Integrability Of Some Three-dimensional Nonlinear Dynamical Systems

The integrability of nonlinear differential equations is a fundamental problem with a long history.The development of the integrable theory of differential equations can help us characterize the topological structure and long time dynamic behavior,and further understand the dynamic mechanism of some important models in the fields of astronomy,physics,economy,and biology.The integrability theory of planar dynamical systems has been widely studied,but the integrability analysis of highdimensional dynamical systems still has great research difficulties,and there are many open problems that are difficult to solve so far.In this work,by using various theoretical methods such as differential Galois theory,Darboux integrability theory,computational algebra and so on,we study integrability and nonintegrability of several important physical models,and try to further develop the integrability of differential equations.The first chapter is an introduction,which gives some background,basic concepts and results related to integrability of differential dynamical systems.In Chapter 2,we are concerned with the segmented disc dynamos with or without friction,which are two basic models describing the self-excitation of a magnetic field and are used to understand the generation of magnetic fields and the reversals in astrophysical bodies.This paper is devoted to giving a contribution to the understanding of their complexity by studying the integrability problem.We first show that,generically,there exists a linear transformation to covert the segmented disc dynamo with friction(SDDF)into the Lorenz system,which yields that the integrability of SDDF can be obtained from the well-known results of the Lorenz system.The existence or non-existence of analytic,polynomial,rational,Darboux,C1 first integrals for both segmented disc dynamo without friction(SDD)and SDDF are discussed.Nonintegrability of SDD and SDDF is also discussed by the differential Galois method without assuming its closeness to an integrable system.To this end,some general non-integrability results are given,which can also be applied to study non-integrability of other three dimensional differential systems.In Chapter 3,we study the integrability and non-integrability of both the NoseHoover oscillator(NH)and the generalized Nose-Hoover oscillator(GNH).The NoseHoover system is a basic primitive model for the molecular dynamics simulations,which describes the equilibrium characterized by canonical distributions at a constant temperature.Its simplest form is the so-called NH model,which is a three-dimensional quadratic polynomial system that admits both regular invariant tori and chaotic trajectories.GNH model is constructed to show the coexistence of invariant tori and topological horseshoe for time-reversible systems in R3.This paper aims to study the integrability of both NH and GNH models.We show that in the case of α=0,both NH and GNH models are integrable by quadratures and the general solutions are given;in the case of α≠0 the non-existence of either global analytic first integrals or Darboux first integrals of two models are discussed and a characterization of Darboux polynomials and exponential factors are given.Both NH and GNH models are not rationally integrable in an extended Liouville sense except for several parameter values by analyzing properties of differential Galois groups.In addition,we also show that the reduced systems of NH and GNH models at the Poincare compactification balls are integrable,which yields complete descriptions of dynamics at infinity for NH and GNH models.In Chapter 4,we propose a computational algebraic method for efficiently computing Darboux polynomials for three-dimensional nonlinear polynomial systems.The method mainly consists of three steps:reduce the systematic parameters and the cofactors;find the compatible condition for systematic parameters,and obtain the values of coefficients of the co-factor;solve the linear system for the coefficients of the Darboux polynomial.As applications,we have discovered new Darboux polynomials for some important physical models,including the Moon-Rand system,Rabinovich-Fabrikant system and Stretch-twist-fold flow.