General stability analysis of synchronized dynamics in coupled systems

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

Volumn 67, Issue 2 2, 2003, Pages 262091-262094

  Chen, Yonghong ^a,c   Rangarajan, Govindan ^b   Ding, Mingzhou ^c  

^a XI'AN JIAOTONG UNIVERSITY   (China)

^b INDIAN INSTITUTE OF SCIENCE   (India)

^c FLORIDA ATLANTIC UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

DYNAMICS; EIGENVALUES AND EIGENFUNCTIONS; LYAPUNOV METHODS; ORDINARY DIFFERENTIAL EQUATIONS; SYNCHRONIZATION;

SYNCHRONIZED DYNAMICS;

SYSTEM STABILITY;

EID: 0038396314     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (43)

1. R.R. Klevecz, J. Bolen, and O. Duran, Int. J. Bifurcation Chaos Appl. Sci. Eng. 2, 941 (1992); D. Hansel and H. Sompolinsky, Phys. Rev. Lett. 68, 718 (1992); D. Hansel, Int. J. Neural Syst. 7, 403 (1996); F. Pasemann, Physica D 128, 236 (1999).
2. R.R. Klevecz, J. Bolen, and O. Duran, Int. J. Bifurcation Chaos Appl. Sci. Eng. 2, 941 (1992); D. Hansel and H. Sompolinsky, Phys. Rev. Lett. 68, 718 (1992); D. Hansel, Int. J. Neural Syst. 7, 403 (1996); F. Pasemann, Physica D 128, 236 (1999).
3. R.R. Klevecz, J. Bolen, and O. Duran, Int. J. Bifurcation Chaos Appl. Sci. Eng. 2, 941 (1992); D. Hansel and H. Sompolinsky, Phys. Rev. Lett. 68, 718 (1992); D. Hansel, Int. J. Neural Syst. 7, 403 (1996); F. Pasemann, Physica D 128, 236 (1999).
4. H.G. Winful and L. Rahman, Phys. Rev. Lett. 65, 1575 (1990); R. Li and T. Erneux, Opt. Commun. 99, 196 (1993); Phys. Rev. A 49, 1301 (1994); R. Roy and K.S. Thornburg, Jr., Phys. Rev. Lett. 72, 2009 (1994); K. Otsuka, R. Kawai, S. Hwong, J. Ko, and J. Chern, ibid. 84, 3049 (2000).
5. C.W. Wu and L.O. Chua, IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. 42, 430 (1995); S. Jankowski, A. Londei, C. Mazur, and A. Lozowski, Int. J. Electron. 79, 823 (1995); G. Filatrella, B. Straughn, and P. Barbara, J. Appl. Phys. 90, 5675 (2001).
6. C.W. Wu and L.O. Chua, IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. 42, 430 (1995); S. Jankowski, A. Londei, C. Mazur, and A. Lozowski, Int. J. Electron. 79, 823 (1995); G. Filatrella, B. Straughn, and P. Barbara, J. Appl. Phys. 90, 5675 (2001).
7. C.W. Wu and L.O. Chua, IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. 42, 430 (1995); S. Jankowski, A. Londei, C. Mazur, and A. Lozowski, Int. J. Electron. 79, 823 (1995); G. Filatrella, B. Straughn, and P. Barbara, J. Appl. Phys. 90, 5675 (2001).
8. C.W. Wu and L.O. Chua, IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. 42, 430 (1995); S. Jankowski, A. Londei, C. Mazur, and A. Lozowski, Int. J. Electron. 79, 823 (1995); G. Filatrella, B. Straughn, and P. Barbara, J. Appl. Phys. 90, 5675 (2001).
9. C.W. Wu and L.O. Chua, IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. 42, 430 (1995); S. Jankowski, A. Londei, C. Mazur, and A. Lozowski, Int. J. Electron. 79, 823 (1995); G. Filatrella, B. Straughn, and P. Barbara, J. Appl. Phys. 90, 5675 (2001).
