Analytical results for fundamental time-delayed feedback systems subjected to multiplicative noise

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 69, Issue 6 1, 2004, Pages

  Frank, T D ^a  

^a UNIVERSITY OF MÜNSTER   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

CORRELATION METHODS; DELAY CIRCUITS; ENGINEERING RESEARCH; FEEDBACK; MATHEMATICAL MODELS; PROBABILITY; PROBABILITY DENSITY FUNCTION; RANDOM PROCESSES; THERMAL NOISE;

AUTOCORRELATION; MULTIPLACATIVE NOISE; TIME-DELAYED FEEDBACK; VARIANCE EFFECTS;

STOCHASTIC CONTROL SYSTEMS;

EID: 42749103061     PISSN: 15393755     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.69.061104     Document Type: Review

References (54)

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2. R. Lang and K. Kobayashi, IEEE J. Quantum Electron. 16, 347 (1980); K. Ikeda, K. Kondo, and O. Akimoto, Phys. Rev. Lett. 49, 1467 (1982); D. Lenstra, Opt. Commun. 81, 209 (1991); E. Niebur, H. G. Schuster, and D. M. Kammen, Phys. Rev. Lett. 67, 2753 (1991); K. Pyragas, Phys. Lett. A 170, 421 (1992); I. Fischer, O. Hess, W. Elsäßer, and E. Göbel, Phys. Rev. Lett. 73, 2188 (1994); J. Foss, A. Longtin, B. Mensour, and J. Milton, ibid. 76, 708 (1996); M. Bestehorn, E. V. Grigorieva, H. Haken, and S. A. Kaschenko, Physica D 145, 110 (2000); C. W. Eurich and J. G. Milton, Phys. Rev. E 54, 6681 (1996); E. Villermaux, Phys. Rev. Lett. 75, 4618 (1995); N. Khrustova, G. Veser, and A. Mikhailov, ibid. 75, 3564 (1995); S. Boccaletti, D. Maza, H. Mancini, R. Genesio, and F. T. Arecchi, ibid. 79, 5246 (1997); M. K. S. Yeung and S. H. Strogatz, ibid. 82, 648 (1999); C. Simmendinger, A. Wunderlin, and A. Pelster, Phys. Rev. E 59, 5344 (1999); T. Ohira and Y. Sato, Phys. Rev. Lett. 82, 2811 (1999); S. Kim, S. H. Park, and H. Pyo, ibid. 82, 1620 (1999); B. F. Redmond, V. G. LeBlanc, and A. Longtin, Physica D 166, 131 (2002); C. Masoller, Phys. Rev. Lett. 88, 034102 (2002); M. Schanz and A. Pelster, Phys. Rev. E 67, 056205 (2003); D. Goldobin, M. Rosenblum, and A. Pikovsky, ibid. 67, 061119 (2003).
3. R. Lang and K. Kobayashi, IEEE J. Quantum Electron. 16, 347 (1980); K. Ikeda, K. Kondo, and O. Akimoto, Phys. Rev. Lett. 49, 1467 (1982); D. Lenstra, Opt. Commun. 81, 209 (1991); E. Niebur, H. G. Schuster, and D. M. Kammen, Phys. Rev. Lett. 67, 2753 (1991); K. Pyragas, Phys. Lett. A 170, 421 (1992); I. Fischer, O. Hess, W. Elsäßer, and E. Göbel, Phys. Rev. Lett. 73, 2188 (1994); J. Foss, A. Longtin, B. Mensour, and J. Milton, ibid. 76, 708 (1996); M. Bestehorn, E. V. Grigorieva, H. Haken, and S. A. Kaschenko, Physica D 145, 110 (2000); C. W. Eurich and J. G. Milton, Phys. Rev. E 54, 6681 (1996); E. Villermaux, Phys. Rev. Lett. 75, 4618 (1995); N. Khrustova, G. Veser, and A. Mikhailov, ibid. 75, 3564 (1995); S. Boccaletti, D. Maza, H. Mancini, R. Genesio, and F. T. Arecchi, ibid. 79, 5246 (1997); M. K. S. Yeung and S. H. Strogatz, ibid. 82, 648 (1999); C. Simmendinger, A. Wunderlin, and A. Pelster, Phys. Rev. E 59, 5344 (1999); T. Ohira and Y. Sato, Phys. Rev. Lett. 82, 2811 (1999); S. Kim, S. H. Park, and H. Pyo, ibid. 82, 1620 (1999); B. F. Redmond, V. G. LeBlanc, and A. Longtin, Physica D 166, 131 (2002); C. Masoller, Phys. Rev. Lett. 88, 034102 (2002); M. Schanz and A. Pelster, Phys. Rev. E 67, 056205 (2003); D. Goldobin, M. Rosenblum, and A. Pikovsky, ibid. 67, 061119 (2003).
