Contracting return maps for some delay differential equations (Q2738690)

The equation \(\dot x(t)=-\mu x(t)+f(x(t-1))\) is certainly among the simplest delay differential equations with instantaneous damping and delayed feedback, where \(\mu >0\) is a given constant and \(f:\mathbb{R}\to \mathbb{R}\) is a given mapping with \(f(0)=0\). Significant progress has been made regarding the existence, uniqueness, and stability of periodic solutions. This progress is regarded as the most essential step towards a complete understanding of the global dynamics of the associated semiflow in the infinite-dimensional phase space \(C=C([-1, 0])\), at least this seems to be the case when \(f\) is monotonic, see, for example, the two classical references [\textit{O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel} and \textit{H.-O. Walther}, Delay equations. Functional-, complex-, and nonlinear analysis. Applied Mathematical Sciences. 110. New York, N.Y.: Springer-Verlag (1995; Zbl 0826.34002) and \textit{J. K. Hale} and \textit{S. M. Verduyn Lunnel}, Introduction to functional differential equations. Applied Mathematical Sciences. 99. New York, N.Y: Springer-Verlag (1993; Zbl 0787.34002)]. See also the recent work of \textit{T. Krisztin, H.-O. Walther} and \textit{J. Wu} [Shape, smoothness and invariant stratification of an attracting set for delayed monotone positive feedback. Fields Institute Monographs. 11. Providence, RI: American Mathematical Society (1999; Zbl 1004.34002)] and \textit{J. Mallet-Paret} and \textit{G. R. Sell} [J. Differ. Equations 125, No. 2, 385-440 (1996; Zbl 0849.34055) and ibid. 441-489 (1996; Zbl 0849.34056)]. It is also important that much of the progress on the periodic solutions have found interesting applications in a number of application problems, including physiology, population dynamics and neural networks, to name a few. Among developed methods for the investigation of the periodic solutions are local/global Hopf bifurcation theory, fixed-point arguments, and general geometric approaches. Each of the above methods has its own advantage and drawback: the local Hopf bifurcation approach provides information about the existence, asymptotic form, direction and stability of periodic solutions but the conclusion remains true only when the parameter is in a small neighborhood of the critical value; the global Hopf bifurcation theory/fixed-point theoretical argument may ensure the global continuation of a branch of periodic solutions and thus obtains the existence of periodic solutions of large amplitudes for a large range of parameter values but this approach seems to generate little information on the stability; the general geometric approach does yield much details of the periodic solutions including the existence, domain of attraction, limiting profiles but the nonlinearities are quite restrictive (monotonicity, for example). NEWLINENEWLINENEWLINEThe paper under review develops an alternative method for the existence and stability of periodic solutions for delay differential equations. It is elegant as it is theoretically simple and straightforward, and it has great potential for applications as its requirements on the nonlinear function \(f\) seems to be quite minimal. The latter is particularly important when the above equation is used as a model for a single neuron with delayed feedback, in this case \(f\) is the signal function. Depending on the contents of the model, either for biological or artificial networks or for electronic implementations, this function can vary from a simple step function (for cellular neural nets), to a rational function (for some biological networks), to a hyperbolic tangent. The existence of noise adds further complication [see \textit{J. Wu}, Introduction to neural dynamics and signal transmission delay. Berlin: W. de Gruyter (2001; Zbl 0977.34069)]. Vaguely, the method involves the construction of a closed convex set and a contractive return mapping along the translation of a solution starting from an initial state in the convex set. This allows the application of the well known contractive mapping principle to obtain a fixed-point, which gives a stable periodic solution to the equation. The beautify is the simplicity of the idea and the use of the information of periodic orbits when \(f\) is a step function (this idea was used by \textit{X. Xie} [J. Dyn. Differ. Equations 3, No. 4, 515-540 (1991; Zbl 0743.34080)] in a different content) to construct the aforementioned convex set, and the technical difficulty is the construction and the associated analysis. Here is some technical detail: the nonlinearity \(f\) is assumed to be a continuous odd function (the symmetry condition is used here mainly for the simplicity of presentation) which outside a neighborhood of \(0\) is close to a step function. This is not necessarily a constant on any nontrivial interval, need not to be monotone nor must the negative feedback condition \(xf(x)<0\) be satisfied for small \(|x|>0\). A closed convex set \(A\) is introduced, of initial data \(\phi \in C\), carefully so that a solution \(x^\phi \) enters \(-A\) at some entry time \(q=q(\phi)\). Then fixed-points of the return map \(R_f: A\ni \phi \mapsto -F_f(q(\phi), \phi)\in A\) define slowly oscillating periodic solutions (solutions with the lengths between consecutive zeros are larger than 1) to the equation. When \(f\) is Lipschitz, it is shown that the return map \(R_f\) becomes Lipschitz continuous, too. Lipschitz constants are estimated in terms of the global Lipschitz constant \(L\) of \(f\) and the Lipschitz constant \(L_\beta \) of the restriction of \(f\) to some interval \([\beta ,\infty)\) with \(\beta >0\). If \(f\) is sufficiently close to a constant multiple of the step function for \(|x|\geq \beta \) and if \(L_\beta \) is small enough then \(R_f\) becomes a contraction, with an attractive fixed-point \(\phi _f\). What generates stability here is that a suitable Lipschitz constant \(L_\beta \) brings solutions \(\overline x\) with \(\beta \leq \overline x(t)\) and \(\beta \leq x(t)\) on \([-1, 0]\) and \(x(t)=\overline x(t)\) closer together on \([t, t+1]\). It is also shown that in case \(f\) is \(C^1\) smooth the attractive fixed-point \(\phi _f\) of \(R_f\) and the associated slowly oscillating periodic solution define a Poincaré return map \(P_f\) with the spectrum of \(DP_f(\phi _f)\) contained in the open unit disk, which means that the periodic orbit is hyperbolic, stable, and exponentially attractive with asymptotic phase. While this article was published only in 2001, the developed method has already been adopted in several recent papers this reviewer has been involved. This includes a joint work in progress with the author for periodic solutions to some systems of delay differential equations, a recent work of the author [Contracting return maps for monotone delayed feedback. Discrete Contin. Dyn. Syst. 7, No. 2, 259-274 (2001)] where a sharp estimate for monotone Lipschitz continuous nonlinearities are obtained, and some extensions of the work to difference equations with delay arising from neural network for associative memory [\textit{Z. Zhou} and \textit{J. Wu}, Attractive periodic orbits in nonlinear discrete-time networks with delayed feedback (to appear)].NEWLINENEWLINEFor the entire collection see [Zbl 0960.00044].