10. H. Fujisaka and T. Yamada, Prog. Theor. Phys. 69, 32 (1983); I. Waller and R. Kapral, Phys. Lett. 105A, 163 (1984); V.S. Afraimovich et al., Sov. Radiophys. 29, 795 (1986); G.L. Oppo and R. Kapral, Phys. Rev. A 36, 5820 (1987); L.A. Bunimovich and Ya.G. Sinai, Nonlinearity 1, 491 (1988); J.F. Heagy, L.M. Pecora, and T.L. Carroll, Phys. Rev. E 50, 1874 (1994); L. Kocarev and U. Parlitz, Phys. Rev. Lett. 76, 1816 (1996); R. Brown and N.F. Rulkov, ibid. 78, 4189 (1997); N.F. Rulkov and M.M. Sushchik, Int. J. Bifurcation Chaos Appl. Sci. Eng. 7, 625 (1997); P.C. Bresloff, Phys. Rev. E 60, 2160 (1999); P. Glendinning, Phys. Lett. A 259, 129 (1999); V.N. Belykh, I.V. Belykh, and M, Hasler, Phys. Rev. E 62, 6332 (2000); V.N. Belykh, I.V. Belykh, and E. Mosekilde, ibid. 63, 036216 (2001).
11. H. Fujisaka and T. Yamada, Prog. Theor. Phys. 69, 32 (1983); I. Waller and R. Kapral, Phys. Lett. 105A, 163 (1984); V.S. Afraimovich et al., Sov. Radiophys. 29, 795 (1986); G.L. Oppo and R. Kapral, Phys. Rev. A 36, 5820 (1987); L.A. Bunimovich and Ya.G. Sinai, Nonlinearity 1, 491 (1988); J.F. Heagy, L.M. Pecora, and T.L. Carroll, Phys. Rev. E 50, 1874 (1994); L. Kocarev and U. Parlitz, Phys. Rev. Lett. 76, 1816 (1996); R. Brown and N.F. Rulkov, ibid. 78, 4189 (1997); N.F. Rulkov and M.M. Sushchik, Int. J. Bifurcation Chaos Appl. Sci. Eng. 7, 625 (1997); P.C. Bresloff, Phys. Rev. E 60, 2160 (1999); P. Glendinning, Phys. Lett. A 259, 129 (1999); V.N. Belykh, I.V. Belykh, and M, Hasler, Phys. Rev. E 62, 6332 (2000); V.N. Belykh, I.V. Belykh, and E. Mosekilde, ibid. 63, 036216 (2001).
12. H. Fujisaka and T. Yamada, Prog. Theor. Phys. 69, 32 (1983); I. Waller and R. Kapral, Phys. Lett. 105A, 163 (1984); V.S. Afraimovich et al., Sov. Radiophys. 29, 795 (1986); G.L. Oppo and R. Kapral, Phys. Rev. A 36, 5820 (1987); L.A. Bunimovich and Ya.G. Sinai, Nonlinearity 1, 491 (1988); J.F. Heagy, L.M. Pecora, and T.L. Carroll, Phys. Rev. E 50, 1874 (1994); L. Kocarev and U. Parlitz, Phys. Rev. Lett. 76, 1816 (1996); R. Brown and N.F. Rulkov, ibid. 78, 4189 (1997); N.F. Rulkov and M.M. Sushchik, Int. J. Bifurcation Chaos Appl. Sci. Eng. 7, 625 (1997); P.C. Bresloff, Phys. Rev. E 60, 2160 (1999); P. Glendinning, Phys. Lett. A 259, 129 (1999); V.N. Belykh, I.V. Belykh, and M, Hasler, Phys. Rev. E 62, 6332 (2000); V.N. Belykh, I.V. Belykh, and E. Mosekilde, ibid. 63, 036216 (2001).
13. H. Fujisaka and T. Yamada, Prog. Theor. Phys. 69, 32 (1983); I. Waller and R. Kapral, Phys. Lett. 105A, 163 (1984); V.S. Afraimovich et al., Sov. Radiophys. 29, 795 (1986); G.L. Oppo and R. Kapral, Phys. Rev. A 36, 5820 (1987); L.A. Bunimovich and Ya.G. Sinai, Nonlinearity 1, 491 (1988); J.F. Heagy, L.M. Pecora, and T.L. Carroll, Phys. Rev. E 50, 1874 (1994); L. Kocarev and U. Parlitz, Phys. Rev. Lett. 76, 1816 (1996); R. Brown and N.F. Rulkov, ibid. 78, 4189 (1997); N.F. Rulkov and M.M. Sushchik, Int. J. Bifurcation Chaos Appl. Sci. Eng. 7, 625 (1997); P.C. Bresloff, Phys. Rev. E 60, 2160 (1999); P. Glendinning, Phys. Lett. A 259, 129 (1999); V.N. Belykh, I.V. Belykh, and M, Hasler, Phys. Rev. E 62, 6332 (2000); V.N. Belykh, I.V. Belykh, and E. Mosekilde, ibid. 63, 036216 (2001).
