# zbMATH — the first resource for mathematics

A proof of Jones’ conjecture. (English) Zbl 1416.34050
In this paper the author completes a proof of a long standing conjecture that the well-known Wright’s differential delay equation $y^{\prime}(t)=-\alpha y(t-1)\left[1+y(t)\right]\tag{1}$ has a unique slowly oscillating periodic solution for every $$\alpha>\frac\pi2$$. A solution of equation (1) is defined as slowly oscillating if the distance between its consecutive zeros is greater than the delay $$\tau=1$$. The existence of slowly oscillating periodic solutions to equation (1) for all $$\alpha>\frac\pi2$$ is a well-known fact that was established in mid fifties – early sixties through the works of E. M. Wright and G. S. Jones. G. S. Jones had also conjectured that such periodic solution is unique for every $$\alpha$$ [J. Math. Anal. Appl. 5, 435–450 (1962; Zbl 0106.29504)].
The earlier work by X. Xie has in particular established the uniqueness of the slowly oscillating periodic solution for all $$\alpha\in[5.67,\infty)$$ [J. Differ. Equations 103, No. 2, 350–374 (1993; Zbl 0780.34054)]. By using similar ideas and a computer assisted proof J. Jaquette et al. have shown the uniqueness of the slowly oscillating periodic solution for every $$\alpha\in[1.9,6.0]$$ [J. Differ. Equations 263, No. 11, 7263–7286 (2017; Zbl 1382.34069)]. The present paper completes the proof of the conjecture by showing the uniqueness of the periodic solution for the remaining range of the parameter values $$\alpha\in(\frac\pi2, 1.9]$$. The proof is computer assisted through a particular method of interval arithmetic developed for the case of Wright’s equation by the author and others in a series of earlier publications. The paper also shows that Wright’s equation has no isolas of slowly oscillating periodic solutions: every slowly oscillating periodic solution belongs to the primary branch that bifurcates off at $$\alpha=\frac\pi2$$ and is a continuous curve for all $$\alpha>\frac\pi2$$ in the $$(\alpha, z)$$-plane, where $$z>0$$ is the amplitude of the slowly oscillating periodic solution that exists at the parameter value $$\alpha$$.

##### MSC:
 34K13 Periodic solutions to functional-differential equations
##### Software:
CkAnalytic; INTLAB
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] Breden, M.; Lessard, J.-P.; Vanicat, M., Global bifurcation diagrams of steady states of systems of PDEs via rigorous numerics: a 3-component reaction-diffusion system, Acta Appl. Math., 128, 1, 113-152, (2013) · Zbl 1277.65088 [3] Chow, S.-N.; Mallet-Paret, J., Integral averaging and bifurcation, J. Differential Equations, 26, 1, 112-159, (1977) · Zbl 0367.34033 [4] Csendes, T.; Ratz, D., Subdivision direction selection in interval methods for global optimization, SIAM J. Numer. Anal., 34, 3, 922-938, (1997) · Zbl 0873.65063 [5] Day, S.; Kalies, W. D., Rigorous computation of the global dynamics of integrodifference equations with smooth nonlinearities, SIAM J. Numer. Anal., 51, 6, 2957-2983, (2013) · Zbl 1288.37030 [6] Fiedler, B.; Mallet-Paret, J., Connections between Morse sets for delay differential equations, J. Reine Angew. Math., 397, 23-41, (1989) · Zbl 0659.34077 [7] Galias, Z.; Zgliczyński, P., Infinite dimensional Krawczyk operator for finding periodic orbits of discrete dynamical systems, Internat. J. Bifur. Chaos, 17, 12, 4261-4272, (2007) · Zbl 1148.37041 [8] Hansen, E.; Walster, G. W., Global Optimization using Interval Analysis: Revised and Expanded, vol. 264, (2003), CRC Press [9] Jaquette, J., MATLAB code available at: [10] Jaquette, J., Counting and Discounting Slowly Oscillating Periodic Solutions to Wright’s Equation, (2018), Rutgers University, PhD thesis [11] Jaquette, J.; Lessard, J.-P.; Mischaikow, K., Stability and uniqueness of slowly oscillating periodic solutions to Wright’s equation, J. Differential Equations, 263, 11, 7263-7286, (2017) · Zbl 1382.34069 [12] 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 [13] 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 [14] Lessard, J.-P.; Mireles James, J. D., Computer assisted Fourier analysis in sequence spaces of varying regularity, SIAM J. Math. Anal., 49, 1, 530-561, (2017) · Zbl 1362.42007 [15] Mallet-Paret, J., Morse decompositions for delay-differential equations, J. Differential Equations, 72, 2, 270-315, (1988) · Zbl 0648.34082 [16] Mallet-Paret, J.; Sell, G. R., The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay, J. Differential Equations, 125, 2, 441-489, (1996) · Zbl 0849.34056 [17] McCord, C.; Mischaikow, K., On the global dynamics of attractors for scalar delay equations, J. Amer. Math. Soc., 9, 4, 1095-1133, (1996) · Zbl 0861.58023 [18] Moore, R. E., A test for existence of solutions to nonlinear systems, SIAM J. Numer. Anal., 14, 4, 611-615, (1977) · Zbl 0365.65034 [19] Moore, R. E.; Kearfott, R. B.; Cloud, M. J., Introduction to Interval Analysis, (2009), SIAM · Zbl 1168.65002 [20] Neumaier, A., Interval Methods for Systems of Equations, vol. 37, (1990), Cambridge University Press [21] Nussbaum, R., Asymptotic analysis of some functional-differential equations, (Cesari, L.; Bednarek, A. R., Dynamical Systems, II, (1982)), 277-301 [22] Nussbaum, R. D., Periodic solutions of analytic functional differential equations are analytic, Michigan Math. J., 20, 249-255, (1973) · Zbl 0291.34052 [23] Nussbaum, R. D., A global bifurcation theorem with applications to functional differential equations, J. Funct. Anal., 19, 4, 319-338, (1975) · Zbl 0314.47041 [24] Regala, B. T., Periodic Solutions and Stable Manifolds of Generic Delay Differential Equations, (1989), Brown University, PhD thesis [25] Rump, S. M., INTLAB—INTerval LABoratory, (Developments in Reliable Computing, (1999), Springer), 77-104 · Zbl 0949.65046 [26] Schichl, H.; Neumaier, A., Exclusion regions for systems of equations, SIAM J. Numer. Anal., 42, 1, 383-408, (2004) · Zbl 1080.65041 [27] Schwandt, H., Almost globally convergent interval methods for discretizations of nonlinear elliptic partial differential equations, SIAM J. Numer. Anal., 23, 2, 304-324, (1986) · Zbl 0602.65034 [28] van den Berg, J. B.; Jaquette, J., A proof of Wright’s conjecture, J. Differential Equations, 264, 12, 7412-7462, (2018) · Zbl 1388.34068 [29] van den Berg, J. B.; Lessard, J.-P., Chaotic braided solutions via rigorous numerics: chaos in the Swift-Hohenberg equation, SIAM J. Appl. Dyn. Syst., 7, 3, 988-1031, (2008) · Zbl 1408.37062 [30] Wright, E. M., A non-linear difference-differential equation, J. Reine Angew. Math., 194, 1-4, 66-87, (1955) · Zbl 0064.34203 [31] Xie, X., Uniqueness and Stability of Slowly Oscillating Periodic Solutions of Differential Delay Equations, (1991), Rutgers University, PhD thesis [32] 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
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.