Amplitude response of coupled oscillators.

*(English)*Zbl 0703.34047Summary: We investigate the interaction of a pair of weakly nonlinear oscillators (e.g. each near a Hopf bifurcation) when the coupling strength is comparable to the attraction of the limit cycles. Changes in amplitude cannot then be ignored, and there are new phenomena. We show that even for simple forms of these equations, there are parameter regimes in which the interaction causes the system to stop oscillating, and the rest state at zero, stabilized by the interaction, is the only stable solution. When the uncoupled oscillators have a local frequency that is independent of amplitude (zero shear), and the coupling is scalar, we give a complete description of the behavior. We then analyze more complicated equations to show the effects of amplitude-dependent frequency in the uncoupled equations (nonzero shear) and nonscalar coupling. For example, when there is shear, there can be bistability between a phase-locked solution and an unlocked “drift” solution, in which the oscillators go at different average frequencies. There can also be bistability between a phase locked solution and an equilibrium, nonoscillatory, solution. The equations for the zero shear case have a pair of degenerate bifurcation points, and the new phenomena in the nonzero shear case arise from a partial unfolding of these singularities. When the coupling is nonscalar, there are a variety of new behaviors, including bistability of an in-phase and an anti-phase solution for two identical oscillators, parameters for which only the anti-phase solution is stable, and “phase-trapping”, i.e., nonlocked solutions in which the oscillators still have the same average frequency. It is shown that whether the coupling is diffusive or direct has many effects on the behavior of the system.

##### MSC:

34C10 | Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations |

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\textit{D. G. Aronson} et al., Physica D 41, No. 3, 403--449 (1990; Zbl 0703.34047)

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[1] | Aronson, D.G.; Doedel, E.J.; Othmer, H.G., An analytical and numerical study of the bifurcations in a system of linearly coupled oscillators, Physica D, 25, 20-104, (1987) · Zbl 0624.34029 |

[2] | Baer, S.; Tier, C., An analysis of a dendritic neuron with an active membrane site, J. math. biol., 23, 137-161, (1986) · Zbl 0587.92012 |

[3] | Bar-Eli, K., On the stability of coupled chemical oscillators, Physica D, 14, 242-252, (1985) |

[4] | Daan, S.; Berde, C., Two coupled oscillators; simulations of the circadian pacemaker in Mammalian activity rhythms, J. theor. biol., 70, 297-314, (1978) |

[5] | Ermentrout, G.B., n:m phase locking of weakly coupled oscillators, J. math. biol., 12, 326-342, (1981) · Zbl 0476.92007 |

[6] | Ermentrout, G.B., The behavior of rings of coupled oscillators, J. math. biol., 23, 55-74, (1985) · Zbl 0583.92002 |

[7] | Ermentrout, G.B., Losing amplitude and saving phase, () · Zbl 0598.92024 |

[8] | Ermentrout, G.B.; Kopell, N., Frequency plateaus in a chain of weakly coupled I, SIAM J. math. anal., 15, 215-237, (1984) · Zbl 0558.34033 |

[9] | Ermentrout, G.B.; Rinzel, J., Beyond a Pacemaker’s entrainment limit: phase walk-through, Am. J. physiol., 246, R102-R106, (1984) |

[10] | Hoppensteadt, F.C.; Keener, J.P., Phase locking of biological clocks, J. math. biol., 15, 339-349, (1982) · Zbl 0489.92006 |

[11] | Kawato, M.; Suzuki, R., Two coupled neural oscillators as a model of the Circadian pacemaker, J. theor. biol., 86, 547-575, (1980) |

[12] | Kopell, N.; Ermentrout, G.B., Symmetry and phase locking in chains of weakly coupled oscillators, Comm. pure. appl. math., 39, 623-660, (1986) · Zbl 0596.92011 |

[13] | Linkens, D.A., The stability of entrainment conditions for RLC coupled van der Pol oscillators used as a model for intestinal electrical rhythms, Bull. math. biol., 39, 359-372, (1977) · Zbl 0354.92015 |

