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。     

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.     

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     

Existence,Uniqueness, and Stability of Generalized Traveling Waves in Time Dependent Monostable Equations

The current paper is devoted to the study of traveling wave solutions of spatially homogeneous monostable reaction diffusion equations with ergodic or recurrent time dependence, which includes periodic and almost periodic time dependence as special cases. Such an equation has two spatially homogeneous and time recurrent solutions with one of them being stable and the other being unstable. Traveling wave solutions are a type of entire solutions connecting the two spatially homogeneous and time recurrent solutions. Recently, the author of the current paper proved that a spatially homogeneous time almost periodic monostable equation has a spreading speed in any given direction. This result can be easily extended to monostable equations with recurrent time dependence. In this paper, we introduce generalized traveling wave solutions for time recurrent monostable equations and show the existence of such solutions in any given direction with average propagating speed greater than or equal to the spreading speed in that direction and non-existence of such solutions of slower average propagating speed. We also show the uniqueness and stability of generalized traveling wave solutions in any given direction with average propagating speed greater than the spreading speed in that direction. Moreover, we show that a generalized traveling wave solution in a given direction with average propagating speed greater than the spreading speed in that direction is unique ergodic in the sense that its wave profile and wave speed are unique ergodic, and if the time dependence of the monostable equation is almost periodic, it is almost periodic in the sense that its wave profile and wave speed are almost periodic.     

Symmetric Periodic Solutions of Parabolic Problems With Discontinuous Hysteresis

We develop a framework for treating the long-term behavior of solutions for parabolic equations in multidimensional domains with discontinuous hysteresis. Bearing in mind the thermostat model, we concentrate in this paper on the prototype heat equation with hysteresis in the boundary condition. We provide an algorithm for constructing all periodic solutions with exactly two switchings on the period and study their stability. Coexistence of several periodic solutions with different stability properties is proved to be possible. A mechanism of appearance and disappearance of periodic solutions is investigated.     

3-periodic traveling wave solutions for a dynamical coupled map lattice

In this paper, the existence of periodic traveling wave solutions with a priori unknown velocity is considered for a coupled map lattice dynamical system. By trasforming our problem into one that involves polynomials, explicit 2- and 3-periodic traveling wave solutions are found, while the other solutions can be computed numerically. Since there does not seem to be any reports on explicit traveling wave solutions, we hope that our results will lead to the discovery of many others.     

Delay Equations with Rapidly Oscillating Stable Periodic Solutions

We prove analytically that there exist delay equations admitting rapidly oscillating stable periodic solutions. Previous results were obtained with the aid of computers, only for particular feedback functions. Our proofs work for stiff equations with several classes of feedback functions. Moreover, we prove that for negative feedback there exists a class of feedback functions such that the larger the stiffness parameter is, the more stable rapidly oscillating periodic solutions there are. There are stable periodic solutions with arbitrarily many zeros per unit time interval if the stiffness parameter is chosen sufficiently large.     

激励Stuart-Landau方程的研究--周期解、稳定性及流动控制

解析得出了有外部激励的Stuart-Landau（S-L）方程的频率锁定周期解,对这些解与外部激励振幅和频率的依赖关系做了详细研究,并用周期系统稳定性理论确定了解的稳定性边界.还对S-L方程所描述的流动控制效果进行了研究,发现由于外部激励的作用,稳定的锁频解可能比原来的饱和解能量减少了,外部的控制最多能使扰动能量减少为原来的一半.     

Periodic solution of a turbidostat model with impulsive state feedback control

In this paper, a turbidostat model with impulsive state feedback control is considered. We obtain sufficient conditions of the global asymptotical stability of the system without impulsive state feedback control. We also obtain that the system with impulsive state feedback control may have order one periodic solution, and the sufficient condition for existence and stability of order one periodic solution is gotten as well. For some special cases, it is shown that in the system an order two periodic solution may exist. Our results show that the control measure is effective and reliable.     

