On maximum periodic solutions of integrodifferential equations of volterra type and their stability

Consideration was given to the question of asymptotic (exponential) stability of the maximum periodic solutions of the integrodifferential equations which have an asymptotically stable linear part and small periodic (exponential maximum periodic) perturbation. Under the unlimitedly increasing time, these solutions tend to the periodic modes. The sufficient conditions for asymptotic stability were indicated. In the resonance case where the linearized equation has a pair of purely imaginary roots with the corresponding oscillation frequency coinciding with the oscillation frequency of the periodic part of small perturbation (time function) and the coefficients of the power series expansion of the nonlinear terms, consideration was given to the problem of existence for the maximum periodic solutions of the integrodifferential equation. Conditions were established for existence of such solutions representable by the power series in the fractional degrees of the small parameter characterizing the value of small perturbation in the equation.     

An averaging approach to chattering

The singularly perturbed relay control systems (SPRCS) as mathematical models of chattering in the small neighborhood of the switching surface in sliding mode systems are examined. Sufficient conditions for existence and stability of fast periodic solutions to the SPRCS are found. It is shown that the slow motions in such SPRCS are approximately described by equations derived from equations for the slow variables of SPRCS by averaging along fast periodic motions. It is shown that In the general case, when the equations of a plant contain relay control nonlinearly, the averaged equations do not coincide with the equivalent control equations or with the Filippov's definition (1988) for the sliding motions in the reduced system; however, in the linear case, they coincide     

On perturbations of systems with multidimensional degeneration

Bifurcation conditions are found for the periodic solutions in systems of differential equations with the perturbation (small disturbance) in the case of existence of joined Floke solutions in a linearized nonperturbed system. For this case a multidimensional analog of the Malkin bifurcation function is built up.     

Boundary value problems for a class of impulsive functional equations

This paper is related to the existence and approximation of solutions for impulsive functional differential equations with periodic boundary conditions. We study the existence and approximation of extremal solutions to different types of functional differential equations with impulses at fixed times, by the use of the monotone method. Some of the options included in this formulation are differential equations with maximum and integro-differential equations. In this paper, we also prove that the Lipschitzian character of the function which introduces the functional dependence in a differential equation is not a necessary condition for the development of the monotone iterative technique to obtain a solution and to approximate the extremal solutions to the equation in a given functional interval. The corresponding results are established for the impulsive case. The general formulation includes several types of functional dependence (delay equations, equations with maxima, integro-differential equations). Finally, we consider the case of functional dependence which is given by nonincreasing and bounded functions.     

离散捕食系统的持久性和全局稳定性

研究一类带有Beddington．DeAngelis功能性反应和时滞的多维离散食饵．捕食系统的持久性和全局稳定性。利用差分方程的比较原理,给出了系统持久性的充分条件;应用估值方法推导得到系统全局稳定性的充分条件;使用不动点理论,证明了系统周期解的存在性和全局稳定性。     

Bifurcations of the horizontally forced spherical pendulum

Various planar motions of the horizontally forced damped spherical pendulum are considered and, in particular, their stability to non-planar perturbations. By making a careful choice of coordinates, all solutions of the planar pendulum can be considered including small amplitude periodic solutions, running oscillations and chaotic solutions. The full nonlinear equations in the chosen coordinates are derived and the symmetries of the system are described. Bifurcation diagrams for various types of solutions are presented. Stability of the chaotic solutions is determined by considering a normal Lyapunov exponent.     

Active filtering of physiological motion in robotized surgery using predictive control

This work presents a predictive-control approach to active mechanical filtering of complex, periodic motions of organs induced by respiration or heart beating in robotized surgery. Two different predictive-control schemes are proposed for the compensation of respiratory motions or cardiac motions. For respiratory motions, the periodic property of the disturbance has been included into the input-output model of the controlled system so as to have the robotic system learn and anticipate perturbation motions. A new cost function is proposed for the unconstrained generalized predictive controller (GPC), where reference tracking is decoupled from the rejection of predictable periodic motions. Cardiac motions are more complex, since they are the combination of two periodic nonharmonic components. An adaptive disturbance predictor is proposed which outputs future predicted disturbance values. These predicted values are used to anticipate the disturbance by using the predictive feature of a regular GPC. Experimental results are presented on a laboratory testbed and in vivo on pigs. They demonstrate the effectiveness of the two proposed methods to compensate complex physiological motion.     

