# zbMATH — the first resource for mathematics

Stability and uniqueness of slowly oscillating periodic solutions to Wright’s equation. (English) Zbl 1382.34069
In this paper by using a computer-assisted proof it is shown that, for any $$\alpha$$ in the interval $$[1.9, 6.0]$$, Wright’s equation $y'(t)=-\alpha y(t-1)[1+y(t)]$ admits a unique slowly oscillating periodic solution $$y$$, i.e., a continuous periodic function $$y(t)$$ with the following property: There exist $$q$$, $$\overline{q}>1$$ such that up to a time translation, $$y(t)>0$$ for $$0<t<q$$, $$y(t)<0$$, for $$q<t<q+\overline{q}$$, and $$y(t+q+\overline{q})=y(t)$$, for all $$t$$.

##### MSC:
 34K13 Periodic solutions to functional-differential equations 34K20 Stability theory of functional-differential equations
INTLAB; Matlab
Full Text:
##### References:
 [1] Bánhelyi, B.; Csendes, T.; Krisztin, T.; Neumaier, A., Global attractivity of the zero solution for Wright’s equation, SIAM J. Appl. Dyn. Syst., 13, 1, 537-563, (2014) · Zbl 1301.34094 [2] Castelli, R.; Lessard, J.-P., Rigorous numerics in Floquet theory: computing stable and unstable bundles of periodic orbits, SIAM J. Appl. Dyn. Syst., 12, 1, 204-245, (2013) · Zbl 1293.37033 [3] Chow, S.-N.; Mallet-Paret, J., Integral averaging and bifurcation, J. Differential Equations, 26, 1, 112-159, (1977) · Zbl 0367.34033 [4] Horst, R.; Tuy, H., Global optimization: deterministic approaches, (2013), Springer Science & Business Media [5] Jones, G. S., The existence of periodic solutions of $$f^\prime(x) = - \alpha f(x - 1) \{1 + f(x) \}$$, J. Math. Anal. Appl., 5, 3, 435-450, (1962) · Zbl 0106.29504 [6] Jones, G. S., On the nonlinear differential-difference equation $$f^\prime(x) = - \alpha f(x - 1) \{1 + f(x) \}$$, J. Math. Anal. Appl., 4, 3, 440-469, (1962) · Zbl 0106.29503 [7] Kaplan, J. L.; Yorke, J. A., On the stability of a periodic solution of a differential delay equation, SIAM J. Math. Anal., 6, 2, 268-282, (1975) · Zbl 0241.34080 [8] Koch, H.; Schenkel, A.; Wittwer, P., Computer-assisted proofs in analysis and programming in logic: a case study, SIAM Rev., 38, 4, 565-604, (1996) · Zbl 0865.68111 [9] Lessard, J.-P., Recent advances about the uniqueness of the slowly oscillating periodic solutions of Wright’s equation, J. Differential Equations, 248, 5, 992-1016, (2010) · Zbl 1200.34078 [10] Minamoto, T.; Nakao, M. T., A numerical verification method for a periodic solution of a delay differential equation, J. Comput. Appl. Math., 235, 3, 870-878, (2010) · Zbl 1202.65086 [11] Moore, R. E., Interval analysis, (1966), Prentice-Hall Inc. Englewood Cliffs, N.J. · Zbl 0176.13301 [12] Nussbaum, R., Asymptotic analysis of some functional-differential equations, (Bednarek, A. R.; Cesari, L., Dynamical Systems, II, (1982)), 277-301 [13] Nussbaum, R. D., A global bifurcation theorem with applications to functional differential equations, J. Funct. Anal., 19, 4, 319-338, (1975) · Zbl 0314.47041 [14] Nussbaum, R. D., The range of periods of periodic solutions of $$x^\prime(t) = - \alpha f(x(t - 1))$$, J. Math. Anal. Appl., 58, 2, 280-292, (1977) · Zbl 0359.34066 [15] Ratschek, H.; Rokne, J., New computer methods for global optimization, (1988), Horwood Chichester · Zbl 0648.65049 [16] Regala, B. T., Periodic solutions and stable manifolds of generic delay differential equations, (1989), Brown University, PhD thesis [17] Rump, S. M., Intlab—interval laboratory, (Developments in Reliable Computing, (1999), Springer), 77-104 · Zbl 0949.65046 [18] Rump, S. M., Verification methods: rigorous results using floating-point arithmetic, Acta Numer., 19, 287-449, (2010) · Zbl 1323.65046 [19] Scholz, D., Deterministic global optimization: geometric branch-and-bound methods and their applications, vol. 63, (2011), Springer Science & Business Media [20] Szczelina, R., Rigorous integration of delay differential equations, (2014), Jagiellonian University, PhD thesis [21] Szczelina, R.; Zgliczyński, P., Algorithm for rigorous integration of delay differential equations and the computer-assisted proof of periodic orbits in the MacKey-Glass equation, (2016), arXiv preprint [22] Tucker, W., Validated numerics: A short introduction to rigorous computations, (2011), Princeton University Press Princeton, NJ · Zbl 1231.65077 [23] J.B. van den Berg, J. Jaquette, A proof of Wright’s conjecture, preprint, 2017. · Zbl 1388.34068 [24] Walther, H.-O., A theorem on the amplitudes of periodic solutions of differential delay equations with applications to bifurcation, J. Differential Equations, 29, 3, 396-404, (1978) · Zbl 0354.34074 [25] Wright, E. M., A non-linear difference-differential equation, J. Reine Angew. Math., 194, 1-4, 66-87, (1955) · Zbl 0064.34203 [26] Xie, X., Uniqueness and stability of slowly oscillating periodic solutions of differential delay equations, (1991), Rutgers University, PhD thesis [27] Xie, X., Uniqueness and stability of slowly oscillating periodic solutions of delay equations with unbounded nonlinearity, J. Differential Equations, 103, 2, 350-374, (1993) · Zbl 0780.34054 [28] Jaquette, J.; Lessard, J.-P.; Mischaikow, K., MATLAB codes to perform the computer-assisted proofs, available at
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.