14. H. Fujisaka and T. Yamada, Prog. Theor. Phys. 69, 32 (1983); I. Waller and R. Kapral, Phys. Lett. 105A, 163 (1984); V.S. Afraimovich et al., Sov. Radiophys. 29, 795 (1986); G.L. Oppo and R. Kapral, Phys. Rev. A 36, 5820 (1987); L.A. Bunimovich and Ya.G. Sinai, Nonlinearity 1, 491 (1988); J.F. Heagy, L.M. Pecora, and T.L. Carroll, Phys. Rev. E 50, 1874 (1994); L. Kocarev and U. Parlitz, Phys. Rev. Lett. 76, 1816 (1996); R. Brown and N.F. Rulkov, ibid. 78, 4189 (1997); N.F. Rulkov and M.M. Sushchik, Int. J. Bifurcation Chaos Appl. Sci. Eng. 7, 625 (1997); P.C. Bresloff, Phys. Rev. E 60, 2160 (1999); P. Glendinning, Phys. Lett. A 259, 129 (1999); V.N. Belykh, I.V. Belykh, and M, Hasler, Phys. Rev. E 62, 6332 (2000); V.N. Belykh, I.V. Belykh, and E. Mosekilde, ibid. 63, 036216 (2001).
15. H. Fujisaka and T. Yamada, Prog. Theor. Phys. 69, 32 (1983); I. Waller and R. Kapral, Phys. Lett. 105A, 163 (1984); V.S. Afraimovich et al., Sov. Radiophys. 29, 795 (1986); G.L. Oppo and R. Kapral, Phys. Rev. A 36, 5820 (1987); L.A. Bunimovich and Ya.G. Sinai, Nonlinearity 1, 491 (1988); J.F. Heagy, L.M. Pecora, and T.L. Carroll, Phys. Rev. E 50, 1874 (1994); L. Kocarev and U. Parlitz, Phys. Rev. Lett. 76, 1816 (1996); R. Brown and N.F. Rulkov, ibid. 78, 4189 (1997); N.F. Rulkov and M.M. Sushchik, Int. J. Bifurcation Chaos Appl. Sci. Eng. 7, 625 (1997); P.C. Bresloff, Phys. Rev. E 60, 2160 (1999); P. Glendinning, Phys. Lett. A 259, 129 (1999); V.N. Belykh, I.V. Belykh, and M, Hasler, Phys. Rev. E 62, 6332 (2000); V.N. Belykh, I.V. Belykh, and E. Mosekilde, ibid. 63, 036216 (2001).
16. H. Fujisaka and T. Yamada, Prog. Theor. Phys. 69, 32 (1983); I. Waller and R. Kapral, Phys. Lett. 105A, 163 (1984); V.S. Afraimovich et al., Sov. Radiophys. 29, 795 (1986); G.L. Oppo and R. Kapral, Phys. Rev. A 36, 5820 (1987); L.A. Bunimovich and Ya.G. Sinai, Nonlinearity 1, 491 (1988); J.F. Heagy, L.M. Pecora, and T.L. Carroll, Phys. Rev. E 50, 1874 (1994); L. Kocarev and U. Parlitz, Phys. Rev. Lett. 76, 1816 (1996); R. Brown and N.F. Rulkov, ibid. 78, 4189 (1997); N.F. Rulkov and M.M. Sushchik, Int. J. Bifurcation Chaos Appl. Sci. Eng. 7, 625 (1997); P.C. Bresloff, Phys. Rev. E 60, 2160 (1999); P. Glendinning, Phys. Lett. A 259, 129 (1999); V.N. Belykh, I.V. Belykh, and M, Hasler, Phys. Rev. E 62, 6332 (2000); V.N. Belykh, I.V. Belykh, and E. Mosekilde, ibid. 63, 036216 (2001).