4. R. Lang and K. Kobayashi, IEEE J. Quantum Electron. 16, 347 (1980); K. Ikeda, K. Kondo, and O. Akimoto, Phys. Rev. Lett. 49, 1467 (1982); D. Lenstra, Opt. Commun. 81, 209 (1991); E. Niebur, H. G. Schuster, and D. M. Kammen, Phys. Rev. Lett. 67, 2753 (1991); K. Pyragas, Phys. Lett. A 170, 421 (1992); I. Fischer, O. Hess, W. Elsäßer, and E. Göbel, Phys. Rev. Lett. 73, 2188 (1994); J. Foss, A. Longtin, B. Mensour, and J. Milton, ibid. 76, 708 (1996); M. Bestehorn, E. V. Grigorieva, H. Haken, and S. A. Kaschenko, Physica D 145, 110 (2000); C. W. Eurich and J. G. Milton, Phys. Rev. E 54, 6681 (1996); E. Villermaux, Phys. Rev. Lett. 75, 4618 (1995); N. Khrustova, G. Veser, and A. Mikhailov, ibid. 75, 3564 (1995); S. Boccaletti, D. Maza, H. Mancini, R. Genesio, and F. T. Arecchi, ibid. 79, 5246 (1997); M. K. S. Yeung and S. H. Strogatz, ibid. 82, 648 (1999); C. Simmendinger, A. Wunderlin, and A. Pelster, Phys. Rev. E 59, 5344 (1999); T. Ohira and Y. Sato, Phys. Rev. Lett. 82, 2811 (1999); S. Kim, S. H. Park, and H. Pyo, ibid. 82, 1620 (1999); B. F. Redmond, V. G. LeBlanc, and A. Longtin, Physica D 166, 131 (2002); C. Masoller, Phys. Rev. Lett. 88, 034102 (2002); M. Schanz and A. Pelster, Phys. Rev. E 67, 056205 (2003); D. Goldobin, M. Rosenblum, and A. Pikovsky, ibid. 67, 061119 (2003).
5. R. Lang and K. Kobayashi, IEEE J. Quantum Electron. 16, 347 (1980); K. Ikeda, K. Kondo, and O. Akimoto, Phys. Rev. Lett. 49, 1467 (1982); D. Lenstra, Opt. Commun. 81, 209 (1991); E. Niebur, H. G. Schuster, and D. M. Kammen, Phys. Rev. Lett. 67, 2753 (1991); K. Pyragas, Phys. Lett. A 170, 421 (1992); I. Fischer, O. Hess, W. Elsäßer, and E. Göbel, Phys. Rev. Lett. 73, 2188 (1994); J. Foss, A. Longtin, B. Mensour, and J. Milton, ibid. 76, 708 (1996); M. Bestehorn, E. V. Grigorieva, H. Haken, and S. A. Kaschenko, Physica D 145, 110 (2000); C. W. Eurich and J. G. Milton, Phys. Rev. E 54, 6681 (1996); E. Villermaux, Phys. Rev. Lett. 75, 4618 (1995); N. Khrustova, G. Veser, and A. Mikhailov, ibid. 75, 3564 (1995); S. Boccaletti, D. Maza, H. Mancini, R. Genesio, and F. T. Arecchi, ibid. 79, 5246 (1997); M. K. S. Yeung and S. H. Strogatz, ibid. 82, 648 (1999); C. Simmendinger, A. Wunderlin, and A. Pelster, Phys. Rev. E 59, 5344 (1999); T. Ohira and Y. Sato, Phys. Rev. Lett. 82, 2811 (1999); S. Kim, S. H. Park, and H. Pyo, ibid. 82, 1620 (1999); B. F. Redmond, V. G. LeBlanc, and A. Longtin, Physica D 166, 131 (2002); C. Masoller, Phys. Rev. Lett. 88, 034102 (2002); M. Schanz and A. Pelster, Phys. Rev. E 67, 056205 (2003); D. Goldobin, M. Rosenblum, and A. Pikovsky, ibid. 67, 061119 (2003).
6. R. Lang and K. Kobayashi, IEEE J. Quantum Electron. 16, 347 (1980); K. Ikeda, K. Kondo, and O. Akimoto, Phys. Rev. Lett. 49, 1467 (1982); D. Lenstra, Opt. Commun. 81, 209 (1991); E. Niebur, H. G. Schuster, and D. M. Kammen, Phys. Rev. Lett. 67, 2753 (1991); K. Pyragas, Phys. Lett. A 170, 421 (1992); I. Fischer, O. Hess, W. Elsäßer, and E. Göbel, Phys. Rev. Lett. 73, 2188 (1994); J. Foss, A. Longtin, B. Mensour, and J. Milton, ibid. 76, 708 (1996); M. Bestehorn, E. V. Grigorieva, H. Haken, and S. A. Kaschenko, Physica D 145, 110 (2000); C. W. Eurich and J. G. Milton, Phys. Rev. E 54, 6681 (1996); E. Villermaux, Phys. Rev. Lett. 75, 4618 (1995); N. Khrustova, G. Veser, and A. Mikhailov, ibid. 75, 3564 (1995); S. Boccaletti, D. Maza, H. Mancini, R. Genesio, and F. T. Arecchi, ibid. 79, 5246 (1997); M. K. S. Yeung and S. H. Strogatz, ibid. 82, 648 (1999); C. Simmendinger, A. Wunderlin, and A. Pelster, Phys. Rev. E 59, 5344 (1999); T. Ohira and Y. Sato, Phys. Rev. Lett. 82, 2811 (1999); S. Kim, S. H. Park, and H. Pyo, ibid. 82, 1620 (1999); B. F. Redmond, V. G. LeBlanc, and A. Longtin, Physica D 166, 131 (2002); C. Masoller, Phys. Rev. Lett. 88, 034102 (2002); M. Schanz and A. Pelster, Phys. Rev. E 67, 056205 (2003); D. Goldobin, M. Rosenblum, and A. Pikovsky, ibid. 67, 061119 (2003).
