DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

ON A BIFURCATION DELAY IN DIFFERENTIAL EQUATIONS
WITH A DELAYED TIME 2nπ
K. TCHIZAWA AND R. MIYAZAKI
In the dynamic Hopf bifurcation it is known that the bifurcation delay occurs. In this
paper we will show that the bifurcation delay is persistent under adding a delayed feedback
control term with a delayed time 2nπ (n is any positive integer) if the period of
the Hopf bifurcating solution is 2π. We will also give some numerical simulation results
which suggest that the length of the bifurcation delay is shorter as n increases.
Copyright © 2006 K. Tchizawa and R. Miyazaki. This is an open access article distributed
under the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Neishtadt [4] shows the existence of a delay for the general Hopf bifurcation. Lobry [2]
gives an introductory explanation of the “delay” phenomenon as follows. Consider the
planar system
x 1 ′ = μx (
1 + x 2 − x 1 x
2
1 + x2) 2 ,
x 2 ′ =−x (
1 + μx 2 − x 2 x
2
1 + x2
2 ) (1.1)
which exhibits a supercritical Hopf bifurcation for μ = 0. More precisely, if μ0, the origin becomes unstable and a stable closed orbit surrounding
the origin with a radius ρ = √ μ appears (see Figure 1.1). Then the closed orbit has
aperiod2nπ. By contrast, consider the parameter μ growing slowly with time in system
(1.1),
x ′ 1 = μx 1 + x 2 − x 1
( x
2
1 + x 2 2) ,
x ′ 2 =−x 1 + μx 2 − x 2
( x
2
1 + x 2 2) ,
μ ′ = ε,
(1.2)
where ε is small, and consider the solution which starts from (x 0 , y 0 ,μ 0 ), where x 2 0 + y 2 0 is
very small and μ 0 < 0. Lobry shows that x 2 + y 2 remains close to 0 until μ is positive, and
from the value μ = 0toμ =−μ 0 the solution remains infinitesimal and then departs very
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 1049–1054