unstable manifolds of the period-6 saddles alone. On one side of each saddle, points spiral into the attracting period-6 points, while on the other side they spiral into the stable period-2 attractor.

Fig. 13
Fig. 13

Period-16 cycle just bifurcated from a period-8 attractor created by a saddle-node bifurcation in F8, the eight composition of the map [Eq. (1)] (p = 5, a = 0.69). The location of one of the unstable period-8 saddles is indicated by the small square. Notice that this period-8 saddle lies much closer to the tip of the lobe than the period-6 orbit in Fig. 11(a). The period-8 attractor undergoes its own period-doubling cascade, culminating in an eight-piece chaotic attractor, which is then destroyed. Again events associated with this attractor occur independently of the period-doubling cascade from the lower-branch fixed point (L), which at this stage has only reached period 2 (labeled with an asterisk).

Fig. 14
Fig. 14

Example of protohorseshoe construction for the period-6 saddle point (p = 5, a = 0.72). The small rectangular region shown in the figure is mapped under F6 to a horseshoelike region spanning the rectangular box. This construction is shown at only one of the unstable saddle points for clarity. A period-12 orbit exists at these parameter values.

Fig. 15
Fig. 15

Blow-up of the protohorseshoe construction for the preceding figure. An invariant attracting region is created within the rectangular box consisting of a saddle and a node. The node period doubles to a chaotic attractor as the horseshoe develops (see the text and Fig. 6). In the present figure the original period-6 attractor has period doubled to period 12 (dots denote two of the twelve periodic points). (b) Same as (a) but with the unstable saddle (S6) and its manifolds included. The manifolds are at a prehomoclinic stage.

Fig. 16
Fig. 16

Fully developed horseshoe is formed after the stable and unstable manifolds reach homoclinic tangency. (a) The saddle point S6 and its stable and unstable manifolds are indicated. (b) One further mapping shows how the horseshoe is mapped into itself, contracting in one direction and expanding in the other. The new region consists of a long, extended filament doubled up within the original horseshoe. Smale16 has shown that the horseshoe contains infinitely many unstable periodic points.

Equations (9)

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g n = a + R exp [ i ( ϕ p 1 + | g n 1 | 2 ) ] g n 1 = F ( g n 1 ) .
a = T E in Δ , ϕ = k L 1 , p = α 0 L / 2 Δ , G n = E n / Δ ,
g n = ( g n 1 a ) exp { i [ ϕ p R 2 / ( R 2 + | g n 1 a | 2 ] } / R .
| g a | = R | g |
cos ( p 1 + | g | 2 ϕ ) = 1 2 R ( 1 + R 2 a 2 | g | 2 ) .
λ 1 = R ( D D 2 1 ) , λ 2 = R ( D + D 3 1 , D = p | g | 2 ( 1 + | g | ) 2 sin ( p 1 + | g | 2 ϕ ) cos ( p 1 + | g | 2 ϕ ) .
F ( W u ) = W u = F 1 ( W u ) , F ( W s ) = W s = F 1 ( W s ) ,
g n = Re i ϕ g n 1 ,
g n = 1 R e i ϕ g n 1

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