17. H. Fujisaka and T. Yamada, Prog. Theor. Phys. 69, 32 (1983); I. Waller and R. Kapral, Phys. Lett. 105A, 163 (1984); V.S. Afraimovich et al., Sov. Radiophys. 29, 795 (1986); G.L. Oppo and R. Kapral, Phys. Rev. A 36, 5820 (1987); L.A. Bunimovich and Ya.G. Sinai, Nonlinearity 1, 491 (1988); J.F. Heagy, L.M. Pecora, and T.L. Carroll, Phys. Rev. E 50, 1874 (1994); L. Kocarev and U. Parlitz, Phys. Rev. Lett. 76, 1816 (1996); R. Brown and N.F. Rulkov, ibid. 78, 4189 (1997); N.F. Rulkov and M.M. Sushchik, Int. J. Bifurcation Chaos Appl. Sci. Eng. 7, 625 (1997); P.C. Bresloff, Phys. Rev. E 60, 2160 (1999); P. Glendinning, Phys. Lett. A 259, 129 (1999); V.N. Belykh, I.V. Belykh, and M, Hasler, Phys. Rev. E 62, 6332 (2000); V.N. Belykh, I.V. Belykh, and E. Mosekilde, ibid. 63, 036216 (2001).
18. H. Fujisaka and T. Yamada, Prog. Theor. Phys. 69, 32 (1983); I. Waller and R. Kapral, Phys. Lett. 105A, 163 (1984); V.S. Afraimovich et al., Sov. Radiophys. 29, 795 (1986); G.L. Oppo and R. Kapral, Phys. Rev. A 36, 5820 (1987); L.A. Bunimovich and Ya.G. Sinai, Nonlinearity 1, 491 (1988); J.F. Heagy, L.M. Pecora, and T.L. Carroll, Phys. Rev. E 50, 1874 (1994); L. Kocarev and U. Parlitz, Phys. Rev. Lett. 76, 1816 (1996); R. Brown and N.F. Rulkov, ibid. 78, 4189 (1997); N.F. Rulkov and M.M. Sushchik, Int. J. Bifurcation Chaos Appl. Sci. Eng. 7, 625 (1997); P.C. Bresloff, Phys. Rev. E 60, 2160 (1999); P. Glendinning, Phys. Lett. A 259, 129 (1999); V.N. Belykh, I.V. Belykh, and M, Hasler, Phys. Rev. E 62, 6332 (2000); V.N. Belykh, I.V. Belykh, and E. Mosekilde, ibid. 63, 036216 (2001).
19. H. Fujisaka and T. Yamada, Prog. Theor. Phys. 69, 32 (1983); I. Waller and R. Kapral, Phys. Lett. 105A, 163 (1984); V.S. Afraimovich et al., Sov. Radiophys. 29, 795 (1986); G.L. Oppo and R. Kapral, Phys. Rev. A 36, 5820 (1987); L.A. Bunimovich and Ya.G. Sinai, Nonlinearity 1, 491 (1988); J.F. Heagy, L.M. Pecora, and T.L. Carroll, Phys. Rev. E 50, 1874 (1994); L. Kocarev and U. Parlitz, Phys. Rev. Lett. 76, 1816 (1996); R. Brown and N.F. Rulkov, ibid. 78, 4189 (1997); N.F. Rulkov and M.M. Sushchik, Int. J. Bifurcation Chaos Appl. Sci. Eng. 7, 625 (1997); P.C. Bresloff, Phys. Rev. E 60, 2160 (1999); P. Glendinning, Phys. Lett. A 259, 129 (1999); V.N. Belykh, I.V. Belykh, and M, Hasler, Phys. Rev. E 62, 6332 (2000); V.N. Belykh, I.V. Belykh, and E. Mosekilde, ibid. 63, 036216 (2001).