两自由度齿轮传动系统全局动力学研究

齿侧间隙的存在,使齿轮传动系统存在着丰富的非线性动力学行为.考虑两自由度齿轮传动系统的动力学模型,利用数值方法分析系统的分岔和混沌动力学行为.通过简单胞映射方法对非线性齿轮系统进行全局分析,得到系统的吸引子,吸引域等全局特性.结果显示:随着激振频率的变化,系统存在多个周期解共存以及周期解与混沌运动共存现象.最后利用系统的相轨线图与庞加莱截面图进行对比分析,在不同的初值条件下,系统呈现出不同的周期运动或混沌运动.利用简单胞映射方法的数值计算结果可以实现在不良参数条件下,通过合理控制系统的初值条件而获得理想的系统响应.     

On the oscillating motion of an Oldroyd-B fluid between two infinite circular cylinders

Exact solutions for some oscillating motions of Oldroyd-B fluids, between two infinite coaxial circular cylinders, are established by means of the Laplace transform. These solutions, presented as sum of the steady-state and transient solutions describe the motion of the fluid at small and large times and reduce to the similar solutions for Maxwell, second grade and Newtonian fluids as limiting cases. After some time, when the transients disappear, the starting solutions tend to the steady-state solutions which are periodic in time and independent of the initial conditions. The required time to obtain the steady-state for the cosine and the sine oscillations of the boundary is determined by graphical illustrations. This time decreases if the frequency of the velocity of the boundary increases.     

Phase noise analysis of oscillators with Sylvester representation for periodic time-varying modulus matrix by regular perturbations

Phase noise analysis of an oscillator is implemented with its periodic time-varying small signal state equations by perturbing the autonomous large signal state equations of the oscillator. In this paper, the time domain steady solutions of os- cillators are perturbed with traditional regular method; the periodic time-varying Jocobian modulus matrices are decomposed with Sylvester theorem, and on the resulting space spanned by periodic vectors, the conditions under which the os- cillator holds periodic steady states with any perturbations are analyzed. In this paper, stochastic calculus is applied to disclose the generation process of phase noise and calculate the phase jitter of the oscillator by injecting a pseudo sinusoi- dal signal in frequency domain, representing the white noise, and a δ correlation signal in time domain into the oscillator. Applying the principle of frequency modulation, we learned how the power-law and the Lorentzian spectrums are formed. Their relations and the Lorentzian spectrums of harmonics are also worked out. Based on the periodic Jacobian modulus matrix, the simple algorithms for Floquet exponents and phase noise are constructed, as well as a simple case is demonstrated. The analysis difficulties and the future directions for the phase noise of oscillators are also pointed out at the end.     

Small Vibrations Superimposed on a Prescribed Rigid Body Motion

A method for analysing flexible multibody systems in which theelastic deformations are small is presented. The motion isconsidered a gross non-linear rigid body motion with small linearvibrations superimposed on it. For periodic gross motion, thisresults in a system of rheolinear differential equations for thedeformations with periodic coefficients. The determination of therequired equations with a program for flexible multibody systemsis discussed which calculates, besides the periodic gross motion,the linearized, or variational, equations of motion. Periodicsolutions are determined with a harmonic balance method, whiletransient solutions are obtained by an averaging method. Thestability of the periodic solutions is considered. The procedurehas a high computational efficiency and leads to more insight intothe structure of solutions. The method is applied to a pendulumwith an elliptical motion of its support point, a slider-crankmechanism with flexible connecting rod, a rotor system, and aCardan drive shaft with misalignment.     

连通式油气悬架三轴重型车辆模型非线性动力学分析

建立了基于连通式油气悬架的三轴重型车辆模型,分别将路面不平度考虑为冲击激励、随机激励和正弦激励,分析了连通式油气悬架的非线性特性对三轴重型车辆振动特性的影响,并分析了连通式油气悬架的抗俯仰性能及抗侧倾性能;将路面不平度考虑为正弦激励,以路面不平度激励频率为参数,通过分叉图、波形图、相图以及庞加莱截面分析了正弦激励作用下三轴重型车辆的非线性动力学响应,仿真结果表明系统在不同激励条件下存在周期运动和混沌运动;连通式油气悬架对重型车辆具有较好的抗侧倾和俯仰特性.     

