On The Application Of Melnikov Method To Determinting Edge Of Chaos Of Mechanical Dynamic System With Muti-Parameters

As a new discipline, the study of chaos has gotten a lot of achievements and great progress since 1960s. It’s becoming interdisciplinary science with other traditional disciplines and has been attracting many attentions. The affects of chaos in many engineering field.can not be ignored. Edge of chaos is a state between periodic and chaotic motion and it’s also the source of complicated behaviors in dynamic system. Therefore, researching on the edge of chaos is helpful to find and grasp the mechanism of chaos arasing, and it’s of great practical and academic significance to the development of chaos science.Currently, most studies on the edge of chaos are limited to numerical simulation and one-parameter; few of them discuss the problem with analytic method for dynamic system with multiparameters. Melnikov method, as a very practical analytical method for studying chaos in a dynamic system, is rarely applied to the edge of chaos although it has got great developments and improvements. In order to explore the application of Melnikov method to determining the dege of chaos in mechanical dynamic system with multiparameter and to lay a firm foundation for the study of chaos arising, three mechanical dynamic systems with similar dynamic equation, Duffing system, nonlinear pendulum system and micro-cantilever dynamic system, are studyed in this paper, and the main research work is as following.1. The theory about homo clinic orbit and heteroclinic orbit, basic properties of Hamiltonian system and Melnikov method are presented to provide the theoretic foundation for the researches of this thesis.2. Based upon the fact that Duffing system, nonlinear pendulum system and micro-cantilever dynamic system all are Hamiltonian system with micro-perturbation, Melnikov method is employed to determine the edge of chaos for these dynamic systems. The relationships between the parameters of the system for the edge of chaos are found out, and a curve near to the edge of chaos can be drawn. All thses analytical solutions on the edge of chaos are verified by the results from simulation of the system. Additionally, some instructive conclusions on the design of the system can be drawn.3. A method to deal with the difficulty in solving the equations of homoclinic orbit and heteroclinic orbit and the intergral of Melnikov function is proposed. It utilized the simulink module in MATLAB. Comparing between the numerical solution and analytic solutions is made, and the results show that they are in good agreement in the permitted range of errors.As a concolusion, Melnikov method is a very practical analytic approach for detreming the edge of chaos of multiparameter in Hamiltonian systems with micro-perturbation. The difficulty in the intergral of Melnikov function can be overcome by utilizing the numerical integration. This concolusion should make some contributions to the further study on the mechanism and law of chaos arising and controlling of chaos.