7. R. Lang and K. Kobayashi, IEEE J. Quantum Electron. 16, 347 (1980); K. Ikeda, K. Kondo, and O. Akimoto, Phys. Rev. Lett. 49, 1467 (1982); D. Lenstra, Opt. Commun. 81, 209 (1991); E. Niebur, H. G. Schuster, and D. M. Kammen, Phys. Rev. Lett. 67, 2753 (1991); K. Pyragas, Phys. Lett. A 170, 421 (1992); I. Fischer, O. Hess, W. Elsäßer, and E. Göbel, Phys. Rev. Lett. 73, 2188 (1994); J. Foss, A. Longtin, B. Mensour, and J. Milton, ibid. 76, 708 (1996); M. Bestehorn, E. V. Grigorieva, H. Haken, and S. A. Kaschenko, Physica D 145, 110 (2000); C. W. Eurich and J. G. Milton, Phys. Rev. E 54, 6681 (1996); E. Villermaux, Phys. Rev. Lett. 75, 4618 (1995); N. Khrustova, G. Veser, and A. Mikhailov, ibid. 75, 3564 (1995); S. Boccaletti, D. Maza, H. Mancini, R. Genesio, and F. T. Arecchi, ibid. 79, 5246 (1997); M. K. S. Yeung and S. H. Strogatz, ibid. 82, 648 (1999); C. Simmendinger, A. Wunderlin, and A. Pelster, Phys. Rev. E 59, 5344 (1999); T. Ohira and Y. Sato, Phys. Rev. Lett. 82, 2811 (1999); S. Kim, S. H. Park, and H. Pyo, ibid. 82, 1620 (1999); B. F. Redmond, V. G. LeBlanc, and A. Longtin, Physica D 166, 131 (2002); C. Masoller, Phys. Rev. Lett. 88, 034102 (2002); M. Schanz and A. Pelster, Phys. Rev. E 67, 056205 (2003); D. Goldobin, M. Rosenblum, and A. Pikovsky, ibid. 67, 061119 (2003).
8. R. Lang and K. Kobayashi, IEEE J. Quantum Electron. 16, 347 (1980); K. Ikeda, K. Kondo, and O. Akimoto, Phys. Rev. Lett. 49, 1467 (1982); D. Lenstra, Opt. Commun. 81, 209 (1991); E. Niebur, H. G. Schuster, and D. M. Kammen, Phys. Rev. Lett. 67, 2753 (1991); K. Pyragas, Phys. Lett. A 170, 421 (1992); I. Fischer, O. Hess, W. Elsäßer, and E. Göbel, Phys. Rev. Lett. 73, 2188 (1994); J. Foss, A. Longtin, B. Mensour, and J. Milton, ibid. 76, 708 (1996); M. Bestehorn, E. V. Grigorieva, H. Haken, and S. A. Kaschenko, Physica D 145, 110 (2000); C. W. Eurich and J. G. Milton, Phys. Rev. E 54, 6681 (1996); E. Villermaux, Phys. Rev. Lett. 75, 4618 (1995); N. Khrustova, G. Veser, and A. Mikhailov, ibid. 75, 3564 (1995); S. Boccaletti, D. Maza, H. Mancini, R. Genesio, and F. T. Arecchi, ibid. 79, 5246 (1997); M. K. S. Yeung and S. H. Strogatz, ibid. 82, 648 (1999); C. Simmendinger, A. Wunderlin, and A. Pelster, Phys. Rev. E 59, 5344 (1999); T. Ohira and Y. Sato, Phys. Rev. Lett. 82, 2811 (1999); S. Kim, S. H. Park, and H. Pyo, ibid. 82, 1620 (1999); B. F. Redmond, V. G. LeBlanc, and A. Longtin, Physica D 166, 131 (2002); C. Masoller, Phys. Rev. Lett. 88, 034102 (2002); M. Schanz and A. Pelster, Phys. Rev. E 67, 056205 (2003); D. Goldobin, M. Rosenblum, and A. Pikovsky, ibid. 67, 061119 (2003).
