%0 Journal Article
%T Vibration, Oscillation, and Escape of the Fiber-Optic Signal under Two-Frequency Perturbations
%A Jiuli Yin
%A Qianqian Xing
%A Liuwei Zhao
%J ISRN Mathematical Physics
%D 2014
%R 10.1155/2014/165250
%X Chaos occurs easily in the nonlinear Schr£¿dinger equation with external perturbations owing to the absence of damping. For the process of information transmission, the perturbation will cause distortion. If we add a suitable controller, it is easy to discover that chaos still appears in the process of propagation of fiber-optic signal when the strength of controller is weak. With the strength of controller increasing, the propagation of fiber-optic signal will arrive at the stable state. As the strength exceeds a certain degree, the propagation of fiber-optic signal system would tend toward the unstable state. Moreover, we consider the parameters¡¯ sensitivity to be controlled. The result demonstrates that the nonlinear term parameter and the two quite different frequencies have less effect on the propagation of fiber-optic signal. Meanwhile, the phenomena of vibration, oscillation, and escape occur in some regions. 1. Introduction The cubic quintic nonlinear Schr£¿dinger (NLS) equation is extensively used in various fields, especially for the process of the fiber-optic signal propagation [1]. Here represents the nondimensional distance along the fiber-optic signal, represents time in a dimensionless form, and and are real valued constants. The dependent variable function is a complex valued function that represents the signal wave profile. In a general way, (1) is a nonlinear partial differential equation, which is completely integrable on the infinite line or periodic boundary conditions in one dimension. As a matter of fact, the propagation of fiber-optic signal must be perturbed with external environment. Extensive systems with external perturbations have been widely investigated by using analytic methods and numerical simulations. A mass of bifurcation sets, the routes to chaos, and Lyapunov exponents are given in [2¨C4]. More attention has been paid to the interaction of external perturbations. The analysis of complex dynamics in three-well forcing or other systems with two external forcings are shown in [5¨C7]. Although these researches have played a certain role in chaos control, there are rarely researches on the fiber-optic signal models with two-frequency perturbations. However, two-frequency perturbations can model the fiber-optic signal under complex external conditions better. Hence, we consider the fiber-optic signal system (1) under two-frequency perturbations where are the amplitudes of the perturbations and are the frequencies; is the velocity of a certain signal. The research purposes of this paper are the two following vital points.
%U http://www.hindawi.com/journals/isrn.mathematical.physics/2014/165250/