Existence Of Periodic Solutions Of Some Classes Of Differential Equations With Constraint Manifolds

It is well known that different kinds of differential equations are widely used to understand and describe practical problems in varies of fields.The periodicity is one of the ubiquitous phenomena in nature,so the research on periodic solutions of differential equations is always the core problems.In deal with practical problems,the abstracted system often is influenced by some constraint conditions,such as physical laws,actual needs,and implicit constraints.These additional constraints will lead to many difficulties,such as singularity in corresponding differential systems.The existence of periodic solutions of such systems is relatively difficult.This paper is concerned with the existence of the periodic solutions of some classes of differential equations with some constraint conditions.The main research work and innovations are as follows:In Chapter 1,we introduce briefly the historical background of differential equations and some research on the periodic solutions of differential equations on manifolds.The basic research methods and necessary preliminary knowledge are listed.At last,we summarize the main results in the rest of chapters.In Chapter 2,the existence of periodic solutions of a class of differential-algebraic equations is studied.With continuation method and topology degree theory,we present our existence theorem.In terms of prior bound for the solutions of differential-algebraic equations,we give some more convenient conditions with guiding functions established by M.Krasnosel'skii.We also give some easy corollaries to estimate the topology degree.It is worth mentioning that our existence theorems about periodic solutions are independent of index of relevant differential-algebraic equations.Therefore,it is also a valid method to find periodic solutions of relevant differential-algebraic equations with high index in practical application.At last,we formulate some differential-algebraic equations which describe a motion of particle on different constraint surfaces to illustrate our existence results intuitively.In Chapter 3,we consider the existence of periodic solution of differential-algebraic equations with perturbation based on which in Chapter 2.We establish a higher order average principle for perturbed differential-algebraic system.It is sufficient to obtain the existence of periodic solutions that the relevant topological degree is estimated.At the same time,we also establish a easily verifiable corollary that if vector field satisfies some conditions on the boundary,the periodic solution is existent.Furthermore,we present a multi-scale version of average principle for the existence of periodic solutions.It enriches our results and widens the applied range of our theory.In like manner,our existence theorems and corollaries are independent of index of relevant differential-algebraic equations.In the last section,we give a numerical simulation for a motion restricted on the given plane with different perturbation parameters.In Chapter 4,the existence of periodic solutions for a class of second order differential equations with constraint conditions is studied.With continuation method and topology degree theory,we present our existence theorem.When the singularity vanishes,the relevant existence theorem is consistent with the classical theorem established by J.Mawhin.Different from the case in first order,we need to obtain the prior bound of solution and its derivative.Bounding functions established by J.Mawhin and Bernstein-Nagumo lemma are used to prove the existence of prior bounds,respectively.At last,we make a numerical simulation for a motion of a particle on the constraint plane to illustrate our results.At the end of this thesis,a summary and prospect are made to determine the future direction.