9. R. Lang and K. Kobayashi, IEEE J. Quantum Electron. 16, 347 (1980); K. Ikeda, K. Kondo, and O. Akimoto, Phys. Rev. Lett. 49, 1467 (1982); D. Lenstra, Opt. Commun. 81, 209 (1991); E. Niebur, H. G. Schuster, and D. M. Kammen, Phys. Rev. Lett. 67, 2753 (1991); K. Pyragas, Phys. Lett. A 170, 421 (1992); I. Fischer, O. Hess, W. Elsäßer, and E. Göbel, Phys. Rev. Lett. 73, 2188 (1994); J. Foss, A. Longtin, B. Mensour, and J. Milton, ibid. 76, 708 (1996); M. Bestehorn, E. V. Grigorieva, H. Haken, and S. A. Kaschenko, Physica D 145, 110 (2000); C. W. Eurich and J. G. Milton, Phys. Rev. E 54, 6681 (1996); E. Villermaux, Phys. Rev. Lett. 75, 4618 (1995); N. Khrustova, G. Veser, and A. Mikhailov, ibid. 75, 3564 (1995); S. Boccaletti, D. Maza, H. Mancini, R. Genesio, and F. T. Arecchi, ibid. 79, 5246 (1997); M. K. S. Yeung and S. H. Strogatz, ibid. 82, 648 (1999); C. Simmendinger, A. Wunderlin, and A. Pelster, Phys. Rev. E 59, 5344 (1999); T. Ohira and Y. Sato, Phys. Rev. Lett. 82, 2811 (1999); S. Kim, S. H. Park, and H. Pyo, ibid. 82, 1620 (1999); B. F. Redmond, V. G. LeBlanc, and A. Longtin, Physica D 166, 131 (2002); C. Masoller, Phys. Rev. Lett. 88, 034102 (2002); M. Schanz and A. Pelster, Phys. Rev. E 67, 056205 (2003); D. Goldobin, M. Rosenblum, and A. Pikovsky, ibid. 67, 061119 (2003).
10. R. Lang and K. Kobayashi, IEEE J. Quantum Electron. 16, 347 (1980); K. Ikeda, K. Kondo, and O. Akimoto, Phys. Rev. Lett. 49, 1467 (1982); D. Lenstra, Opt. Commun. 81, 209 (1991); E. Niebur, H. G. Schuster, and D. M. Kammen, Phys. Rev. Lett. 67, 2753 (1991); K. Pyragas, Phys. Lett. A 170, 421 (1992); I. Fischer, O. Hess, W. Elsäßer, and E. Göbel, Phys. Rev. Lett. 73, 2188 (1994); J. Foss, A. Longtin, B. Mensour, and J. Milton, ibid. 76, 708 (1996); M. Bestehorn, E. V. Grigorieva, H. Haken, and S. A. Kaschenko, Physica D 145, 110 (2000); C. W. Eurich and J. G. Milton, Phys. Rev. E 54, 6681 (1996); E. Villermaux, Phys. Rev. Lett. 75, 4618 (1995); N. Khrustova, G. Veser, and A. Mikhailov, ibid. 75, 3564 (1995); S. Boccaletti, D. Maza, H. Mancini, R. Genesio, and F. T. Arecchi, ibid. 79, 5246 (1997); M. K. S. Yeung and S. H. Strogatz, ibid. 82, 648 (1999); C. Simmendinger, A. Wunderlin, and A. Pelster, Phys. Rev. E 59, 5344 (1999); T. Ohira and Y. Sato, Phys. Rev. Lett. 82, 2811 (1999); S. Kim, S. H. Park, and H. Pyo, ibid. 82, 1620 (1999); B. F. Redmond, V. G. LeBlanc, and A. Longtin, Physica D 166, 131 (2002); C. Masoller, Phys. Rev. Lett. 88, 034102 (2002); M. Schanz and A. Pelster, Phys. Rev. E 67, 056205 (2003); D. Goldobin, M. Rosenblum, and A. Pikovsky, ibid. 67, 061119 (2003).
11. R. Lang and K. Kobayashi, IEEE J. Quantum Electron. 16, 347 (1980); K. Ikeda, K. Kondo, and O. Akimoto, Phys. Rev. Lett. 49, 1467 (1982); D. Lenstra, Opt. Commun. 81, 209 (1991); E. Niebur, H. G. Schuster, and D. M. Kammen, Phys. Rev. Lett. 67, 2753 (1991); K. Pyragas, Phys. Lett. A 170, 421 (1992); I. Fischer, O. Hess, W. Elsäßer, and E. Göbel, Phys. Rev. Lett. 73, 2188 (1994); J. Foss, A. Longtin, B. Mensour, and J. Milton, ibid. 76, 708 (1996); M. Bestehorn, E. V. Grigorieva, H. Haken, and S. A. Kaschenko, Physica D 145, 110 (2000); C. W. Eurich and J. G. Milton, Phys. Rev. E 54, 6681 (1996); E. Villermaux, Phys. Rev. Lett. 75, 4618 (1995); N. Khrustova, G. Veser, and A. Mikhailov, ibid. 75, 3564 (1995); S. Boccaletti, D. Maza, H. Mancini, R. Genesio, and F. T. Arecchi, ibid. 79, 5246 (1997); M. K. S. Yeung and S. H. Strogatz, ibid. 82, 648 (1999); C. Simmendinger, A. Wunderlin, and A. Pelster, Phys. Rev. E 59, 5344 (1999); T. Ohira and Y. Sato, Phys. Rev. Lett. 82, 2811 (1999); S. Kim, S. H. Park, and H. Pyo, ibid. 82, 1620 (1999); B. F. Redmond, V. G. LeBlanc, and A. Longtin, Physica D 166, 131 (2002); C. Masoller, Phys. Rev. Lett. 88, 034102 (2002); M. Schanz and A. Pelster, Phys. Rev. E 67, 056205 (2003); D. Goldobin, M. Rosenblum, and A. Pikovsky, ibid. 67, 061119 (2003).
