Quasi-periodic Solutions Of Josephson System With Two External Forcing Terms

This Josephson equation with two external forcing terms has rarely been studied up to now. But, the effect of the Josephson equation is widely used in many fields, for example, measuring the volatility of the magnetic field of the earth’s surface; measuring the changes of heart and brain magnetic field, and so on.In this thesis, quasi-periodic solutions of Josephson equation with two ex-ternal forcing terms are investigated by using the KAM theory and Newton-type method. Our main work is to discuss if the original system in the vicinity of the origin has a quasi periodic solution.Through a series of reversible transformations, we get a normal form which can be analysed by using KAM theory, and then the normal form is reduced by the Newton method. In the reduction process, we have to solve homological equations. In the iterative procedure there will be small divisor problems. Then the parameters must meet the Diophantine condition and we have to estimate the measure. In this way, the convergence of the iterative transformations implies the existence of quasi-periodic solutions. Finally for most of arameters belonging to the considered parameter set, the normal form exists quasi-periodic solutions near the origin.