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Vibro-impact dynamics near a strong resonance point

Acta Mechanica Sinica 2007年第3期23卷 329-341页

作者： Guanwei Luo Yanlong Zhang Jianhua Xie Jiangang Zhang School of Mechatronic Engineering Lanzhou Jiaotong UniversityLanzhou 730070 China Department of Engineering Mechanics Southwest Jiaotong University Chengdu 610031 China School of Mathematics Physics and Software EngineeringLanzhou Jiaotong University Lanzhou 730070 China

Two typical vibratory systems with impact are considered, one of which is a two-degree-of-freedom vibratory system impacting an unconstrained rigid body, the other impacting a rigid amplitude stop. Such models play an important role in the studies of dynamics of mechanical systems with repeated impacts. Two-parameter bifurcations of fixed points in the vibro-impact systems, associated with 1 ： 4 strong resonance, are analyzed by using the center manifold and normal form method for maps. The single-impact periodic motion and Poincaré map of the vibro-impact systems are derived analytically. Stability and local bifurcations of a single-impact periodic motion are analyzed by using the Poincaré map. A center manifold theorem technique is applied to reduce the Poincaré map to a two-dimensional one, and the normal form map for 1：4 resonance is obtained. Local behavior of two vibro-impact systems, near the bifurcation points for 1：4 resonance, are studied. Near the bifurcation point for 1：4 strong resonance there exist a Neimark-Sacker bifurcation of period one single-impact motion and a tangent （fold） bifurcation of period 4 four-impact motion, etc. The results from simulation show some interesting features of dynamics of the vibro-impact systems： namely, the ＂heteroclinic＂ circle formed by coinciding stable and unstable separatrices of saddles, Tin, Ton and Tout type tangent （fold） bifurcations, quasi-periodic impact orbits associated with period four four-impact and period eight eight-impact motions, etc. Different routes of period 4 four-impact motion to chaos are obtained by numerical simulation, in which the vibro-impact systems exhibit very complicated quasiperiodic impact motions.

关键词： Vibration Impact Resonance periodicmotion Bifurcation

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Competition mechanism of multiple four-wave mixing in highly nonlinear fiber： spatial instability and satellite characteristics

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Frontiers of Optoelectronics 2012年第4期5卷 414-428页

作者： Liang ZHAO Junqiang SUN Xinliang ZHANG Cong CHEN Wuhan Foreign Languages School Wuhan 430022China Wuhan National Laboratory for Optoelectronics School of Optoelectronic Science and EngineeringHuazhong University of Science and TechnologyWuhan 430074China Naval University of Engineering Wuhan 430033China

Competition mechanism in multiple four-wave mixing （MFWM） processes is demonstrated theoretically. Provided considering only two waves injected into a highly nonlinear fiber （HNLF）, there are three modes displaying comprehensive dynamic behaviors, such as fixed points, periodic motion, and chaotic motion. Especially, Mode C of MFWM is emphasized by analyzing its phase-space trajectory to demonstrate nonlinear wave- wave interactions. The study shows that, when the phase- space trajectory approaches or gets through a saddle point, a dramatic power depletion for the injected wave can be realized, with the representative point moving chaotically, but when phase-space trajectories are distributed around a center point, the power for the injected wave is retained almost invariable, with the representative point moving periodically. Finally, the evolvement of satellite wave over an optical fiber is investigated by comparing it with the interference pattern in Young＇s double-slit experiment.

关键词： highly nonlinear fiber (HNLF) periodicmotion representative point Young's double-slit experi-ment

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