Slowly oscillating periodic solutions for a nonlinear second order differential equation with state-dependent delay

Zhang, Ling; Guo, Shangjiang

doi:10.1090/proc/13714

Slowly oscillating periodic solutions for a nonlinear second order differential equation with state-dependent delay
HTML articles powered by AMS MathViewer

by Ling Zhang and Shangjiang Guo

Proc. Amer. Math. Soc. 145 (2017), 4893-4903

DOI: https://doi.org/10.1090/proc/13714

Published electronically: June 22, 2017

PDF | Request permission

Abstract:

In this paper, a second order differential equation with state-dependent delay is investigated. The existence of slowly oscillating periodic solutions is established by using Browder’s theorem on the existence of a non-ejective fixed point.

References

Wolfgang Alt, Periodic solutions of some autonomous differential equations with variable time delay, Functional differential equations and approximation of fixed points (Proc. Summer School and Conf., Univ. Bonn, Bonn, 1978) Lecture Notes in Math., vol. 730, Springer, Berlin, 1979, pp. 16–31. MR 547978
U. an der Heiden, Periodic solutions of a nonlinear second-order differential equation with delay, J. Math. Anal. Appl. 70 (1979), no. 2, 599–609. MR 543597, DOI 10.1016/0022-247X(79)90068-4
U. an der Heiden, A. Longtin, M. C. Mackey, J. G. Milton, and R. Scholl, Oscillatory modes in a nonlinear second-order differential equation with delay, J. Dynam. Differential Equations 2 (1990), no. 4, 423–449. MR 1073472, DOI 10.1007/BF01054042
O. Arino, K. P. Hadeler, and M. L. Hbid, Existence of periodic solutions for delay differential equations with state dependent delay, J. Differential Equations 144 (1998), no. 2, 263–301. MR 1616960, DOI 10.1006/jdeq.1997.3378
S. J. Bhatt and C. S. Hsu, Stability criteria for second-order dynamical systems with time lag, Trans. ASME Ser. E. J. Appl. Mech. 33 (1966), 113–118. MR 207245, DOI 10.1115/1.3624967
Eugene Boe and Hsueh-Chia Chang, Transition to chaos from a two-torus in a delayed feedback system, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 1 (1991), no. 1, 67–81. MR 1104542, DOI 10.1142/S0218127491000063
F. G. Boese and P. van den Driessche, Stability with respect to the delay in a class of differential-delay equations, Canad. Appl. Math. Quart. 2 (1994), no. 2, 151–175. MR 1285907
Felix E. Browder, A further generalization of the Schauder fixed point theorem, Duke Math. J. 32 (1965), 575–578. MR 203719
Sue Ann Campbell, Jacques Bélair, Toru Ohira, and John Milton, Limit cycles, tori, and complex dynamics in a second-order differential equation with delayed negative feedback, J. Dynam. Differential Equations 7 (1995), no. 1, 213–236. MR 1321711, DOI 10.1007/BF02218819
J. Chuma and P. van den Driessche, A general second-order transcendental equation, Appl. Math. Notes 5 (1980), no. 3-4, 85–96. MR 592643
Kenneth L. Cooke and Zvi Grossman, Discrete delay, distributed delay and stability switches, J. Math. Anal. Appl. 86 (1982), no. 2, 592–627. MR 652197, DOI 10.1016/0022-247X(82)90243-8
R. B. Grafton, A periodicity theorem for autonomous functional differential equations, J. Differential Equations 6 (1969), 87–109. MR 243176, DOI 10.1016/0022-0396(69)90119-3
Aiyu Hou and Shangjiang Guo, Stability and Hopf bifurcation in van der Pol oscillators with state-dependent delayed feedback, Nonlinear Dynam. 79 (2015), no. 4, 2407–2419. MR 3317452, DOI 10.1007/s11071-014-1821-3
Qingwen Hu and Jianhong Wu, Global Hopf bifurcation for differential equations with state-dependent delay, J. Differential Equations 248 (2010), no. 12, 2801–2840. MR 2644135, DOI 10.1016/j.jde.2010.03.020
G. Stephen Jones, The existence of periodic solutions of $f^{\prime } (x)=-\alpha f(x-1)\{1+f(x)\}$, J. Math. Anal. Appl. 5 (1962), 435–450. MR 141837, DOI 10.1016/0022-247X(62)90017-3
Y. Kuang and H. L. Smith, Slowly oscillating periodic solutions of autonomous state-dependent delay equations, Nonlinear Anal. 19 (1992), no. 9, 855–872. MR 1190871, DOI 10.1016/0362-546X(92)90055-J
N. MacDonald, Biological delay systems: linear stability theory, Cambridge Studies in Mathematical Biology, vol. 8, Cambridge University Press, Cambridge, 1989. MR 996637
P. Magal and O. Arino, Existence of periodic solutions for a state dependent delay differential equation, J. Differential Equations 165 (2000), no. 1, 61–95. MR 1771789, DOI 10.1006/jdeq.1999.3759
John Mallet-Paret, Roger D. Nussbaum, and Panagiotis Paraskevopoulos, Periodic solutions for functional-differential equations with multiple state-dependent time lags, Topol. Methods Nonlinear Anal. 3 (1994), no. 1, 101–162. MR 1272890, DOI 10.12775/TMNA.1994.006
Roger D. Nussbaum, Periodic solutions of some nonlinear, autonomous functional differential equations. II, J. Differential Equations 14 (1973), 360–394. MR 372370, DOI 10.1016/0022-0396(73)90053-3
Roger D. Nussbaum, Periodic solutions of some nonlinear autonomous functional differential equations, Ann. Mat. Pura Appl. (4) 101 (1974), 263–306. MR 361372, DOI 10.1007/BF02417109
E. M. Wright, A non-linear difference-differential equation, J. Reine Angew. Math. 194 (1955), 66–87. MR 72363, DOI 10.1515/crll.1955.194.66
Ling Zhang and Shangjiang Guo, Hopf bifurcation in delayed van der Pol oscillators, Nonlinear Dynam. 71 (2013), no. 3, 555–568. MR 3015261, DOI 10.1007/s11071-012-0681-y