Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-18T05:48:37.408Z Has data issue: false hasContentIssue false
  • English
  • Français

Periodic solutions of planar systems with two delays

Published online by Cambridge University Press:  14 November 2011

Shigui Ruan  and
Junjie Wei
Shigui Ruan
Affiliation:
Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J5
Junjie Wei
Affiliation:
Department of Mathematics, Northeast Normal University, Changchun, Jilin 130024, People's, Republic of China
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we consider a planar system with two delays:

ẋ1(t) = −a0x1(t) + a1F1 (x1(t − τ1), x2(τ−t2)).

ẋ2(t) = −b0x2(t) + b1F2 (x1(t − τ1), x2(t − τxs2)).

Firstly, linearized stability and local Hopf bifurcations are studied. Then, existence conditions for non-constant periodic solutions are derived using degree theory methods. Finally, a simple neural network model with two delays is analysed as an example.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 129 , Issue 5 , 1999 , pp. 1017 - 1032
Copyright
Copyright © Royal Society of Edinburgh 1999

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Baptistini, M. and Táboas, P.. On the existence and global bifurcation of periodic solutions to planar differential delay equations. J. Diff. Eqns 127 (1996), 391–425.CrossRefGoogle Scholar
2Chow, S.-N.. Existence of periodic solutions of autonomous functional differential equations. J. Diff. Eqns 15 (1974). 350–378.CrossRefGoogle Scholar
3Chow, S.-N. and Hale, J. K.. Periodic solutions of autonomous equations. J. Math. Analysis Appl. 66 (1978), 495–506.CrossRefGoogle Scholar
4Chow, S.-N. and Mallet-Paret, J.. The Fuller index and global Hopf bifurcation. J. Diff. Eqns 29 (1978), 66–85.CrossRefGoogle Scholar
5Cooke, K. L. and Grossman, Z.. Discrete delay, distributed delay and stability switches. J. Math. Analysis Appl. 86 (1982), 592–627.CrossRefGoogle Scholar
6Dieudonné, J.. Foundations of modern analysis (New York: Academic Press, 1960).Google Scholar
7Erbe, L. H., Krawcewicz, W., Gęba, K. and Wu, J.. S 1-degree and global Hopf bifurcation theory of functional-differential equations. J. Diff. Eqns 98 (1992), 277–298.CrossRefGoogle Scholar
8Godoy, S. M. S. and Reis, J. G. dos. Stability and existence of periodic solutions of a functional-differential equation. J. Math. Analysis Appl. 198 (1996), 381–398.CrossRefGoogle Scholar
9Gopalsamy, K. and Leung, I.. Delay induced periodicity in a neural netlet of excitation and inhibition. Physica D 89 (1996), 395–426.CrossRefGoogle Scholar
10Grafton, R.. A periodicity theorem for autonomous functional differential equations. J. Diff. Eqns 6 (1969), 87–109.CrossRefGoogle Scholar
11Hadeler, K. P. and Tomiuk, J.. Periodic solutions of differential-difference equations. Arch. Ration. Mech. Analysis 65 (1977). 87–95.CrossRefGoogle Scholar
12Hale, J. K. and Lunel, S. M. V.. Introduction to functional differential equations (New York: Springer 1993).CrossRefGoogle Scholar
13Heiden, U. an der. Periodic solutions of a nonlinear second order differential equation with delay. J. Math. Analysis Appl. 70 (1979), 599–609.CrossRefGoogle Scholar
14Huang, Q., Wei, J., Wu, J. and Zou, X.. Direction and stability of bifurcating periodic solutions in predator–prey systems with discrete delay. In Differential equations and control theory (ed. Deng, Z., Liang, Z., Lu, G. and Ruan, S.). pp. 107–119 (New York: Marcel Dekker, 1996).Google Scholar
15Jones, G.. The existence of periodic solutions of f'(t) = –af (x – 1)[1 + f(x)], J. Math. Analysis Appl. 5 (1962), 435–450.CrossRefGoogle Scholar
16Leung, A.. Periodic solutions for a prey–predator differential delay equation. J. Diff. Eqns 26 (1977), 391–403.CrossRefGoogle Scholar
17Mallet-Paret, J. and Nussbaum, R. D.. Global continuation and asymptotic behavior for periodic solutions of a differential-delay equation. Ann. Math. Pura Appl. 145 (1986), 33–128.CrossRefGoogle Scholar
18Mallet-Paret, J. and Nussbaum, R. D.. A differential-delay equation arising in optics and physiology. SIAM J. Math. Analysis 20 (1989), 249–292.CrossRefGoogle Scholar
19Nussbaum, R. D.. Periodic solutions of some nonlinear autonomous functional differential equations. II. J. Diff. Eqns 14 (1973), 360–394.CrossRefGoogle Scholar
20Nussbaum, R. D.. Periodic solutions of some nonlinear autonomous functional differential equations. Ann. Math. Pura Appl. 101 (1974), 263–306.CrossRefGoogle Scholar
21Nussbaum, R. D.. Differential delay equations with two delays. Mem. Am. Math. Soc. 16 (1978), 1–62.Google Scholar
22Olien, L. and Bélair, J.. Bifurcations, stability, and monotonicity properties of a delayed neural network model. Physica D 102 (1997), 349–363.CrossRefGoogle Scholar
23Táboas, P.. Periodic solutions of a planar delay equation. Proc. R. Soc. Edinb. A 116 (1990), 85–101CrossRefGoogle Scholar
24Wei, J. and Ruan, S.. Stability and bifurcation in a neural network model with two delays. Physica D 130 (1999), 255–272.CrossRefGoogle Scholar
25Wu, J.. Theory and applications of partial functional differential equations (New York: Springer, 1996).CrossRefGoogle Scholar
26Zhao, T., Kuang, Y. and Smith, H. L.. Global existence of periodic solutions in a class of delayed Gause-type predator–prey systems. Nonlinear Analysis 28 (1997), 1373–1394.CrossRefGoogle Scholar