12. R. Lang and K. Kobayashi, IEEE J. Quantum Electron. 16, 347 (1980); K. Ikeda, K. Kondo, and O. Akimoto, Phys. Rev. Lett. 49, 1467 (1982); D. Lenstra, Opt. Commun. 81, 209 (1991); E. Niebur, H. G. Schuster, and D. M. Kammen, Phys. Rev. Lett. 67, 2753 (1991); K. Pyragas, Phys. Lett. A 170, 421 (1992); I. Fischer, O. Hess, W. Elsäßer, and E. Göbel, Phys. Rev. Lett. 73, 2188 (1994); J. Foss, A. Longtin, B. Mensour, and J. Milton, ibid. 76, 708 (1996); M. Bestehorn, E. V. Grigorieva, H. Haken, and S. A. Kaschenko, Physica D 145, 110 (2000); C. W. Eurich and J. G. Milton, Phys. Rev. E 54, 6681 (1996); E. Villermaux, Phys. Rev. Lett. 75, 4618 (1995); N. Khrustova, G. Veser, and A. Mikhailov, ibid. 75, 3564 (1995); S. Boccaletti, D. Maza, H. Mancini, R. Genesio, and F. T. Arecchi, ibid. 79, 5246 (1997); M. K. S. Yeung and S. H. Strogatz, ibid. 82, 648 (1999); C. Simmendinger, A. Wunderlin, and A. Pelster, Phys. Rev. E 59, 5344 (1999); T. Ohira and Y. Sato, Phys. Rev. Lett. 82, 2811 (1999); S. Kim, S. H. Park, and H. Pyo, ibid. 82, 1620 (1999); B. F. Redmond, V. G. LeBlanc, and A. Longtin, Physica D 166, 131 (2002); C. Masoller, Phys. Rev. Lett. 88, 034102 (2002); M. Schanz and A. Pelster, Phys. Rev. E 67, 056205 (2003); D. Goldobin, M. Rosenblum, and A. Pikovsky, ibid. 67, 061119 (2003).
13. R. Lang and K. Kobayashi, IEEE J. Quantum Electron. 16, 347 (1980); K. Ikeda, K. Kondo, and O. Akimoto, Phys. Rev. Lett. 49, 1467 (1982); D. Lenstra, Opt. Commun. 81, 209 (1991); E. Niebur, H. G. Schuster, and D. M. Kammen, Phys. Rev. Lett. 67, 2753 (1991); K. Pyragas, Phys. Lett. A 170, 421 (1992); I. Fischer, O. Hess, W. Elsäßer, and E. Göbel, Phys. Rev. Lett. 73, 2188 (1994); J. Foss, A. Longtin, B. Mensour, and J. Milton, ibid. 76, 708 (1996); M. Bestehorn, E. V. Grigorieva, H. Haken, and S. A. Kaschenko, Physica D 145, 110 (2000); C. W. Eurich and J. G. Milton, Phys. Rev. E 54, 6681 (1996); E. Villermaux, Phys. Rev. Lett. 75, 4618 (1995); N. Khrustova, G. Veser, and A. Mikhailov, ibid. 75, 3564 (1995); S. Boccaletti, D. Maza, H. Mancini, R. Genesio, and F. T. Arecchi, ibid. 79, 5246 (1997); M. K. S. Yeung and S. H. Strogatz, ibid. 82, 648 (1999); C. Simmendinger, A. Wunderlin, and A. Pelster, Phys. Rev. E 59, 5344 (1999); T. Ohira and Y. Sato, Phys. Rev. Lett. 82, 2811 (1999); S. Kim, S. H. Park, and H. Pyo, ibid. 82, 1620 (1999); B. F. Redmond, V. G. LeBlanc, and A. Longtin, Physica D 166, 131 (2002); C. Masoller, Phys. Rev. Lett. 88, 034102 (2002); M. Schanz and A. Pelster, Phys. Rev. E 67, 056205 (2003); D. Goldobin, M. Rosenblum, and A. Pikovsky, ibid. 67, 061119 (2003).
