Morse decompositions for delay-differential equations. (English) Zbl 0648.34082

Mallet-Paret, John

J. Differ. Equations 72, No. 2, 270-315 (1988).

The author deals with global dynamical properties of the scalar delay- differential equation (1) \(\dot x(t)=-f(x(t),x(t-1)),\) where f satisfies appropriate conditions, in particular, a negative feedback condition in the delay. Main results of the paper are that the dynamical system for (1) possesses a global integer value Lypunov function which gives rise to a Morse decomposition of the attractor. These results are then used to obtain information about the asymptotic oscillatory properties of solutions to arbitrary initial value problems. The relation between periodic solutions of (1) and the Morse decomposition is also explored.

Reviewer: J.Ohriska

MSC:

34K99	Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
37C70	Attractors and repellers of smooth dynamical systems and their topological structure

Keywords:

global dynamical properties; scalar delay-differential equation; global integer value Lypunov function; attractor; Morse decomposition

References:

[1]	Bellman, R.; Cooke, K.L., Differential-difference equations, (1963), Academic Press New York · Zbl 0115.30102
[2]	Billotti, J.E.; LaSalle, J.P., Periodic dissipative processes, Bull. amer. math. soc., 6, 1082-1089, (1971) · Zbl 0274.34061
[3]	Chow, S.-N., Existence of periodic solutions of autonomous functional differential equations, J. differential equations, 15, 350-378, (1974) · Zbl 0295.34055
[4]	Chow, S.-N.; Diekmann, O.; Mallet-Paret, J., Stability, multiplicity, and global continuation of symmetric periodic solutions of a nonlinear Volterra integral equation, Japan J. appl. math., 2, 433-469, (1985) · Zbl 0596.45009
[5]	Chow, S.-N.; Green, D., Some results on singular delay-differential equations, (), 161-182
[6]	Chow, S.-N.; Mallet-Paret, J., The fuller index and global Hopf bifurcation, J. differential equations, 29, 66-85, (1978) · Zbl 0369.34020
[7]	Chow, S.-N.; Mallet-Paret, J., Singularly perturbed delay-differential equations, (), 7-12
[8]	Conley, C., Isolated invariant sets and the Morse index, () · Zbl 0397.34056
[9]	Farmer, J.D., Chaotic attractors of an infinite-dimensional dynamical system, Physica D, 4, 366-393, (1982) · Zbl 1194.37052
[10]	{\scB. Fiedler and J. Mallet-Paret}, Connections between Morse sets for delay-differential equations, submitted. · Zbl 0659.34077
[11]	Hadeler, K.P.; Tomiuk, J., Periodic solutions of difference-differential equations, Arch. rational mech. anal., 65, 87-95, (1977) · Zbl 0426.34058
[12]	Hale, J.K., Theory of functional differential equations, (1977), Springer-Verlag New York/Berlin · Zbl 0425.34048
[13]	Hale, J.K.; Lopes, O., Fixed point theorems and dissipative processes, J. differential equations, 13, 391-402, (1973) · Zbl 0256.34069
[14]	Henry, D., Small solutions of linear autonomous functional differential equations, J. differential equations, 8, 494-501, (1970) · Zbl 0228.34044
[15]	Jones, G.S., The existence of periodic solutions of f′(x) = −αf(x −1){1 + f(x)}, J. math. anal. appl., 5, 435-450, (1962) · Zbl 0106.29504
[16]	Jones, G.S., Periodic motions in Banach space and applications to functional differential equations, Contrib. differential equations, 3, 75-106, (1964)
[17]	Kaplan, J.L.; Yorke, J.A., On the stability of a periodic solution of a differential delay equation, SIAM J. math. anal., 6, 268-282, (1975) · Zbl 0241.34080
[18]	Kaplan, J.L.; Yorke, J.A., On the nonlinear differential delay equationx′(t)=−f(x(t), x(t − 1)), J. differential equations, 23, 293-314, (1977) · Zbl 0307.34070
[19]	Mackey, M.C.; Glass, L., Oscillation and chaos in physiological control systems, Science, 197, 287-289, (1977) · Zbl 1383.92036
[20]	Mallet-Paret, J., Morse decompositions and global continuation of periodic solutions for singularly perturbed delay equations, (), 351-365
[21]	Mallet-Paret, J.; Nussbaum, R.D., Global continuation and complicated trajectories for periodic solutions of a differential-delay equation, (), 155-167, Part 2
[22]	Mallet-Paret, J.; Nussbaum, R.D., Global continuation and asymptotic behaviour for periodic solutions of a differential-delay equation, Ann. mat. pura appl., 145, 33-128, (1986) · Zbl 0617.34071
[23]	Mallet-Paret, J.; Nussbaum, R.D., A bifurcation gap for a singularly perturbed delay equation, (), 263-286
[24]	{\scJ. Mallet-Paret and R. D. Nussbaum}, A differential-delay equation arising in optics and physiology, submitted. · Zbl 0676.34043
[25]	Matano, H., Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation, J. fac. sci. univ. Tokyo sect. IA math., 29, 401-441, (1982) · Zbl 0496.35011
[26]	Myschkis, A.D., Lineare differentialgleichungen mit nacheilendem argument, (1955), Deutscher Verlag Wiss., Berlin · Zbl 0067.31802
[27]	Nickel, K., Gestaltaussagenu¨ber lo¨sungen parabolischer differentialgleichungen, J. reine angew. math., 211, 78-94, (1962)
[28]	Nussbaum, R.D., The fixed point index for local condensing maps, Ann. mat. pura appl., 89, 217-258, (1971) · Zbl 0226.47031
[29]	Nussbaum, R.D., A global bifurcation theorem with applications to functional differential equations, J. funct. anal., 19, 319-338, (1975) · Zbl 0314.47041
[30]	Nussbaum, R.D., Periodic solutions of nonlinear autonomous functional differential equations, Springer lecture notes in math., 730, 283-325, (1979)
[31]	Nussbaum, R.D., Uniqueness and nonuniqueness for periodic solutions of x′(t)= −αg(x(t − 1)), J. differential equations, 34, 24-54, (1979) · Zbl 0404.34057
[32]	Peters, H., Comportement chaotique d’une equation differentielle retarde´e, C. R. acad. sci. Paris ser. A, 290, 1119-1122, (1980) · Zbl 0435.34063
[33]	Rybakowski, K.P., An introduction to the homotopy index theory in noncompact spaces, () · Zbl 0799.34060
[34]	Saupe, D., Global bifurcation of periodic solutions to some autonomous differential delay equations, Appl. math. comput., 13, 185-211, (1983) · Zbl 0522.34067
[35]	Saupe, D., Accelerated PL-continuation methods and periodic solutions of parametrized differential-delay equations, ()
[36]	Schwartzman, S., Asymptotic cycles, Ann. math., 66, 270-284, (1957) · Zbl 0207.22603
[37]	Smoller, J., Shock waves and reaction-diffusion equations, (1983), Springer-Verlag New York/Berlin · Zbl 0508.35002
[38]	Walther, H.-O., On instability, ω-limit sets and periodic solutions to nonlinear autonomous differential delay equations, Springer lecture notes in math., 730, 489-503, (1979)
[39]	Walther, H.-O., On density of slowly oscillating solutions of x˙(t)=−f(x(t − 1)), J. math. anal. appl., 79, 127-140, (1981) · Zbl 0451.34063