The Study Of Global Bifurcation In Nonlinear Systems Based On The Padeapproximation

Homoclinic (heteroclinic) orbits are the basic concept in the research of bifurcation theory and chaos theory. Rupture of homoclinic (heteroclinic) orbits can lead to chaos. The study of the homoclinic (heteroclinic) bifurcation in nonlinear dynamics systems is important for the study of the global behavior of system. In the exiting analysis technique, such as the Melnikov method, which is often used to study the homoclinic (heteroclinic) bifurcation, is common for a limited range, inadequate accuracy and so on. Considering the importance of analytic method in the research of global dynamic behavior in nonlinear system, it is necessary to improve the exiting analysis technique. The Pade approximants are based on the rational function approximants, which have been applied with numerical mathematics, quantum mechanics and control theory. Now by combining this method and the homoclinic (heteroclinic) bifurcation, it is helpful to improve the global dynamics of nonlinear dynamic systems analysis. The major research includes the flowing parts:1） The conventional quasi-Pade approximants were developed. The method was applied for the conservative, autonomous and non-autonomous system with certain strongly cubic nonlinear oscillators, in which the parameters in the front of the linear and nonlinear terms were adopted. The homoclinic (heteroclinic) solutions and the critical parameter value of homoclinic (heteroclinic) bifurcation of those equations were deserved. At the same time, two heteroclinic solutions function were constructed and the new convergence condition was introduced in order to analysis the heteroclinic orbits in the conservative system with certain strongly cubic nonlinear oscillators, and the calculation accuracy was improved and the computation was raduced.2） On the basis of the above-mentioned, and considering the linear and nonlinear parameters, firstly the Pade approximants were used to analysis the global bifurcation problem of the nonlinear oscillators. The rule of selecting the initial point of Pade approximants under the disturbance was proved. Using the conservative and autonomous system with cubic nonlinear term as the research object, homoclinic (heteroclinic) solutions under the direct effect of the unknown parameters and the disturbance parameters were derived, also the bifurcation parameter were successfully obtained. Respectively the characteristics of the conservative and autonomous with certain strongly cubic and square nonlinear oscillators were analyzed. The homoclinic solutions and the critical parameter value of homoclinic bifurcation of the equations were deserved. Simultaneously, it was theoretically proved that using Pade approximants to study the homoclinic (heteroclinic) solutions of the non-autonomous system with certain strongly cubic nonlinear the analytical expression of homoclinic (heteroclinic) orbits was derived.3） The new convergence condition in the process of the calculation was introduced to improve the calculation accuracy. On the other hand, it was proved theoretically that Pade approximants was used to solve the differential equations with the disturbance parameter with no need of limiting the value range of parameter, which is correct and accurate. A new method was presented to solve the differential equations in certain strongly nonlinear systems.