20. H. Fujisaka and T. Yamada, Prog. Theor. Phys. 69, 32 (1983); I. Waller and R. Kapral, Phys. Lett. 105A, 163 (1984); V.S. Afraimovich et al., Sov. Radiophys. 29, 795 (1986); G.L. Oppo and R. Kapral, Phys. Rev. A 36, 5820 (1987); L.A. Bunimovich and Ya.G. Sinai, Nonlinearity 1, 491 (1988); J.F. Heagy, L.M. Pecora, and T.L. Carroll, Phys. Rev. E 50, 1874 (1994); L. Kocarev and U. Parlitz, Phys. Rev. Lett. 76, 1816 (1996); R. Brown and N.F. Rulkov, ibid. 78, 4189 (1997); N.F. Rulkov and M.M. Sushchik, Int. J. Bifurcation Chaos Appl. Sci. Eng. 7, 625 (1997); P.C. Bresloff, Phys. Rev. E 60, 2160 (1999); P. Glendinning, Phys. Lett. A 259, 129 (1999); V.N. Belykh, I.V. Belykh, and M, Hasler, Phys. Rev. E 62, 6332 (2000); V.N. Belykh, I.V. Belykh, and E. Mosekilde, ibid. 63, 036216 (2001).
21. H. Fujisaka and T. Yamada, Prog. Theor. Phys. 69, 32 (1983); I. Waller and R. Kapral, Phys. Lett. 105A, 163 (1984); V.S. Afraimovich et al., Sov. Radiophys. 29, 795 (1986); G.L. Oppo and R. Kapral, Phys. Rev. A 36, 5820 (1987); L.A. Bunimovich and Ya.G. Sinai, Nonlinearity 1, 491 (1988); J.F. Heagy, L.M. Pecora, and T.L. Carroll, Phys. Rev. E 50, 1874 (1994); L. Kocarev and U. Parlitz, Phys. Rev. Lett. 76, 1816 (1996); R. Brown and N.F. Rulkov, ibid. 78, 4189 (1997); N.F. Rulkov and M.M. Sushchik, Int. J. Bifurcation Chaos Appl. Sci. Eng. 7, 625 (1997); P.C. Bresloff, Phys. Rev. E 60, 2160 (1999); P. Glendinning, Phys. Lett. A 259, 129 (1999); V.N. Belykh, I.V. Belykh, and M, Hasler, Phys. Rev. E 62, 6332 (2000); V.N. Belykh, I.V. Belykh, and E. Mosekilde, ibid. 63, 036216 (2001).
22. H. Fujisaka and T. Yamada, Prog. Theor. Phys. 69, 32 (1983); I. Waller and R. Kapral, Phys. Lett. 105A, 163 (1984); V.S. Afraimovich et al., Sov. Radiophys. 29, 795 (1986); G.L. Oppo and R. Kapral, Phys. Rev. A 36, 5820 (1987); L.A. Bunimovich and Ya.G. Sinai, Nonlinearity 1, 491 (1988); J.F. Heagy, L.M. Pecora, and T.L. Carroll, Phys. Rev. E 50, 1874 (1994); L. Kocarev and U. Parlitz, Phys. Rev. Lett. 76, 1816 (1996); R. Brown and N.F. Rulkov, ibid. 78, 4189 (1997); N.F. Rulkov and M.M. Sushchik, Int. J. Bifurcation Chaos Appl. Sci. Eng. 7, 625 (1997); P.C. Bresloff, Phys. Rev. E 60, 2160 (1999); P. Glendinning, Phys. Lett. A 259, 129 (1999); V.N. Belykh, I.V. Belykh, and M, Hasler, Phys. Rev. E 62, 6332 (2000); V.N. Belykh, I.V. Belykh, and E. Mosekilde, ibid. 63, 036216 (2001).
23. A.S. Pikovsky, Z. Phys. B: Condens, Matter 55, 149 (1984); K. Kaneko, Prog. Theor. Phys. 72, 480 (1984); K. Kaneko, Physica D 23, 436 (1986); N. Chatterjee and N. Gupte, Phys. Rev. E 53, 4457 (1996); D.J. Gauthier and J.C. Bienfang, Phys. Rev. Lett. 77, 1751 (1996); M. Ding and W. Yang, Phys. Rev. E 56, 4009 (1997); K. Zhu and T. Chen, ibid. 63, 067201 (2001); J. Jost and M.P. Joy, ibid. 65, 016201 (2002).
