Simplest Normal Form And Study On Bifurcation And Chaos Of Electromechanical Coupled Nonlinear Dynamical System

Normal form theory is one of the useful tools in the study of nonlinear dynamics and stresses a profound influence on complex dynamic theory such as bifurcation and chaos dynamics. The simplest normal form is the normal form that can be used to deeply simplify the original differential equations based on conventional normal form and nonlinear transformation theory. The study and the application of the simplest normal form are developing to high-dimension. The computation of the simplest normal form is very complicated. But, nonlinear dynamical systems can be simplified by the simplest normal form method and the nonlinear dynamical behavior of these systems near critical equilibrium can be obtained more easily. Aiming at the simplest normal form and the dynamical behavior of the electromechanical coupled nonlinear dynamical system, the research contents and the innovative contributions of this dissertation are as follows(1) The coefficients of the simplest normal forms of high-dimensional generalized Hopf and high-dimensional Hopf bifurcations systems are discussed using the adjoint operator method. A particular nonlinear scaling and an inner product are introduced and the central manifold equations are simplified. Theorems are established for the explicit expression of the simplest normal forms in terms of the coefficients of the conventional normal forms of Hopf and generalized Hopf bifurcations systems. Symbolic program is designed to perform the calculation of the coefficients of the simplest normal forms using Mathematica. The original ordinary differential equation is required in the input and the simplest normal form can be obtained as the output. Finally, the simplest normal form of 6-dimensional generalized Hopf and 5-dimensional Hopf bifurcation system are discussed by executing the program. The outputs show that the 5th-order and 9th-order terms remain in 6-dimensional generalized Hopf and the 3rd-order and 5th-order terms remain in 5-dimensional Hopf bifurcation system.(2) The coefficients of the simplest normal forms of both high-dimensional Neimark-Sacker and generalized Neimark-Sacker bifurcation systems are discussed. On the basis of the simplest normal form theory, using appropriate nonlinear transformations, direct computation and the second complete induction, the central manifold equations are further reduced to the simplest normal forms which only contain two nonlinear terms. Theorems are established for the explicit expression of the simplest normal forms in terms of the coefficients of the conventional normal forms of Neimark-Sacker and generalized Neimark-Sacker bifurcations systems.(3) Applying a reversible linear transformation and a near-identity transformation, the simplest normal forms for high-dimensional nonlinear dynamical system is studied without calculating its traditional normal form. Using a reversible linear transformation, the matrix of the linear part for the nonlinear dynamical system is topologically equivalent to the block diagonal matrix that adapts to the demand of the practical research: companion matrixes distribute on the diagonal line and the remaining elements are zero. In order to obtain the simplest normal form, we use lower order nonlinear terms in the normal form for the simplifications of higher order terms. In the simplest normal form, the nonlinear coefficient matrix contains non-zero elements only in the row corresponding to the last row of each companion matrix and zero elements in the remaining rows. The general program with the Mathematica language is provided, which can compute the simplest normal form of an arbitrary nonlinear dynamical system easily. For nonlinear dynamical systems of 2-dimensional, 3-dimensional, 4-dimensional, 6-dimensional and 7-dimension, the simplest normal forms up to order 4 are discussed by executing the program.(4) Applying a reversible linear transformation and a near-identity nonlinear transformation with small parameters, the calculating method according to the simplest normal forms of nonlinear dynamical systems with perturbation parameters is obtained. The general program with the Mathematica language is provided, which can compute the simplest normal form of the mentioned above nonlinear dynamical system easily.(5) Applying a reversible linear transformation and a near-identity nonlinear transformation with parameters, the simplest normal form of an electromechanical coupled nonlinear dynamical system is obtained. Furthermore, the universal unfolding and the relationship between its parameters and the parameters of the original nonlinear dynamical system are obtained. The codimension-two bifurcation is analyzed and the effects of various parameters to the dynamical behavior of the system mentioned above are revealed, which lay a theoretical foundation for the parameter design, stable operation and fault diagnosis of a real system. The numerical simulating results of the electromechanical coupled nonlinear dynamical system are obtained, which verify the corresponding theoretical analysis result.(6) Based on the Silnikov criterion, the chaotic characters of mechanically and electrically coupled nonlinear 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 space trajectory, the Lyapunov exponent and the Lyapunov dimension are investigated via numerical simulation, which show chaotic attractor existed in the non-linear dynamical systems.