Boundary layer phenomena for differential-delay equations with state-dependent time lags, I.

AbstractIn this paper we begin a study of the differential-delay equation % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0Jc9yq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepS0he9vr0-vr% 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabew7aLjqadI% hagaqbaiaacIcacaWG0bGaaiykaiabg2da9iabgkHiTiaadIhacaGG% OaGaamiDaiaacMcacqGHRaWkcaWGMbGaaiikaiaadIhacaGGOaGaam% iDaiabgkHiTiaadkhacaGGPaGaaiykaiaacYcacaqGGaGaaeiiaiaa% dkhacqGH9aqpcaWGYbGaaiikaiaadIhacaGGOaGaamiDaiaacMcaca% GGPaaaaa!5192! $$\varepsilon x'(t) = - x(t) + f(x(t - r)), r = r(x(t))$$ .We prove the existence of periodic solutions for 0<ɛ<ɛ0, where ɛ0 is an optimal positive number. We investigate regularity and monotonicity properties of solutions x(t) which are defined for all t and of associated functions like η(t)=t−r(x(t)). We begin the development of a Poincaré-Bendixson theory and phase-plane analysis for such equations. In a companion paper these results will be used to investigate the limiting profile and corresponding boundary layer phenomena for periodic solutions as ɛ approaches zero.

[1]  R. D. Driver,et al.  A two-body problem of classical electrodynamics: the one-dimensional case☆ , 1963 .

[2]  R. D. Driver,et al.  A Functional-Differential System of Neutral Type Arising in a Two-Body Problem of Classical Electrodynamics , 1963 .

[3]  Note on uniqueness for a one-dimensional two-body problem of classical electrodynamics , 1967 .

[4]  RobertF Brown,et al.  The Lefschetz fixed point theorem , 1971 .

[5]  R. Nussbaum.  Some asymptotic fixed point theorems , 1972 .

[6]  Roger D. Nussbaum,et al.  Periodic solutions of analytic functional differential equations are analytic. , 1973 .

[7]  Roger D. Nussbaum,et al.  Periodic solutions of some nonlinear autonomous functional differential equations , 1974 .

[8]  Roger D. Nussbaum,et al.  A global bifurcation theorem with applications to functional differential equations , 1975 .

[9]  James A. Yorke,et al.  On the Stability of a Periodic Solution of a Differential Delay Equation , 1975 .

[10]  B. D. Coleman,et al.  Periodic Solutions of Certain Nonlinear Integral Equations with a Time Lag , 1976 .

[11]  K. P. Hadeler,et al.  Periodic solutions of difference-differential equations , 1977 .

[12]  James A. Yorke,et al.  On the nonlinear differential delay equation x′(t) = −f(x(t), x(t − 1)) , 1977 .

[13]  Shui-Nee Chow,et al.  Integral averaging and bifurcation , 1977 .

[14]  L. Glass,et al.  Oscillation and chaos in physiological control systems. , 1977, Science.

[15]  R. Nussbaum.  Periodic solutions of nonlinear autonomous functional differential equations , 1979 .

[16]  Roger D. Nussbaum,et al.  Uniqueness and nonuniqueness for periodic solutions of x′(t) = −g(x(t − 1)) , 1979 .

[17]  Harlan W. Stech,et al.  The Hopf bifurcation: A stability result and application , 1979 .

[18]  K. Ikeda.  Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system , 1979 .

[19]  S. P. Blythe,et al.  Nicholson's blowflies revisited , 1980, Nature.

[20]  K. Ikeda,et al.  Optical Turbulence: Chaotic Behavior of Transmitted Light from a Ring Cavity , 1980 .

[21]  K. Schmitt,et al.  Periodic solutions of some nonlinear delay-differential equations , 1980 .

[22]  H. M. Gibbs,et al.  Bifurcation gap in a hybrid optically bistable system , 1982 .

[23]  H. M. Gibbs,et al.  BIFURCATIONS TO CHAOS IN OPTICAL BISTABILITY , 1982 .

[24]  K. Ikeda,et al.  Successive Higher-Harmonic Bifurcations in Systems with Delayed Feedback , 1982 .

[25]  Alternate paths to chaos in optical bistability , 1983 .

[26]  Shui-Nee Chow,et al.  Singularly Perturbed Delay-Differential Equations , 1983 .

[27]  J. Mallet-Paret.  Morse Decompositions and Global Continuation of Periodic Solutions for Singularly Perturbed Delay Equations , 1983 .

[28]  John Mallet-Paret,et al.  Global continuation and asymptotic behaviour for periodic solutions of a differential-delay equation , 1986 .

[29]  J. Mallet-Paret,et al.  A BIFURCATION GAP FOR A SINGULARLY PERTURBED DELAY EQUATION , 1986 .

[30]  J. Mallet-Paret.  Morse Decompositions for delay-differential equations , 1988 .

[31]  André Longtin,et al.  Complex oscillations in the human pupil light reflex with “mixed” and delayed feedback , 1988 .

[32]  Michael C. Mackey,et al.  Consumer memory and price fluctuations in commodity markets: An integrodifferential model , 1989 .

[33]  André Longtin,et al.  Nonlinear oscillations, noise and chaos in neural delayed feedback , 1989 .

[34]  A Longtin,et al.  Modelling autonomous oscillations in the human pupil light reflex using non-linear delay-differential equations. , 1989, Bulletin of Mathematical Biology.

[35]  Michael C. Mackey,et al.  Commodity price fluctuations: Price dependent delays and nonlinearities as explanatory factors , 1989 .

[36]  J. Mallet-Paret,et al.  Connections between Morse sets for delay-differential equations. , 1989 .

[37]  John Mallet-Paret,et al.  A differential-delay equation arising in optics and physiology , 1989 .

[38]  J. Mallet-Paret,et al.  The Poincare-Bendixson theorem for monotone cyclic feedback systems , 1990 .

[39]  Michael C. Mackey,et al.  Feedback, delays and the origin of blood cell dynamics , 1990 .

[40]  John Mallet-Paret,et al.  Multiple transition layers in a singularly perturbed differential-delay equation , 1993, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.