14. R. Lang and K. Kobayashi, IEEE J. Quantum Electron. 16, 347 (1980); K. Ikeda, K. Kondo, and O. Akimoto, Phys. Rev. Lett. 49, 1467 (1982); D. Lenstra, Opt. Commun. 81, 209 (1991); E. Niebur, H. G. Schuster, and D. M. Kammen, Phys. Rev. Lett. 67, 2753 (1991); K. Pyragas, Phys. Lett. A 170, 421 (1992); I. Fischer, O. Hess, W. Elsäßer, and E. Göbel, Phys. Rev. Lett. 73, 2188 (1994); J. Foss, A. Longtin, B. Mensour, and J. Milton, ibid. 76, 708 (1996); M. Bestehorn, E. V. Grigorieva, H. Haken, and S. A. Kaschenko, Physica D 145, 110 (2000); C. W. Eurich and J. G. Milton, Phys. Rev. E 54, 6681 (1996); E. Villermaux, Phys. Rev. Lett. 75, 4618 (1995); N. Khrustova, G. Veser, and A. Mikhailov, ibid. 75, 3564 (1995); S. Boccaletti, D. Maza, H. Mancini, R. Genesio, and F. T. Arecchi, ibid. 79, 5246 (1997); M. K. S. Yeung and S. H. Strogatz, ibid. 82, 648 (1999); C. Simmendinger, A. Wunderlin, and A. Pelster, Phys. Rev. E 59, 5344 (1999); T. Ohira and Y. Sato, Phys. Rev. Lett. 82, 2811 (1999); S. Kim, S. H. Park, and H. Pyo, ibid. 82, 1620 (1999); B. F. Redmond, V. G. LeBlanc, and A. Longtin, Physica D 166, 131 (2002); C. Masoller, Phys. Rev. Lett. 88, 034102 (2002); M. Schanz and A. Pelster, Phys. Rev. E 67, 056205 (2003); D. Goldobin, M. Rosenblum, and A. Pikovsky, ibid. 67, 061119 (2003).
15. R. Lang and K. Kobayashi, IEEE J. Quantum Electron. 16, 347 (1980); K. Ikeda, K. Kondo, and O. Akimoto, Phys. Rev. Lett. 49, 1467 (1982); D. Lenstra, Opt. Commun. 81, 209 (1991); E. Niebur, H. G. Schuster, and D. M. Kammen, Phys. Rev. Lett. 67, 2753 (1991); K. Pyragas, Phys. Lett. A 170, 421 (1992); I. Fischer, O. Hess, W. Elsäßer, and E. Göbel, Phys. Rev. Lett. 73, 2188 (1994); J. Foss, A. Longtin, B. Mensour, and J. Milton, ibid. 76, 708 (1996); M. Bestehorn, E. V. Grigorieva, H. Haken, and S. A. Kaschenko, Physica D 145, 110 (2000); C. W. Eurich and J. G. Milton, Phys. Rev. E 54, 6681 (1996); E. Villermaux, Phys. Rev. Lett. 75, 4618 (1995); N. Khrustova, G. Veser, and A. Mikhailov, ibid. 75, 3564 (1995); S. Boccaletti, D. Maza, H. Mancini, R. Genesio, and F. T. Arecchi, ibid. 79, 5246 (1997); M. K. S. Yeung and S. H. Strogatz, ibid. 82, 648 (1999); C. Simmendinger, A. Wunderlin, and A. Pelster, Phys. Rev. E 59, 5344 (1999); T. Ohira and Y. Sato, Phys. Rev. Lett. 82, 2811 (1999); S. Kim, S. H. Park, and H. Pyo, ibid. 82, 1620 (1999); B. F. Redmond, V. G. LeBlanc, and A. Longtin, Physica D 166, 131 (2002); C. Masoller, Phys. Rev. Lett. 88, 034102 (2002); M. Schanz and A. Pelster, Phys. Rev. E 67, 056205 (2003); D. Goldobin, M. Rosenblum, and A. Pikovsky, ibid. 67, 061119 (2003).
