Affine-periodic Solutions For Perturbed Systems

Perturbation methods for differential equations became important when scientists in the 18th century were trying to use Newton 's theory of gravitation to observe the motion of planets.To explain the deviations one considered effects as the influence of satellites and large planets.These considerations led to the formulation of perturbed two-body motion.Since Lagrange[39]and Laplace[40]use the averaging method,it is always a powerful tool to study the perturbed periodic systems.However,many phenomena have the symmetry of space in addition to the periodicity of time.In this thesis,we will study a class of systems with both time periodicity and space symmetry,and we call them the affine periodic ones.For a periodic differential equation,a natural problem is to find periodic solutions,correspondingly,the concern of this dissertation is whether there exists a solution with the same symmetric structure in an affine periodic system which is called an affine periodic solution.In the first chapter,we briefly introduce the origin and development of the perturbed system and the averaging method,and give the results of using the higher order averaging method to find the periodic solutions in recent years.We introduce Mawhin's coincidence degree theory and Krasnosel'skii-Perov's existence theorem which is an interesting existence theorem of periodic solutions obtained by topological theory.At the end of Chapter one,we give the definition of the affine periodic system and introduce some relevant works and our main results.In the second chapter,we give the averaging method of perturbed affine periodic systems.For a perturbed periodic system,if the averaging function of the vector field has a non-degenerate singular point,the system will have a peri-odic solution.This is the classical first-order averaging method.For perturbed affine periodic systems,we find that the existence of the periodic solution is related to the properties of the affine matrix.If the difference between the i-dentity matrix and the affine matrix is invertible,when the parameter is small enough,the system naturally has an affine periodic solution.Since the affine matrix of periodic system is the identity matrix,this property is not reflected in the periodic case.When the difference between the identity matrix and the affine matrix is not invertible,we give the first-order averaging method and the higher order averaging method respectively.Consider the projection of the averaging function for the first order perturbed field in the kernel space of the difference between the identity matrix and the affine matrix,if the topological degree is not zero,the system will also have affine periodic solutions.This can be regarded as a natural generalization of the averaging method for classical periodic systems,since the results are consistent with the first order averaging method of periodic systems when the affine matrix is equal to the identity matrix.Compared to the first-order perturbed systems,besides nonzero topologi-cal degree,we also need some additional properties of the perturbed functions for higher order case.Because that the method we use is based on the co-incidence degree theory of Mawhin,we will use the homotopy invariance of topological degree theory,and the value of the homotopy map on the bound-ary should not be zero.According to the different properties of the functions,we give two different results.Compared with the existing results,our results are completely new,even in the periodic case.Recently,the higher order av-eraging principle for periodic system is obtained by using Poincare's method.Expanding the solution with the small parameter and finding the fixed point of Poincare's map by using the topological degree theory of perturbed functions,then the periodic solution of the system is obtained.Compared with it,we need some additional conditions on the vector fields.However,we only need lower smooth properties of the system,and the averaging function we use is just the sum of each order.Instead of calculating solutions for higher order variational equations,our method is much easier.In the third chapter,we give the Krasnosel'skii-Perov's existence theorem for affine periodic systems.In the 50s and 60s of last century,Krasnosel' skii and Perov[35,36]considered the periodic systems on a bounded region.They proved that if the solutions that start from the boundary will not return to the initial point during a periodic time and the topological degree of the vector field at zero time is not equal to zero,the system will have a periodic solution.We extend the results to affine periodic systems from two aspects.Firstly,we use the idea of coincidence degree to reduce it to the kernel space of the difference between the identity matrix and the affine matrix.We prove that after a transformation,the solution with the initial point whose projection on the kernel space is on the boundary will not return during a periodic time,and the solution which returns in a periodic time can not reach the boundary.Moreover,the topological degree of the vector field at zero time is not equal to zero,then the system will have an affine periodic solution.In the second result,we consider the system on the whole space.We prove that if there exists a continuous matrix function,the value of which is exactly the affine matrix at T moment,and the difference between the value at zero time and the identity matrix is invertible.Moreover,if the solution starts from the boundary does not return in a periodic time after a transformation by the matrix function at the same time,then the system will have an affine periodic solution.The condition of Krasnosel' skii-Perov's existence theorem is usually diffi-cult to verify in practice.In the second section of the third chapter,we give a result for perturbed systems that the condition is relatively easy to test.