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.     

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.     

STABILITY ANALYSIS OF LINEAR AND NONLINEAR PERIODIC CONVECTION IN THERMOHALINE DOUBLE－DIFFUSIVE SYSTEMS

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The adjoining cell mapping and its recursive unraveling,part I: Description of adaptive and recursive algorithms

A new type of cell mapping, referred to as an adjoining cell mapping, is developed in this paper for autonomous dynamical systems employing the cellular state space. It is based on an adaptive time integration employed to compute an associated cell mapping for the system. This technique overcomes the problem of determining an appropriate duration of integration time for the simple cell mapping method. Employing the adjoining mapping principle, the first type of algorithm developed here is an adaptive mapping unraveling algorithm to determine equilibria and limit cycles of the dynamical system in a way similar to that of the simple cell mapping. In addition, it is capable of providing useful information regarding the behavior of dynamical systems possessing pathological dynamics and of systems with rapidly changing vector field. The adjoining property inherent in the adjoining cell mapping method, in general, permits development of new recursive algorithms for unraveling dynamics. The required computer memory for a practical implementation of such algorithms is considerably less than that required by the simple cell mapping algorithm since they allow for a recursive partitioning of state space for trajectory analysis. The second type of algorithm developed in this paper is a recursive unraveling algorithm based on adaptive integration and recursive partitioning of state space into blocks of cells with a view toward its practical implementation. It can find equilibria of the system in the same manner as the simple cell mapping method but is more efficient in locating periodic solutions.     

A numerical method on estimation of stable regions of rotor systems supported on lubricated bearings

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非线性振动系统周期运动及其稳定性的数值研究

§1引言确定型非线性振动系统的运动可分类如下: 1.非定常运动;2.定常运动:(1)周期运动,(2)各态历经运动,(3)浑沌运动。其中非定常运动是一暂态过程,会随着时间的增长逐步衰减乃至实际上消失。定常运动中的各态历经运动,指系统至少有两个互不通约(即其比值为无理数)的振动频率,因此运动虽然局      

冲击动力系统的鲁棒稳定性分析

考虑冲击动力系统的ｋ－ｐ周期运动的鲁棒稳定性问题。首先，根据微分方程的解、冲击条件和衔接条件，应用迭代法给出了系统存在ｋ－ｐ周期运动的充分必要条件，并利用稳定性的等价原理，通过周期运动的扰动差分方程导出其稳定条件；然后，着重对含有不确定参数的冲击动力系统的ｋ－ｐ周期运动的稳定性进行了分析，得出了鲁棒稳定的充分条件，文末给出了用于阐明理论结果的算例。     

Modal interaction and bifurcations in two degree of freedom duffing oscillators

A methodology is first presented for analyzing long time response of periodically exited nonlinear oscillators. Namely, a systematic procedure is employed for determining periodic steady state response, including harmonic and superharmonic components. The stability analysis of the located periodic motions is also performed, utilizing results of Froquet theory. This methodology is then applied to a special class of two degree of freedom nonlinear oscillators, subjected to harmonic excitation. The numberical results presented in the second part of this study illustrate effects caused by the interaction of the modes as well as effects of the nonlinearities on the steady state response of these oscillators. In addition, sequences of bifurcations are analyzed for softening systems, leading to unbounded response of the model examined. Finally, the importance of higher harmonics on the response of systems with strongly nonlinear characteristics is investigated.     

多自由度非线性随机系统的响应与稳定性

响应与稳定性分析一直是随机动力学研究的热点, 发展预测随机响应及判定系统响应性态的方法具有重要的科学意义与广阔的应用前景. 本文综述了有关多自由度非线性随机系统的响应与稳定性的研究. 首先简介用于随机系统响应预测的Fokker-Planck-Kolmogorov方程法、随机平均法、等效线性化法、等效非线性系统法和Monte Carlo模拟法, 评述其优缺点, 进而讨论了多自由度非线性随机系统响应的精确平稳解、近似瞬态解的研究现状. 然后介绍了随机系统稳定性分析的两类方法, 即Lyapunov函数法及Lyapunov指数法,并综述了多自由度非线性随机系统稳定性分析的研究现状. 最后给出几点发展建议.     

An enriched multiple scales method for harmonically forced nonlinear systems

This article explores enrichment to the method of Multiple Scales, in some cases extending its applicability to periodic solutions of harmonically forced, strongly nonlinear systems. The enrichment follows from an introduced homotopy parameter in the system governing equation, which transitions it from linear to nonlinear behavior as the value varies from zero to one. This same parameter serves as a perturbation quantity in both the asymptotic expansion and the multiple time scales assumed solution form. Two prototypical nonlinear systems are explored. The first considered is a classical forced Duffing oscillator for which periodic solutions near primary resonance are analyzed, and their stability is assessed, as the strengths of the cubic term, the forcing, and a system scaling factor are increased. The second is a classical forced van der Pol oscillator for which quasiperiodic and subharmonic solutions are analyzed. For both systems, comparisons are made between solutions generated using (a) the enriched Multiple Scales approach, (b) the conventional Multiple Scales approach, and (c) numerical simulations. For the Duffing system, important qualitative and quantitative differences are noted between solutions predicted by the enriched and conventional Multiple Scales. For the van der Pol system, increased solution flexibility is noted with the enriched Multiple Scales approach, including the ability to seek subharmonic (and superharmonic) solutions not necessarily close to the linear natural frequency. In both nonlinear systems, comparisons to numerical simulations show strong agreement with results from the enriched technique, and for the Duffing case in particular, even when the system is strongly nonlinear.     

