一种非线性系统周期解的延拓算法

提出了一种非线性系统周期解的延拓算法。指出了非线性系统周期解在分岔点处由于雅可比矩阵奇异而导致一般延拓方法延拓失败问题;然后基于推广的打靶法的思想,将普通延拓算法推广,提出了一种用于周期解延拓的算法。对于非线性动力系统,该算法可以在已知某一参数下的周期解的基础上,求解出在一定参数范围内非线性动力系统的解随参数的连续变化情况。应用该方法对非线性柔性转子-轴承系统的周期解与参数的依赖关系进行了求解,验证了方法的有效性。     

Quantitative methodology for stability analysis of nonlinear rotor systems

IntroductionRotor-bearings systems applied widely in industry are nonlinear dynamic systems of multi-degree-of-freedom.Synchronous vibration is its typical motion under unavoidable unbalance.Subharmonic,quasi-periodic and chaotic vibrations,caused by the …     

The floquet theory for quasi-periodic linear systems

In this paper we establish the Floquet theory for the quasi-perio-dic systemwhere A(u_1,u_2,…,u_m)is an n×n periodic matrix function of u_1,u_2.…,u_mwith period 2π,and it is of C~τ,τ=(N 1)τ_0,τ_0=2(m 1).N=(1/2)n(n 1).Meanwhile,we define the characteristic exponential roots β_1,β_2,…,β_nof(0.1),and assume thatwhere K(ω),K(ω,β)>0.k_μ,j_v.are integers,all the integers k_1,k_2,…,k_m.are not zero,i~2=-1,Then there exists aquasi-periodic linear transformation,which carries(0.1)into a li-near system with constant coefficients.     

A method of stability analysis of nonlinear large-scale systems

Results of stability analysis of some classes of perturbed equations of motion are presented. Sufficient conditions for the asymptotic stability in the large and instability of the equilibrium state of large-scale systems are established using a quadratic Lyapunov function. Some results are illustrated by examples of fourth-order systems     

Parallel numerical continuation of periodic responses of local nonlinear systems

Nonlinear Dynamics - The aim of this paper is to develop an efficient and robust parallel numerical continuation method for periodic solutions of local nonlinear systems. In this method, a...     

The stability of limit cycles in nonlinear systems

In this paper, a necessary condition is first presented for the existence of limit cycles in nonlinear systems, then four theorems are presented for the stability, instability, and semistabilities of limit cycles in second order nonlinear systems. Necessary and sufficient conditions are given in terms of the signs of first and second derivatives of a continuously differentiable positive function at the vicinity of the limit cycle. Two examples considering nonlinear systems with familiar limit cycles are presented to illustrate the theorems.     

Lyapunov stability for continuous-time multidimensional nonlinear systems

This paper deals with the stability of continuous-time multidimensional nonlinear systems in the Roesser form. The concepts from 1D Lyapunov stability theory are first extended to 2D nonlinear systems and then to general continuous-time multidimensional nonlinear systems. To check the stability, a direct Lyapunov method is developed. While the direct Lyapunov method has been recently proposed for discrete-time 2D nonlinear systems, to the best of our knowledge what is proposed in this paper are the first results of this kind on stability of continuous-time multidimensional nonlinear systems. Analogous to 1D systems, a sufficient condition for the stability is the existence of a certain type of the Lyapunov function. A new technique for constructing Lyapunov functions for 2D nonlinear systems and general multidimensional systems is proposed. The proposed method is based on the sum of squares (SOS) decomposition, therefore, it formulates the Lyapunov function search algorithmically. In this way, polynomial nonlinearities can be handled exactly and a large class of other nonlinearities can be treated introducing some auxiliary variables and constrains.     

Control of nonlinear systems exhibiting chaos to desired periodic or quasi-periodic motions

Nonlinear Dynamics - A novel method in the design of controllers to drive general nonlinear systems to desired periodic or quasi-periodic motions is presented in this paper. The viability of the...     

New methods to find solutions and analyze stability of equilibrium of nonholonomic mechanical systems

A large proportion of constrained mechanical systems result in nonlinear ordinary differential equations, for which it is quite difficult to find analytical solutions. The initial motions method proposed by Whittaker is effective to deal with such problems for various constrained mechanical systems, including the nonholonomic systems discussed in the first part of this paper, where in addition to differential equations of motion, nonholonomic constraints apply. The final equations of motion for these systems are obtained in the form of corresponding power series. Also, an alternative, direct method to determine the initial values of higher-order derivatives \({\ddot{q}}_0 ,{{\dddot{q}{} }}_{\!0} ,\ldots \) is proposed, being different from that of Whittaker. The second part of this work analyzes the stability of equilibrium of less complex, nonholonomic mechanical systems represented by gradient systems. We discuss the stability of equilibrium of such systems based on the properties of the gradient system. The advantage of this novel method is its avoidance of the difficulty of directly establishing Lyapunov functions aimed at such unsteady nonlinear systems. Finally, these theoretical considerations are illustrated through four examples.     

The MLP method for subharmonic and ultraharmonic resonance solutions of strongly nonlinear systems

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Perturbation methods for nonlinear autonomous discrete-time dynamical systems

Two perturbation methods for nonlinear autonomous discrete-time dynamical systems are presented. They generalize the classical Lindstedt-Poincaré and multiple scale perturbation methods that are valid for continuous-time systems. The Lindstedt-Poincaré method allows determination of the periodic or almost-periodic orbits of the nonlinear system (limit cycles), while the multiple scale method also permits analysis of the transient state and the stability of the limit cycles. An application to the discrete Van der Pol equation is also presented, for which the asymptotic solution is shown to be in excellent agreement with the exact (numerical) solution. It is demonstrated that, when the sampling step tends to zero the asymptotic transient and steady-state discrete-time solutions correctly tend to the asymptotic continuous-time solutions.     

Existence and uniqueness of periodic solutions for fourth-order nonlinear periodic systems

In this paper, we study the existence, uniqueness and stability of the periodic solutions for fourth-order nonlinear nonhomogeneous periodic systems with slowly changing coefficients by using the method of Liapunor Function. We obtain some sufficient conditions which guarantee the existence, uniqueness and asymptotic stability of the periodic solutions of these systems and estimate the extent to which the coefficients are allowed to change.