16. R. Lang and K. Kobayashi, IEEE J. Quantum Electron. 16, 347 (1980); K. Ikeda, K. Kondo, and O. Akimoto, Phys. Rev. Lett. 49, 1467 (1982); D. Lenstra, Opt. Commun. 81, 209 (1991); E. Niebur, H. G. Schuster, and D. M. Kammen, Phys. Rev. Lett. 67, 2753 (1991); K. Pyragas, Phys. Lett. A 170, 421 (1992); I. Fischer, O. Hess, W. Elsäßer, and E. Göbel, Phys. Rev. Lett. 73, 2188 (1994); J. Foss, A. Longtin, B. Mensour, and J. Milton, ibid. 76, 708 (1996); M. Bestehorn, E. V. Grigorieva, H. Haken, and S. A. Kaschenko, Physica D 145, 110 (2000); C. W. Eurich and J. G. Milton, Phys. Rev. E 54, 6681 (1996); E. Villermaux, Phys. Rev. Lett. 75, 4618 (1995); N. Khrustova, G. Veser, and A. Mikhailov, ibid. 75, 3564 (1995); S. Boccaletti, D. Maza, H. Mancini, R. Genesio, and F. T. Arecchi, ibid. 79, 5246 (1997); M. K. S. Yeung and S. H. Strogatz, ibid. 82, 648 (1999); C. Simmendinger, A. Wunderlin, and A. Pelster, Phys. Rev. E 59, 5344 (1999); T. Ohira and Y. Sato, Phys. Rev. Lett. 82, 2811 (1999); S. Kim, S. H. Park, and H. Pyo, ibid. 82, 1620 (1999); B. F. Redmond, V. G. LeBlanc, and A. Longtin, Physica D 166, 131 (2002); C. Masoller, Phys. Rev. Lett. 88, 034102 (2002); M. Schanz and A. Pelster, Phys. Rev. E 67, 056205 (2003); D. Goldobin, M. Rosenblum, and A. Pikovsky, ibid. 67, 061119 (2003).
17. R. Lang and K. Kobayashi, IEEE J. Quantum Electron. 16, 347 (1980); K. Ikeda, K. Kondo, and O. Akimoto, Phys. Rev. Lett. 49, 1467 (1982); D. Lenstra, Opt. Commun. 81, 209 (1991); E. Niebur, H. G. Schuster, and D. M. Kammen, Phys. Rev. Lett. 67, 2753 (1991); K. Pyragas, Phys. Lett. A 170, 421 (1992); I. Fischer, O. Hess, W. Elsäßer, and E. Göbel, Phys. Rev. Lett. 73, 2188 (1994); J. Foss, A. Longtin, B. Mensour, and J. Milton, ibid. 76, 708 (1996); M. Bestehorn, E. V. Grigorieva, H. Haken, and S. A. Kaschenko, Physica D 145, 110 (2000); C. W. Eurich and J. G. Milton, Phys. Rev. E 54, 6681 (1996); E. Villermaux, Phys. Rev. Lett. 75, 4618 (1995); N. Khrustova, G. Veser, and A. Mikhailov, ibid. 75, 3564 (1995); S. Boccaletti, D. Maza, H. Mancini, R. Genesio, and F. T. Arecchi, ibid. 79, 5246 (1997); M. K. S. Yeung and S. H. Strogatz, ibid. 82, 648 (1999); C. Simmendinger, A. Wunderlin, and A. Pelster, Phys. Rev. E 59, 5344 (1999); T. Ohira and Y. Sato, Phys. Rev. Lett. 82, 2811 (1999); S. Kim, S. H. Park, and H. Pyo, ibid. 82, 1620 (1999); B. F. Redmond, V. G. LeBlanc, and A. Longtin, Physica D 166, 131 (2002); C. Masoller, Phys. Rev. Lett. 88, 034102 (2002); M. Schanz and A. Pelster, Phys. Rev. E 67, 056205 (2003); D. Goldobin, M. Rosenblum, and A. Pikovsky, ibid. 67, 061119 (2003).
18. R. Lang and K. Kobayashi, IEEE J. Quantum Electron. 16, 347 (1980); K. Ikeda, K. Kondo, and O. Akimoto, Phys. Rev. Lett. 49, 1467 (1982); D. Lenstra, Opt. Commun. 81, 209 (1991); E. Niebur, H. G. Schuster, and D. M. Kammen, Phys. Rev. Lett. 67, 2753 (1991); K. Pyragas, Phys. Lett. A 170, 421 (1992); I. Fischer, O. Hess, W. Elsäßer, and E. Göbel, Phys. Rev. Lett. 73, 2188 (1994); J. Foss, A. Longtin, B. Mensour, and J. Milton, ibid. 76, 708 (1996); M. Bestehorn, E. V. Grigorieva, H. Haken, and S. A. Kaschenko, Physica D 145, 110 (2000); C. W. Eurich and J. G. Milton, Phys. Rev. E 54, 6681 (1996); E. Villermaux, Phys. Rev. Lett. 75, 4618 (1995); N. Khrustova, G. Veser, and A. Mikhailov, ibid. 75, 3564 (1995); S. Boccaletti, D. Maza, H. Mancini, R. Genesio, and F. T. Arecchi, ibid. 79, 5246 (1997); M. K. S. Yeung and S. H. Strogatz, ibid. 82, 648 (1999); C. Simmendinger, A. Wunderlin, and A. Pelster, Phys. Rev. E 59, 5344 (1999); T. Ohira and Y. Sato, Phys. Rev. Lett. 82, 2811 (1999); S. Kim, S. H. Park, and H. Pyo, ibid. 82, 1620 (1999); B. F. Redmond, V. G. LeBlanc, and A. Longtin, Physica D 166, 131 (2002); C. Masoller, Phys. Rev. Lett. 88, 034102 (2002); M. Schanz and A. Pelster, Phys. Rev. E 67, 056205 (2003); D. Goldobin, M. Rosenblum, and A. Pikovsky, ibid. 67, 061119 (2003).