24. A.S. Pikovsky, Z. Phys. B: Condens, Matter 55, 149 (1984); K. Kaneko, Prog. Theor. Phys. 72, 480 (1984); K. Kaneko, Physica D 23, 436 (1986); N. Chatterjee and N. Gupte, Phys. Rev. E 53, 4457 (1996); D.J. Gauthier and J.C. Bienfang, Phys. Rev. Lett. 77, 1751 (1996); M. Ding and W. Yang, Phys. Rev. E 56, 4009 (1997); K. Zhu and T. Chen, ibid. 63, 067201 (2001); J. Jost and M.P. Joy, ibid. 65, 016201 (2002).
25. A.S. Pikovsky, Z. Phys. B: Condens, Matter 55, 149 (1984); K. Kaneko, Prog. Theor. Phys. 72, 480 (1984); K. Kaneko, Physica D 23, 436 (1986); N. Chatterjee and N. Gupte, Phys. Rev. E 53, 4457 (1996); D.J. Gauthier and J.C. Bienfang, Phys. Rev. Lett. 77, 1751 (1996); M. Ding and W. Yang, Phys. Rev. E 56, 4009 (1997); K. Zhu and T. Chen, ibid. 63, 067201 (2001); J. Jost and M.P. Joy, ibid. 65, 016201 (2002).
26. A.S. Pikovsky, Z. Phys. B: Condens, Matter 55, 149 (1984); K. Kaneko, Prog. Theor. Phys. 72, 480 (1984); K. Kaneko, Physica D 23, 436 (1986); N. Chatterjee and N. Gupte, Phys. Rev. E 53, 4457 (1996); D.J. Gauthier and J.C. Bienfang, Phys. Rev. Lett. 77, 1751 (1996); M. Ding and W. Yang, Phys. Rev. E 56, 4009 (1997); K. Zhu and T. Chen, ibid. 63, 067201 (2001); J. Jost and M.P. Joy, ibid. 65, 016201 (2002).
27. A.S. Pikovsky, Z. Phys. B: Condens, Matter 55, 149 (1984); K. Kaneko, Prog. Theor. Phys. 72, 480 (1984); K. Kaneko, Physica D 23, 436 (1986); N. Chatterjee and N. Gupte, Phys. Rev. E 53, 4457 (1996); D.J. Gauthier and J.C. Bienfang, Phys. Rev. Lett. 77, 1751 (1996); M. Ding and W. Yang, Phys. Rev. E 56, 4009 (1997); K. Zhu and T. Chen, ibid. 63, 067201 (2001); J. Jost and M.P. Joy, ibid. 65, 016201 (2002).
28. A.S. Pikovsky, Z. Phys. B: Condens, Matter 55, 149 (1984); K. Kaneko, Prog. Theor. Phys. 72, 480 (1984); K. Kaneko, Physica D 23, 436 (1986); N. Chatterjee and N. Gupte, Phys. Rev. E 53, 4457 (1996); D.J. Gauthier and J.C. Bienfang, Phys. Rev. Lett. 77, 1751 (1996); M. Ding and W. Yang, Phys. Rev. E 56, 4009 (1997); K. Zhu and T. Chen, ibid. 63, 067201 (2001); J. Jost and M.P. Joy, ibid. 65, 016201 (2002).
29. A.S. Pikovsky, Z. Phys. B: Condens, Matter 55, 149 (1984); K. Kaneko, Prog. Theor. Phys. 72, 480 (1984); K. Kaneko, Physica D 23, 436 (1986); N. Chatterjee and N. Gupte, Phys. Rev. E 53, 4457 (1996); D.J. Gauthier and J.C. Bienfang, Phys. Rev. Lett. 77, 1751 (1996); M. Ding and W. Yang, Phys. Rev. E 56, 4009 (1997); K. Zhu and T. Chen, ibid. 63, 067201 (2001); J. Jost and M.P. Joy, ibid. 65, 016201 (2002).
