This paper presents an investigation on the phenomenon of delayed bifurcation in time-delayed slow-fast differential systems.Here the two delayed's have different meanings. The delayed bifurcation means that the bifurcation does not happen immediately at the bifurcation point as the bifurcation parameter passes through some bifurcation point, but at some other point which is above the bifurcation point by an obvious distance. In a time-delayed system, the evolution of the system depends not only on the present state but also on past states. In this paper, the time-delayed slow-fast system is firstly simplified to a slow-fast system without time delay by means of the center manifold reduction, and then the so-called entry-exit function is defined to characterize the delayed bifurcation on the basis of Neishtadt's theory. It shows that delayed Hopf bifurcation exists in time-delayed slow-fast systems, and the theoretical prediction on the exit-point is in good agreement with the numerical calculation, as illustrated in the two illustrative examples.
time delay, delayed bifurcation, Hopf bifurcation, slow-fast systems, exit-point, entry-exit function
Citation:Zheng Y G, Wang Z H. Delayed Hopf bifurcation in time-delayed slow-fast systems.
Boron is selectively implanted on the surface of an n-type silicon wafer to form a p-type area surrounded by an n-type area. The wafer is then put into a buffered oxide etch solution. It is found that the n-type area can be selectively etched without illumination, with an etching rate lower than 1 nm min(-1), while the p-type area can be selectively etched under illumination with a much higher etching rate. The possible mechanism of the etching phenomenon is discussed. A simple fabrication process of silicon nanowires is proposed according to the above phenomenon. In this process only traditional micro-electromechanical system technology is used. Dimensions of the fabricated nanowire can be controlled well. A 50 nm wide and 50 nm thick silicon nanowire has been formed using this method.
The local dynamics around the trivial solution of an optoelectronic time-delay feedback system is investigated in the paper, and the effect of the feedback strength on the stability is addressed. The linear stability analysis shows that as the feedback strength varies, the system undergoes exactly two times of stability switch from a stable status to an unstable status or vice versa, and at each of the two end points of the stable interval, a Hopf bifurcation occurs. To gain insight of the bifurcated periodic solution, the Lindstedt-Poincaré method that involves easy computation, rather than the center manifold reduction that involves a great deal of tedious computation as done in the literature, is used to calculate the bifurcated periodic solution, and to determine the direction of the bifurcation. Two case studies are made to demonstrate the efficiency of the method.
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