Weak coupling of strongly nonlinear, weakly dissipative identical oscillators

References

• Aronson , D.G , Ermentrout, , G. B and Kopell , N . 1990 . Amplitude response of coupled oscillators . PhysicaD , 41 : 403 – 449 .
   Google Scholar
• Ashwin , P and Swift , J.W . 1992 . The dynamics of n identical oscillators with symmetric coupling . Journal of Nonlinear Science , 2 : 69 – 108 .
   Web of Science ®Google Scholar
• Ashwin , P , Guaschi , J and Phelps , J.M . 1991 . Rotation sets and phase locking in an electronic three oscillator system . PhysicaD , 66 : 392 – 411 .
   Google Scholar
• Baesens , C , Guckenheimer , J , Kim , S and MacKay , R. S . 1991 . Three coupled oscillators:mode-locking,global bifurcations and toroidal chaos . PhysicaD , 24 : 387 – 475 .
   Google Scholar
• Bogoliubov , N. N . 1961 . :Asymptotic Methods in Nonlinear Oscillations , Delhi : Hindustan Publishing .
   Google Scholar
• Bourland , F.J and Haberman , R . 1961 . The modulated phase shift for strongly nonlinear slowly varying and weakly damped oscillators . SIAM Journal of Applied Mathematics , 48 : 737 – 748 .
   Web of Science ®Google Scholar
• Bourland , F.J , Haberman , R and Kath , W.L . 1991 . Averaging methods for the phase shift of . SIAM Journal of Applied Mathematics , 51 : 1150 – 1167 .
   Web of Science ®Google Scholar
• Carr , J . Applications of Centre Manifold Theory , Vol. 35 , New York : Springer . Applied Mathematical Science
   Google Scholar
• Dangelmayr , G , Kirby , M , E , Allgower , K , Bohmer and M , Golubitsky . 1991,1992 . “ On diffusively coupled oscillators ” . In Bifurcation and Symmetry , Vol. 104 , 85 – 97 . Marburg,Birkhauser , Basel : ISNM .
   Google Scholar
• Ermentrout , G.B and Kopell , N . 1984 . Frequency plateaus in a chain of weakly coupled oscillators . SIAM Journal of Mathematical Analysis , 15 : 215 – 237 .
   Web of Science ®Google Scholar
• Ermentrout , G.B and Kopell , N . 1991 . Multiple pulse interactions and averaging in systems of coupled neural oscillators . Journal of Mathematical Biology , 29 : 195 – 217 .
   Web of Science ®Google Scholar
• Guckenheimer , j and Holmes , F . 1983 . Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields , Vol. 42 , New York : Springer . Applied Mathmatical Science
   Google Scholar
• Hirsch , M , Pugh , C and M . 1977 . Invariant Manifolds , Vol. 583 , Springer , Berlin : LNM .
   Google Scholar
• Holmes , P and Rand , D . 1980 . Phase portraits and bifurcations of a non-linear oscillator . Inter- national Journal of Nonlinear Mechanics , 15 : 449 – 458 .
   Web of Science ®Google Scholar
• Kevorkian , J . 1987 . Perturbation techniques for oscillatory systems with slowly varying coefficients . SIAM Review , 29 : 391 – 461 .
   Web of Science ®Google Scholar
• Kevorkian , J and Cole , J.D . 1981 . Perturbation Methods in Applied Mathematics Vol. 34 , Springer , New York Applied Mathe- matical Science
   Google Scholar
• Kielhofer , H . 1976 . On the Lyapunov stability of stationary solutions of semilinear parabolic differ- ential equations . Journal oj Differential Equations , 22 : 193 – 208 .
   Web of Science ®Google Scholar
• Kopell , N and Ermentrout , G.B . 1986 . Symmetry and phase locking in chains of weakly coupled oscillators . Communications in Pure and Applied Mathematics , 39 : 623 – 660 .
   Web of Science ®Google Scholar
• Luke , J.C . 1965 . A perturbation method for nonlinear dispersive wave problems . Proceedings of the Royal Society of London A , 292 : 403 – 412 .
   Google Scholar
• Poliashenko , M and S.R . 1992 . Chaos due to homoclinic and heteroclinic orbits in two coupled oscillators with nonisochronism . Physical Review A , 46 : 5271 – 5274 .
   PubMed Web of Science ®Google Scholar
• Sanders , J.A and Cushman , R . 1986 . Limit cycles in the Josephson equation . SIAM Journal of Mathe- matical Analysis , 17 : 495 – 510 .
   Web of Science ®Google Scholar
• Sanders , J.A . 1985 . Averaging Methods in Nonlinear Dynamical Systems Vol. 59 , Springer , New york Applied Mathematical Science
   Google Scholar
• Sherman , A and Rinzel , J . 1992 . Rhythmogenic effects of weak electrotonic coupling in neuronal models . Proceedings of the National Academy of Science of the USA , 89 : 2471 – 2474 .
   PubMed Web of Science ®Google Scholar