30. A.S. Pikovsky, Z. Phys. B: Condens, Matter 55, 149 (1984); K. Kaneko, Prog. Theor. Phys. 72, 480 (1984); K. Kaneko, Physica D 23, 436 (1986); N. Chatterjee and N. Gupte, Phys. Rev. E 53, 4457 (1996); D.J. Gauthier and J.C. Bienfang, Phys. Rev. Lett. 77, 1751 (1996); M. Ding and W. Yang, Phys. Rev. E 56, 4009 (1997); K. Zhu and T. Chen, ibid. 63, 067201 (2001); J. Jost and M.P. Joy, ibid. 65, 016201 (2002).
31. L.M. Pecora and T.L. Carroll, Phys. Rev. Lett. 80, 2109 (1998); K.S. Fink, G. Johnson, T.L. Carroll, and L.M. Pecora, Phys. Rev. E 61, 5080 (2000); M. Barahona and L.M. Pecora, Phys. Rev. Lett. 89, 054101 (2002).
32. L.M. Pecora and T.L. Carroll, Phys. Rev. Lett. 80, 2109 (1998); K.S. Fink, G. Johnson, T.L. Carroll, and L.M. Pecora, Phys. Rev. E 61, 5080 (2000); M. Barahona and L.M. Pecora, Phys. Rev. Lett. 89, 054101 (2002).
33. L.M. Pecora and T.L. Carroll, Phys. Rev. Lett. 80, 2109 (1998); K.S. Fink, G. Johnson, T.L. Carroll, and L.M. Pecora, Phys. Rev. E 61, 5080 (2000); M. Barahona and L.M. Pecora, Phys. Rev. Lett. 89, 054101 (2002).
34. G. Hu, J. Yang, and W. Liu, Phys. Rev. E 58, 4440 (1998); M. Zhan, G. Hu, and J. Yang, ibid. 62, 2963 (2000).
35. G. Hu, J. Yang, and W. Liu, Phys. Rev. E 58, 4440 (1998); M. Zhan, G. Hu, and J. Yang, ibid. 62, 2963 (2000).
36. M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra (Academic Press, New York, 1974).
37. max is computed from typical initial conditions, the stability here is with respect to the blowout bifurcation, Bubbling transition can occur while the parameters are still within the bound derived in this work. For details on these phenomena see E. Ott and J.C. Sommerer, Phys. Lett. A 188, 39 (1994); P. Ashwin, J. Buescu, and I. Stewart, Phys. Lett. 193, 126 (1994); S.C. Venkataramani et al., Phys. Rev. Lett. 77, 5361 (1996); Phys. Rev. E 54, 1346 (1996).
38. max is computed from typical initial conditions, the stability here is with respect to the blowout bifurcation, Bubbling transition can occur while the parameters are still within the bound derived in this work. For details on these phenomena see E. Ott and J.C. Sommerer, Phys. Lett. A 188, 39 (1994); P. Ashwin, J. Buescu, and I. Stewart, Phys. Lett. 193, 126 (1994); S.C. Venkataramani et al., Phys. Rev. Lett. 77, 5361 (1996); Phys. Rev. E 54, 1346 (1996).
39. max is computed from typical initial conditions, the stability here is with respect to the blowout bifurcation, Bubbling transition can occur while the parameters are still within the bound derived in this work. For details on these phenomena see E. Ott and J.C. Sommerer, Phys. Lett. A 188, 39 (1994); P. Ashwin, J. Buescu, and I. Stewart, Phys. Lett. 193, 126 (1994); S.C. Venkataramani et al., Phys. Rev. Lett. 77, 5361 (1996); Phys. Rev. E 54, 1346 (1996).
40. max is computed from typical initial conditions, the stability here is with respect to the blowout bifurcation, Bubbling transition can occur while the parameters are still within the bound derived in this work. For details on these phenomena see E. Ott and J.C. Sommerer, Phys. Lett. A 188, 39 (1994); P. Ashwin, J. Buescu, and I. Stewart, Phys. Lett. 193, 126 (1994); S.C. Venkataramani et al., Phys. Rev. Lett. 77, 5361 (1996); Phys. Rev. E 54, 1346 (1996).
41. R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University Press, Cambridge, 1985).
42. G. H. Golub and C. F. Van Loan, Matrix Computations (Johns Hopkins University Press, Baltimore, 1996).
43. G. Rangarajan and M. Ding, Phys Lett. A 296, 204 (2002).