The Research On Analytical Approximate Method In Solving Strongly Nonlinear Oscillator And Its Application In Infinite-dimensional Dynamical System

With the development of science and technology and the emergence of new materials and structure, mechanical systems are becoming increasingly complex. The requirements of vibration isolation for precision laboratory are higher than before. The anti vibration design of high-rise buildings, large-span spatial structures and bridges becomes more and more complicated. The wind resistant design of cable membrane structure becomes more and more difficult. The above-mentioned problems make the researches of nonlinear vibration are becoming increasingly important. In these few decades, many reseachers focus their attentions on study new theorems and methods of nonlinear vibration. However, because of the characteristics of nonlinear differential equations, a method which can obtain exact solutions of all kinds of nonlinear differential equations is still absent. This makes the quantitative methods of nonlinear differential equations have the traits of diversity and restriction simultaneously. Thus, the studies on innovation and modification the methods in solving nonlinear differential equations are still the hot topic. This paper emphasizes on this hot topic, and proposes two kinds of modified methods. One of them is called the quadratic generalized harmonic function perturbation method, the other is generalized Padé approximation and its derivative methods. The main work of this paper can be summarized as follows.First, the research situation of bifurcations and quantitative methods of nonlinear dynamics are introduced.Second, by constructing a novel description of solutions of nonlinear oscillators and simplifying the solving procedure, a quadratic generalized harmonic function perturbation method which can be considered as a modified method of generalized harmonic function Lindstedt-Poincaré method is proposed. Via this method, the homoclinic and heteroclinic bifurcations of Helmholtz-Duffing oscillator and Duffing-Harmonic-van de Pol oscillator are investigated. The critical values of the homoclinic and heteroclinic bifurcation parameters are predicted. Meanwhile, the analytical solutions of limit cycle and homo-heteroclinic orbits of these oscillators are also attained. To illustrate the accuracy of the present method, all the abovementioned results are compared with those of Runge-Kutta method, which shows that the proposed method is effective and feasible.Third, based on the classical Padé approximation method, a generalized Padé approximation method is proposed. Two novel kinds of generalized Padé approximants are constructed for determining homo-heteroclinic solutions and periodic solutions of nonlinear oscillators. Via the proposed method, the homoheteroclinic solutions and periodic solutions of strongly nonlinear autonomous oscillator with high order polynomial function, rational function and irrational function potential are obtained. This method spreads the scope of classical Padé approximate method, and also provides a novel idea of applying Padé approximate method in studying nonlinear vibration. In addition, The precision of the solutions is high when the nonlinear parameters or initial amplitude are large. Both theriotical analysis and calculation indicate that the proposed method is not restricted to solving some certain systems. It can be utilized in many kinds of systems, which means that the proposed method is generally applicable. So the investigation in generalized Padé approximation is meaningful.Fourth, by combining the generalized Padé approximation method and the wellknown Lindstedt-Poincaré method, a novel technique, referred to as the generalized Padé-Lindstedt-Poincaré method, is proposed for determining homo-heteroclinic orbits of nonlinear autonomous oscillators. The accucacy of using generalized Padé approximation method to solve self-excited oscillator is insufficient. Meanwhile, the elliptic function perturbation methods are restricted in solving some certain strongly nonliear oscillators. Furthemore, the solutions obtain from generalized harmonic function perturbation methods are implicit solution. In order to make up for the deficiencies of the above-mentioned methods, a combining method, the generalized Padé-Lindstedt-Poincaré method, is proposed. Based on this method, the oscillators with high order polynomial function and rational function potential are investigated. The homo-heteroclinic bifurcations of these oscillators are predicted. Meanwhile, the analytical solutions to these oscillators are also calculated. To illustrate the accuracy of the present method, the solutions obtained in this paper are compared with those of the Runge-Kutta method, which shows the method proposed in this paper is both effective and feasible. Thus the proposed method can be considered to be a supplement of the perturbation-based method.Fifth, the generalized Padé approximation method and the quadratic generalized harmonic function perturbation method are applied in solve some classes of Infinitedimensional dynamical systems. Based on the generalized Padé approximation method, the solitary wave solutions of modified Zakharov-Kuznetso equation, the generalized Pochhammer-Chree equation and the generalized Drinfeld-Sokolov equation are obtained. Based on the quadratic generalized harmonic function perturbation method, the analytical limit cycle solution of a model for biological invasions is obtained. Meanwhile, the relationship between initial value of limit cycle and the control parameter are attained. Based on this relationship, the initial values of limit cycle with different parameters are predicted. The comparisons between our results and the numerical results show that the proposed method is feasible and reliable.