19. R. Lang and K. Kobayashi, IEEE J. Quantum Electron. 16, 347 (1980); K. Ikeda, K. Kondo, and O. Akimoto, Phys. Rev. Lett. 49, 1467 (1982); D. Lenstra, Opt. Commun. 81, 209 (1991); E. Niebur, H. G. Schuster, and D. M. Kammen, Phys. Rev. Lett. 67, 2753 (1991); K. Pyragas, Phys. Lett. A 170, 421 (1992); I. Fischer, O. Hess, W. Elsäßer, and E. Göbel, Phys. Rev. Lett. 73, 2188 (1994); J. Foss, A. Longtin, B. Mensour, and J. Milton, ibid. 76, 708 (1996); M. Bestehorn, E. V. Grigorieva, H. Haken, and S. A. Kaschenko, Physica D 145, 110 (2000); C. W. Eurich and J. G. Milton, Phys. Rev. E 54, 6681 (1996); E. Villermaux, Phys. Rev. Lett. 75, 4618 (1995); N. Khrustova, G. Veser, and A. Mikhailov, ibid. 75, 3564 (1995); S. Boccaletti, D. Maza, H. Mancini, R. Genesio, and F. T. Arecchi, ibid. 79, 5246 (1997); M. K. S. Yeung and S. H. Strogatz, ibid. 82, 648 (1999); C. Simmendinger, A. Wunderlin, and A. Pelster, Phys. Rev. E 59, 5344 (1999); T. Ohira and Y. Sato, Phys. Rev. Lett. 82, 2811 (1999); S. Kim, S. H. Park, and H. Pyo, ibid. 82, 1620 (1999); B. F. Redmond, V. G. LeBlanc, and A. Longtin, Physica D 166, 131 (2002); C. Masoller, Phys. Rev. Lett. 88, 034102 (2002); M. Schanz and A. Pelster, Phys. Rev. E 67, 056205 (2003); D. Goldobin, M. Rosenblum, and A. Pikovsky, ibid. 67, 061119 (2003).
20. R. Lang and K. Kobayashi, IEEE J. Quantum Electron. 16, 347 (1980); K. Ikeda, K. Kondo, and O. Akimoto, Phys. Rev. Lett. 49, 1467 (1982); D. Lenstra, Opt. Commun. 81, 209 (1991); E. Niebur, H. G. Schuster, and D. M. Kammen, Phys. Rev. Lett. 67, 2753 (1991); K. Pyragas, Phys. Lett. A 170, 421 (1992); I. Fischer, O. Hess, W. Elsäßer, and E. Göbel, Phys. Rev. Lett. 73, 2188 (1994); J. Foss, A. Longtin, B. Mensour, and J. Milton, ibid. 76, 708 (1996); M. Bestehorn, E. V. Grigorieva, H. Haken, and S. A. Kaschenko, Physica D 145, 110 (2000); C. W. Eurich and J. G. Milton, Phys. Rev. E 54, 6681 (1996); E. Villermaux, Phys. Rev. Lett. 75, 4618 (1995); N. Khrustova, G. Veser, and A. Mikhailov, ibid. 75, 3564 (1995); S. Boccaletti, D. Maza, H. Mancini, R. Genesio, and F. T. Arecchi, ibid. 79, 5246 (1997); M. K. S. Yeung and S. H. Strogatz, ibid. 82, 648 (1999); C. Simmendinger, A. Wunderlin, and A. Pelster, Phys. Rev. E 59, 5344 (1999); T. Ohira and Y. Sato, Phys. Rev. Lett. 82, 2811 (1999); S. Kim, S. H. Park, and H. Pyo, ibid. 82, 1620 (1999); B. F. Redmond, V. G. LeBlanc, and A. Longtin, Physica D 166, 131 (2002); C. Masoller, Phys. Rev. Lett. 88, 034102 (2002); M. Schanz and A. Pelster, Phys. Rev. E 67, 056205 (2003); D. Goldobin, M. Rosenblum, and A. Pikovsky, ibid. 67, 061119 (2003).
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53. Proof: if B(2)>b(1) then ã(2)<ã(1) and b̃(2)>b̃(1) which implies that (-ã/b̃)(2)>(-ã/b̃)(1). Since cos(x) decreases monotonically in [0, π], we have arccos(-ã/b̃)(2)φ(1) for B(2)>B(1). In sum, we have τ*(2)<τ*(1) for B(2)>B(1).
54. Proof: if B(2)>B(1) then ã(2)<ã(1) and b̃(2) > b̃(l) which implies (ã/b̃)(2) < (ã/b̃)(1). Since sinh(x) is a monotonically increasing function, we have arcsinh(ã/b̃)(2) < arcsinh(ã/b̃)(1). Furthermore, we have φ<0 and φ(2) > φ(1) for B(2) > B(1). In sum, we have τ*(2) < τ*(1) for B(2)>B(1).