The Integrability And Non-integrability Of Several Kinds Of Differential Dynamical Systems

The integrability of differential equations is not only classical,but also modern.Due to its wide and important applications in many fields such as astromechanics,classical mechanics,quantum mechanics and geodesic flow on Riemannian manifolds,it has attracted keen interest of mathematicians and theoretical physicists,and become a frontier and hot topic in scientific research.In this article,the integrability and non-integrability of several kinds of differential dynamical systems are studied by using differential Galois theory and the Darboux theory of integrability.The first chapter is an introduction which summarizes the integrability and non-integrability theory of differential dynamical systems.In Chapter 2,by using Morales-Ramis theory and Kovacic's algorithm,we in-vestigate the non-integrability of some Hamiltonian systems in the sense of Liouville,including the planar Hamiltonian with Nelson potential,double-well potential,and the perturbed elliptic oscillators Hamiltonian.Although these three kinds of systems are simple in structure,they do contain abundant dynamic behavior,and the numer-ical results show that these systems are chaotic in a large range of their parameter.Specifically,the following results are obtained.1.The Nelson system is non-integrable in the sense of Liouville;2.When ???0,the double-well potential Hamiltonian system is non-integrable in the sense of Liouville;3.When ??0,the perturbed elliptic oscillators Hamiltonian system is non-integrable in the sense of Liouville.To some extent these results explain the complexity of dynamic behavior for the systems from the point of view of non-integrability.In Chapter 3,we study the non-integrability of the three-dimensional polynomial system below which contains many classical three-dimensional differential systems.We discuss the existence and nonexistence of analytic first integral,the nonexistence of local C1 first integral,while the rational non-integrability in the sense of Bogoyavlenskij.In Chapter 4,we study the Darboux polynomials of Maxwell-Bloch system By using the theory of weight homogeneous polynomials and the method of character-istic curve,we characterize all the irreducible Darboux polynomials of Maxwell-Bloch system in the cases of zero intensity of field and non-zero intensity of field respectively.Besides that,the polynomial and rational first integrals of Maxwell-Bloch system are obtained.