On the stability of the shear flow of a viscoelastic fluid with slip along the fixed wall

In this paper we solve the time-dependent shear flow of an Oldroyd-B fluid with slip along the fixed wall. We use a non-linear slip model relating the shear stress to the velocity at the wall and exhibiting a maximum and a minimum. We assume that the material parameters in the slip equation are such that multiple steady-state solutions do not exist. The stability of the steady-state solutions is investigated by means of a one-dimensional linear stability analysis and by numerical calculations. The instability regimes are always within or coincide with the negative-slope regime of the slip equation. As expected, the numerical results show that the instability regimes are much broader than those predicted by the linear stability analysis. Under our assumptions for the slip equation, the Newtonian solutions are stable everywhere. The interval of instability grows as one moves from the Newtonian to the upper-convected Maxwell model. Perturbing an unstable steady-state solution leads to periodic solutions. The amplitude and the period of the oscillations increase with elasticity.     

Stability of Periodic Solutions¶of Conservation Laws with Viscosity:¶Analysis of the Evans Function

Nonclassical conservation laws with viscosity arising in multiphase fluid and solid mechanics exhibit a rich variety of traveling-wave phenomena, including homoclinic (pulse-type) and periodic solutions along with the standard heteroclinic (shock, or front-type) solutions. Here, we investigate stability of periodic traveling waves within the abstract Evans-function framework established by R. A. Gardner. Our main result is to derive a useful stability index analogous to that developed by Gardner and Zumbrun in the traveling-front or -pulse context, giving necessary conditions for stability with respect to initial perturbations that are periodic on the same period T as the traveling wave; moreover, we show that the periodic-stability index has an interpretation analogous to that of the traveling-front or -pulse index in terms of well-posedness of an associated Riemann problem for an inviscid medium, now to be interpreted as allowing a wider class of measure-valued solutionsor, alternatively, in terms of existence and nonsingularity of a local “mass map” from perturbation mass to potential time-asymptotic T-periodic states. A closely related calculation yields also a complementary long-wave stability criterion necessary for stability with respect to periodic perturbations of arbitrarily large period NT, N → ∞. We augment these analytical results with numerical investigations analogous to those carried out by Brin in the traveling-front or -pulse case, approximating the spectrum of the linearized operator about the wave.The stability index and long-wave stability criterion are explicitly evaluable in the same planar, Hamiltonian cases as is the index of Gardner and Zumbrun, and together yield rigorous results of instability similar to those obtained previously for pulse-type solutions; this is established through a novel dichotomy asserting that the two criteria are in certain cases logically exclusive. In particular, we obtain results bearing on the nature and mechanism for formation of highly oscillatory Turing-like patterns observed numerically by Frid and Liu and ?ani? and Peters in models of multiphase flow. Specifically, for the van der Waals model considered by Frid and Liu, we show instability of all periodic waves such that the period increases with amplitude in the one-parameter family of nearby periodic orbits, and in particular of large- and small-amplitude waves; for the standard, double-well potential, this yields instability of all periodic waves.Likewise, for a quadratic-flux model like that considered by ?ani? and Peters, we show instability of large-amplitude waves of the type lying near observed patterns, and of all small-amplitude waves; our numerical results give evidence that intermediate-amplitude waves are unstable as well. These results give support for an alternative mechanism for pattern formation conjectured by Azevedo, Marchesin, Plohr, and Zumbrun, not involving periodic waves.     

Stability and global Hopf bifurcation in toxic phytoplankton–zooplankton model with delay and selective harvesting

Consider that some zooplankton can be harvested for food and some phytoplankton can liberate toxin; a toxin producing phytoplankton–zooplankton model with delay and selective harvesting is proposed and investigated. We discuss the stability of equilibria and perform the analysis of Hopf bifurcation. More precisely, the global asymptotical stability of equilibria is investigated by the Lyapunov method and Dulac theorem. In addition, the computing formulas of stability and direction of the Hopf bifurcating periodic solutions are also given. Furthermore, we prove that there exists at least one positive periodic solution as a time delay varies in some regions by using the global Hopf-bifurcation result of Wu (Trans. Am. Math. Soc. 350:4799–4838, 1998) for functional differential equations. Finally, the impact of harvesting is discussed along with numerical results to provide some support to the analytical findings.     

