Differential Galois Theory And Non-integrability Of Nonlinear Dynamical Systems

In the late 1980 s,Picard and Vessiot extended Galois theory of the algebraic equations to the homogeneous linear differential equations,and established the socalled differential Galois theory.In the 1990 s,based on the differential Galois theory and Ziglin theory,Morales-Ruiz and Ramis et.al presented a criteria to determine the non-integrability of analytic Hamiltonian systems,and obtained many significant results.In this thesis,we will use the differential Galois theory to study the integrability and non-integrability of non-linear dynamical systems,discuss the relationship between the non-integrability of dynamical systems and complex behaviors,the integrability of dynamical systems and the weak Painlev?e property.Full text chapters contains five chapters.In Chapter 2,we introduce two different definitions of local first integrals for stochastic differential equations,and provide equivalent algebraic characterizations of them,respectively.Moreover,we present the necessary conditions for the existence of local first integrals for stochastic differential equations,which can be regarded as an extension of the Poincar?e non-integrability theorem from ordinary differential equations to stochastic differential equations.Some examples are given to illustrate our results.In Chapter 3,we apply the differential Galoisian approach to non-integrability of a class of 3D equations in mathematical physics,including the Lorenz equations,the Shimizu-Morioka system and the generalized Rikitake system.Our main results show that all these considered systems are,in fact,non-integrable in nearly all parameters,which are in accord with that these systems admit chaotic behaviors for a large range of their parameters.Consider the Lorenz system(?)When(?) and (?),(?) arbitrary,it is completely integrable with two functional independent first integrals [J.Phys.A.38(2005)2681–2686].We study the integrability of the Lorenz system when(?)take the remaining values.For the case of (?),we show that it dose not admit any meromorphic first integrals if (?) is not an odd number.For the case of (?),we present necessary conditions of the Lorenz system processing formal first integrals or time-dependent formal first integral in the form of(?),respectively.Consider the Shimizu-Morioka system(?)Firstly,we propose a linear scaling in time and coordinates which converts the ShimizuMorioka system into a special case of the Rucklidge system when (?),and discuss the relationship between the Shimizu-Morioka system and the Rucklidge system.Secondly,based on this observation,Darboux integrability of the Shimizu-Morioka system with(?)is trivially derived from the corresponding results on the Rucklidge system.When (?),we investigate Darboux integrability by using the Gr¨obner basis in algebraic geometry.Finally,in the case (?),we prove it is not rationally integrable for almost all parameter values and in the case (?),we show that it is not algebraically integrable.Consider the generalized Rikitake system(?)For the integrable case,we derive a family of integrable deformations of the generalized Rikitake system by altering its constants of motion,and give two classes of HamiltonPoisson structures which implies these integrable deformations,including the generalized Rikitake system,are bi-Hamiltonian and have infinitely many Hamilton-Poisson realizations.We show that the generalized Rikitake system is not rationally integrable in an extended Liouville sense for almost all parameter values and also discuss the non-existence of analytic first integrals.In Chapter 4,we present some necessary conditions for quasihomogeneous systems to be completely integrable via Kowalevski exponents.Then,as an application,we partially solve a conjecture by Goriely [J.Math.Phys.37(1996),1871-1893].Additionally,we show that in order that a homogeneous Newton system is integrable in the generalized Liouville sense,all possible Kowalevski's exponents must be rational number.In Chapter 5,we prove that if an 9)-dimensional divergence-free system has 9)-2functionally independent meromorphic first integrals,then the identity component of the differential Galois group of the variational equation along a particular solution is solvable.Based on this result,we study the polynomial first integrals or polynomial integrability of Karabut system.The Karabut system is introduced to find the Witting series which characterizes the solitary wave in a fluid of finite depth.We show that the 5-dimensional Karabut system has two and only two functionally independent polynomial first integrals,which improves the result in [Nonlinear Anal.32(2016)91–97] from the point of view of partial integrability.