Poincare型胞映射分析方法及其应用

本文用Poincare型胞映射方法对平衡及不平．衡轴承转子非线性动力系统的全局特性进行了分析研究,同时求得了一定状态空间内系统存在的周期解及其在各不同Poincare截面上的吸引域,得到了一些新的现象和规律,并通过对平衡及不平衡轴承转子系统的全局特性异同的比较,说明了要建立既适用于平衡轴承转子系统又适用于不平衡轴承转子系统的非线性稳定性准则应注意的几个问题     

Stability analysis of duffing oscillator with time delayed and/or fractional derivatives

The periodic motions of the fractional order and/or delayed nonlinear systems are investigated in the frequency domain using a harmonic balance method with the analytical gradients of the nonlinear quality constraints and the sensitivity information of the Fourier coefficients can also obtained. The properties of fractional order derivatives and trigonometric functions are utilized to construct the fractional order derivatives, delayed and product operational matrices. The operational matrices are used to derive the analytical formulae of nonlinear systems of algebraic equations. The stability of periodic solutions for the delayed nonlinear systems is identified by an eigenvalue analysis of quasi-polynomials characteristic equations. Sensitivity analysis is performed to study the influence of the structural parameters on the system responses. Finally, three numerical examples are presented to illustrate the validity and feasibility of the developed method. It is concluded that the proposed methodology has the potential to facilitate highly efficient optimization, as well as sensitivity and uncertainty analysis of nonlinear systems with fractional derivatives and/or time delayed.     

A numerical treatment of the periodic solutions of non-linear vibration systems

Direct numerical integration can be used to find the periodicsolutions for the equations of motion of nonlinear vibrationsystems.The initial conditions are iterated so that theycoincide With the terminal conditions.The time interval ofthe integration(i.e.,the period)and certain parameters ofthe equations of motion can be included in the iterations.Theintegration method has a variable stoplength.This Sbooting method can produce periodic solutions witha shorter computex time.The only error occurs in the numeri-cal integration and it can therefore be estimated and madesmall enough.Using this method one can treat a variety ofvibration problems.such as free conservative.forced.para-meter-excited and self-sustained vibrations with one or se-veral degrees-of-freedom.Unstable solutions and those Whichare sensitive to parameter Changes can also be calculated.Thestability of the solutions is investigated based on the thecryof differential equations with periodic coefficients.The ex-trapolation method and the proc     

Steady state response analysis for fractional dynamic systems based on memory-free principle and harmonic balancing

A semi-analytic approach is proposed to analyze steady state responses of dynamic systems containing fractional derivatives. A major purpose is to efficiently combine the harmonic balancing (HB) technique and Yuan–Agrawal (YA) memory-free principle. As steady solutions being expressed by truncated Fourier series, a simple yet efficient way is suggested based on the YA principle to explicitly separate the Caputo fractional derivative as periodic and decaying non-periodic parts. Neglecting the decaying terms and applying HB procedures result into a set of algebraic equations in the Fourier coefficients. The linear algebraic equations are solved exactly for linear systems, and the non-linear ones are solved by Newton–Raphson plus arc-length continuation algorithm for non-linear problems. Both periodic and triple-periodic solutions obtained by the presented method are in excellent agreement with those by either predictor–corrector (PC) or YA method. Importantly, the presented method is capable of detecting both stable and unstable periodic solutions, whereas time-stepping integration techniques such as YA and PC can only track stable ones. Together with the Floquet theory, therefore, the presented method allows us to address the bifurcations in detail of the steady responses of fractional Duffing oscillator. Symmetry breakings and cyclic-fold bifurcations are found and discussed for both periodic and triple-periodic solutions.     

A Perturbation Method for Nonlinear Two-Dimensional Maps

A new perturbation method for a weakly nonlinear two-dimensional discrete-time dynamical system is presented. The proposed technique generalizes the asymptotic perturbation method that is valid for continuous-time systems and detects periodic or almost-periodic orbits and their stability. Two equations for the amplitude and the phase of solutions are derived and their fixed points correspond to limit cycles for the starting nonlinear map. The method is applied to various nonlinear (autonomous or not) two-dimensional maps. For the autonomous maps we derive the conditions for the appearance of a supercritical Hopf bifurcation and predict the characteristics of the corresponding limit cycle. For the nonautonomous maps we show amplitude-response and frequency-response curves. Under appropriate conditions, we demonstrate the occurrence of saddle-node bifurcations of cycles and of jumps and hysteresis effects in the system response (cusp catastrophe). Modulated motion can be observed for very low values of the external excitation and an infinite-period bifurcation occurs if the external excitation increases. Analytic approximate solutions are in good agreement with numerically obtained solutions.     