Existence and stability of traveling waves in periodic media governed by a bistable nonlinearity

We prove the existence of multidimensional traveling wave solutions of the bistable reaction-diffusion equation with periodic coefficients under the condition that these coefficients are close to constants. In the case of one space dimension, we prove their asymptotic stability.     

Zero–Hopf bifurcation in a hyperchaotic Lorenz system

We characterize the zero–Hopf bifurcation at the singular points of a parameter codimension four hyperchaotic Lorenz system. Using averaging theory, we find sufficient conditions so that at the bifurcation points two periodic solutions emerge and describe the stability of these orbits.     

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      

Analysis of transient response of saturated porous elastic soil under cyclic loading using element-free Galerkin method

A two-dimensional numerical procedure is presented to analyse the transient response of saturated porous elastic soil layer under cyclic loading. The procedure is based on the element-free Galerkin method and incorporated into the periodic conditions (temporal and spatial periodicity). Its shape function is constructed by moving least-square approximants, essential boundary conditions are implemented through Lagrange multipliers and the periodic conditions are implemented through a revised variational formulation. Time domain is discretized through the Crank–Nicolson scheme. Analytical solutions are developed to assess the effectiveness and accuracy of the current procedure in one and two dimensions. For only temporal periodic problems, a one-dimensional transient problem of finite thickness soil layer is analysed for sinusoidal surface loading. For both temporal and spatial periodic problems, a typical two-dimensional wave-induced transient problem with the seabed of finite thickness is analysed. Finally, a moving boundary problem is analysed. It is found that the current procedure is simple, efficient and accurate in predicting the response of soil layer under cyclic loading.     

The monotonicity and critical periods of periodic waves?of?the �� 6 field model

In this paper, we study the relationship between period and energy of periodic traveling wave solutions for the ?? ⁶ field model. The various topological phase portraits with periodic annulus are given by using standard phase portrait analytical technique. Some analytic behaviors (convexity, monotonicity and number of critical periods) of the period functions associated with periodic waves are investigated. We prove that the period function has exactly one critical period under certain conditions. Moreover, the numerical simulation is made. The results show that our theoretical analysis agrees with the numerical simulation.     

Stability of Equilibrium Solutions of Autonomous and Periodic Hamiltonian Systems with <Emphasis Type="Italic">n</Emphasis>-Degrees of Freedom in the Case of Single Resonance

This paper concerns with the study of the stability of an equilibrium solution of an analytic Hamiltonian system in a neighborhood of the equilibrium point with n-degrees of freedom, in the autonomous and periodic case under the presence of a single resonance. Our Main Theorem generalizes several results existing in the literature and we also give a geometrical interpretation of the hypotheses involved there. In particular, our Main Theorem provides necessary and sufficient conditions for the stability of the equilibrium solutions under the existence of a single resonance, depending on the coefficients of the Hamiltonian function.     

Sensitive Dependence on Initial Conditions of Strongly Nonlinear Periodic Orbits of the Forced Pendulum

We employ nonsmooth transformations of the independent coordinate to analytically construct families of strongly nonlinear periodic solutions of the harmonically forced nonlinear pendulum. Each family is parametrized by the period of oscillation, and the solutions are based on piecewise constant generating solutions. By examining the behavior of the constructed solutions for large periods, we find that the periodic orbits develop sensitive dependence on initial conditions. As a result, for small perturbations of the initial conditions the response of the system can jump from one periodic orbit to another and the dynamics become unpredictable. An analytical procedure is described which permits the study of the generation of periodic orbits as the period increases. The periodic solutions constructed in this work provide insight into the sensitive dependence on initial conditions of chaotic trajectories close to transverse intersections of invariant manifolds of saddle orbits of forced nonlinear oscillators.