[14] | Neu, J.C., Coupled chemical oscillators, SIAM J. appl. math., 37, 307-315, (1979) · Zbl 0417.34063 |

[15] | Rand, R.H.; Holmes, P.J., Bifurcation of periodic motions in two weakly coupled van der Pol oscillators, Int. J. nonlin. mech., 15, 387-399, (1980) · Zbl 0447.70028 |

[16] | Cohen, A.H.; Holmes, P.J.; Rand, R.H., The nature of coupling between segmental oscillators of the lamprey spinal generator, J. math. biol., 13, 345-369, (1982) · Zbl 0476.92003 |

[17] | Vincent, M.St., Entrainment of a limit cycle oscillator with shear by large amplitude forcing, SIAM J. appl. math., 19, 648-666, (1988) · Zbl 0658.34032 |

[18] | Smale, S., A mathematical model of two cells via Turing’s equation, (), 17-26, No. 6 · Zbl 0333.92002 |

[19] | Torre, V., A theory of synchronization of two heart pace-maker cells, J. theor. biol., 61, 55-71, (1976) |

[20] | Winfree, A.T., The geometry of biological time, (1980), Springer Berlin · Zbl 0856.92002 |

[21] | Fenichel, N., Persistence and smoothness of invariant manifolds for flows, Ind. univ. math. J., 21, 193-226, (1971) · Zbl 0246.58015 |

[22] | Schreiber, I.; Marek, M., Strange attractors in coupled reaction-diffusion cells, Physica D, 15, 258-292, (1982) |

[23] | Holmes, P.; Rand, R., Bifurcations of the forced van der Pol oscillator, Quart. appl. math., 35, 495-509, (1978) · Zbl 0375.34031 |

[24] | Chakraborty, T.; Rand, R., The transition from phase locking to drift in a system of two weakly coupled van der Pol oscillators, Int. J. nonlin. mech., 23, 369-376, (1988) · Zbl 0667.70028 |

[25] | Turing, A.M., The chemical basis for morphogenesis, Phil. trans. R. soc. London ser. B, 237, 37-72, (1952) · Zbl 1403.92034 |

[26] | G.B. Ermentrout and N. Kopell, Oscillator death in systems of coupled oscillators, SIAM J. Appl. Math., to appear. · Zbl 0686.34033 |

[27] | Crowley, M.; Epstein, I., Experimental and theoretical studies of a coupled chemical oscillator: phase death, multistability and in- and out-of-phase entrainment, J. phys. chem., 93, 2496-2502, (1989) |

[28] | Guckenheimer, J.; Holmes, P., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, (1983), Springer Berlin · Zbl 0515.34001 |

[29] | Hale, J., Ordinary differential equations, (1969), Wiley-Interscience New York · Zbl 0186.40901 |

[30] | Gambaudo, J.M.; Glendinning, P.; Tresser, C., The gluing bifurcation I. symbolic dynamics of the closed curves, Nonlinearity, 1, 203-213, (1988) · Zbl 0656.58023 |

[31] | Sattinger, D.H., Group theoretic methods in bifurcation theory, () · Zbl 0414.58013 |

[32] | Chow, S.-N.; Mallet-Paret, J., Integral averaging and bifurcation, J. diff. eqs., 26, 112-159, (1977) · Zbl 0367.34033 |

[33] | Chow, S.-N.; Hale, J., Methods of bifurcation theory, (1982), Springer Berlin |

[34] | Yamaguchi, Y.; Shimizu, H., Theory of self-organization in the presence of native frequency distributions and external noise, Physica D, 11, 212-226, (1984) · Zbl 0582.92006 |

[35] | Ermentrout, G.B., Oscillator death in populations of “all to all” coupled nonlinear oscillars, Physica D, 41, 219-231, (1990) · Zbl 0693.34040 |

[36] | R. Mirollo and S. Strogatz, Amplitude death in an array of limit cycle oscillators, preprint. · Zbl 1086.34525 |

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