Analysis And Application Of The Normal Form In Nonlinear Dynamical System

The normal form theory is one of the useful tools in the fields of dynamical system, ordinary differential equations and nonlinear vibration. Normal form theory plays an important role in the study of dynamical behavior of nonlinear systems near the dynamic equilibrium points because it greatly simplifies the analysis and formulations. This simple form can be used conveniently for analyzing the dynamical behavior of the original system in the vicinity of equilibrium. However, it is not a simple task to calculate the normal form for some given ordinary differential equations. The normal forms of Hopf and generalized Hopf bifurcations, as well as their applications have been extensively studied by many researchers. The research contents and the innovative contributions of this dissertation are as follows:(1) The Hopf Bifurcations have been studied by normal form theory and the matrix representation method of dynamic system. On the basis of normal form theory, the Hopf bifurcation systems are further simplified to the simplest normal forms. The normal forms of generalized Hopf bifurcations have been extensively studied. Theorems are presented to show that the conventional normal form of generalized Hopf bifurcations is further simplified to the simplest normal form there are at most two terms remaining in the amplitude equation of the simplest normal form up to 2k+1 order. There are two kinds of the simplest normal forms.(2) The normal forms of generalized Neimark-Sacker bifurcations have been extensively studied using normal form theory of dynamic system. In this paper, five theorems are presented to show that the conventional Neimark-Sacker bifurcations can be further simplified. We calculate the simplest normal forms of generalized Neimark-Sacker bifurcations. There are two kinds of the simplest normal forms. These algebra expression formulas are given.(3) Study Hopf bifurcations by normal form theory in the Furuta pendulum system. We calculate the normal forms of the Hopf bifurcation systems. The stability of the limit cycle is discussed. The space trajectories are investigated via numerical simulation, which are aslo verified the validity of our analysis. Based on the Silnikov criterion, the chaotic characters of the dynamical systems are discussed. Using Cardano formula and series solution of differential equation, eigenvalue problem and the existence of homoclinic orbit are studied respectively. Furthermore, a rigorous proof for the existence of Silnikov-sense Smale horseshoes chaos is presented and some conditions which lead to the chaos are obtained.(4) Study a new three-dimensional continuous autonomous chaotic system. The new Van del Pol Jerk system contains a cubic terms and six system parameters. Basic dynamic properties of the new system are studied by means of theoretical analysis and numerical simulation. Based on the Silnikov criterion, the chaotic characters of the dynamical systems are discussed. Using Cardano formula and series solution of differential equation, eigenvalue problem and the existence of homoclinic orbit are studied respectively. Furthermore, a rigorous proof for the existence of Silnikov-sense Smale horseshoes chaos is presented and some conditions which lead to the chaos are obtained. The formation mechanism shows that this chaotic system has impulsive homoclinic chaos, and numerical simulation demonstrates that there is a route to chaos.(5) Study a class of multimolecules saturated reaction model. By using the qualitative theory of ordinary differential equations, completely discuss the existence, nonexistence and uniqueness of limit cycles of the system. By using the normal form theory, the Hopf bifurcation behavior of the dynamic system is exploited. We compared qualitative property with the system with saturated reaction speed.