A Simple Approach to the 1:1 Resonance Bifurcation in Follower-Load Problems

We consider the problem of 1:1 resonance in autonomous, timereversible systems. We first present an abstract treatment for n-dimensionalsecond-order systems, and then apply our method to two simplemechanical examples involving follower loads. As the magnitude of the follower load is increased past a criticalvalue, the trivial solution loses stability as the real-valuedfrequencies of the linearized system first coalesce and then splitapart with complex-conjugate values. In Hamiltonian systems this isusually referred to as the Hamiltonian–Hopf bifurcation. Some novelfeatures of our analysis are the direct exploitation of reversibilityand a Liapunov–Schmidt reduction of the second-order (Newtonian)equations of motion, the latter of which requires no complexification.The analysis of the resulting two-parameter, one-dimensionalbifurcation equation yields the possibility that families ofnontrivial periodic solutions may exist for load values in excess of the critical value.     

存在间隙的多自由度系统的周期运动及Robust稳定性

研究一类存在间隙的多自由度振动系统的动态响应．系统由线性元件构成，但其中一个元件的最大位移不能超过由刚性平面约束所确定的阀值．应用模态矩阵方法将系统解耦，并根据碰撞条件和由碰撞规律所确定的衔接条件求得系统的周期运动及其稳定条件．将Ｌｙａｐｕｎｏｖ方法应用于周期运动的扰动差分方程，导出了含不确定参数的碰撞振动系统周期运动的鲁棒（Ｒｏｂｕｓｔ）稳定性条件．文末用一个二自由度系统阐明了方法的有效性     

Modified Parker-Sochacki method for solving nonlinear oscillators

A new approach is presented for solving nonlinear oscillatory systems. Parker-Sochacki method (PSM) is combined with Laplace-Padé resummation method to obtain approximate periodic solutions for three nonlinear oscillators. The first one is Duffing oscillator with quintic nonlinearity which has odd nonlinearity. The second one is Helmholtz oscillator which has even nonlinearity. The last one is a strongly nonlinear oscillator, namely; relativistic harmonic oscillator which has a fractional order nonlinearity. Solutions are also obtained using Runge-Kutta numerical method (RKM) and Lindstedt-Poincare method (LPM). However, the LPM could not be used to solve the relativistic harmonic oscillator since it is a strongly nonlinear oscillator. The comparison between these solutions shows that the convergence zone for the Parker-Sochacki with Laplace-Padé method (PSLPM) is remarkably increased compared to PSM method. It also shows that the PSLPM solutions are in excellent agreement with LPM solutions for Duffing oscillator and are superior to LPM solutions in case of Helmholtz oscillator. The PSLPM succeeded to give an accurate periodic solution for the relativistic harmonic oscillator. For a wide range of solution domain, comparing PSLPM with RKM prove the correctness of the PSLPM method. Hence, the PSLPM method can be used with satisfied confidence to solve a broad class of nonlinear oscillators.     

On the hyperchaotic complex Lü system

The aim of this paper is to introduce the new hyperchaotic complex Lü system. This system has complex nonlinear behavior which is studied and investigated in this work. Numerically the range of parameter values of the system at which hyperchaotic attractors exist is calculated. This new system has a whole circle of equilibria and three isolated fixed points, while the real counterpart has only three isolated ones. The stability analysis of the trivial fixed point is studied. Its dynamics is more rich in the sense that our system exhibits both chaotic and hyperchaotic attractors, as well as periodic and quasi-periodic solutions and solutions that approach fixed points. The nonlinear control method based on Lyapunov function is used to synchronize the hyperchaotic attractors. The control of these attractors is studied. Different forms of hyperchaotic complex Lü systems are constructed using the state feedback controller and complex periodic forcing.     

Stability of Semi-Discrete Shock Profiles by Means of an Evans Function in Infinite Dimensions

Semi-discrete shock profiles are traveling wave solutions of hyperbolic systems of conservation laws under discretization in space. The existence of semi-discrete shocks has been investigated in earlier papers. Here the spectral stability of those nonlinear waves is addressed, and formulated in terms of a variational delay differential operator. Constructing a generalized Evans function, in infinite dimensions, it is shown how to derive stability criteria. Some examples are given when the criterion is fully explicit, e.g., for extreme Lax shocks. Additionally, connection is made with the alternative approach proposed by Chow, Mallet-Paret, and Shen (Journal of Differential Equations 1998), regarding the stability of traveling waves in general Lattice Dynamical Systems.