Slow periodic motions in variable structure systems

Singularly perturbed relay control systems (SPRCS) with stable periodic motion in reduced systems are studied here. The slow motions integral manifold of such systems consists of parts that correspond to different values of control and the solutions of SPRCS contain the jumps from one part of the slow manifold to the other. The theorems about existence and stability of the slow periodic solutions are proved. An algorithm of asymptotic representation for this periodic solutions using a boundary layer method is suggested.     

不可压缩超弹性球体中微孔运动的分岔和混沌

针对一类径向横观各向同性不可压缩neo Hookean材料组成的球体,研究了周期扰动载荷作用下球体中心处微孔的动力学行为.根据平衡微分方程和初边值条件等导出描述微孔径向对称运动的强非线性的非自治常微分方程,通过对微分方程解的定性分析,讨论了微孔的定性行为：(1) 在常值载荷作用下,讨论了材料参数和结构参数对系统平衡点的影响,特别分析了微孔的二次转向分岔;通过对系统势阱的分析,讨论了微孔在常值载荷作用下的周期和振幅跳跃现象.(2) 在周期扰动载荷作用下,利用时程曲线,Poincaré截面和最大Lyapunov指数分别讨论了二次转向分岔情形下微孔的拟周期和混沌运动,给出了系统产生混沌的条件,并进一步分析了周期扰动载荷对微孔混沌运动的影响.     

Features of motion of dynamic systems with disturbances in the neighborhood of manifolds of switchings

The method of computation of control in real time of a linear system with disturbance is suggested. The system of linear algebraic equations is obtained, which links the deviations of phase coordinates to the deviations of initial conditions of the normalized conjugate system and to the deviation of the finite moment. The calculations reduce to the sequence of the solutions of systems of linear algebraic equations and the integration of a matrix differential equation over transfer intervals of the control switching moments and the finite moment of time. The correction of switching moments and the finite moment of control in the accompaniment of the phase trajectory of motion of a controllable object is considered. Simple constructive conditions of the origin of the sliding mode, motions of the representative point over manifolds of switchings, and changes of the control structure in accompanying the phase trajectory of the system motion are obtained. The convergence of the computational method is proved.     

Slow periodic motions with internal sliding modes in variable structure systems

Singularly perturbed relay systems (SPRS) in which the reduced systems have the stable periodic motions with internal sliding modes are studied. The slow motion integral manifold of such systems consists of the parts which correspond to the different values of relay control and the solutions may contain the jumps from one part of the slow manifold to another. For such systems a theorem about existence and stability of the periodic solutions is proved. An algorithm of asymptotic representation for this periodic solutions using boundary layer method is presented. It is proved that in the neighbourhood of the break away point the asymptotic representation starts with the first order boundary layer function.     

Optimal rejection of persistent bounded disturbances

In this paper, we formulate the problem of optimal disturbance rejection in the case where the disturbance is generated as the output of a stable system in response to an input which is assumed to be of unit amplitude, but is otherwise arbitrary. The objective is to choose a controller that minimizes the maximum amplitude of the plant output in response to such a disturbance. Mathematically, this corresponds to requiring uniformly good disturbance rejection over all time. Since the problem of optimal tracking is equivalent to that of optimal disturbance rejection if a feedback controller is used (see [7, sect. 5.6]), the theory presented here can also be used to design optimal controllers that achieve uniformly good tracking over all time rather than a tracking error whose L2-norm is small, as is the case with the currently popularH_{infty}theory. The present theory is a natural counterpart to the existing theory of optimal disturbance rejection (the so-calledH_{infty}theory) which is based on the assumption that the disturbance to be rejected is generated by a stable system whose input is square-integrable and has unit energy. It is shown that the problem studied here has quite different features from its predecessor. Complete solutions to the problem are given in several important cases, including those where the plant is minimum phase or when it has only a single unstable zero. In other cases, procedures are given for obtaining bounds on the solution and for obtaining suboptimal controllers.     

Sensitivity of certain dynamic systems with respect to a small delay

A class of non-linear differential equations of order one with pure delay is examined. It is found that the ‘effective’ order of an equation of this class need not be constant for all values of the delay parameter, even if the latter is small. In the autonomous case periodic solutions are determined and an estimate of the domain of influence of a constant steady state is made. In the non-autonomous case it is found that the periodic and almost periodic solutions are of more varied types than for analoguous differential equations without pure delay.     

介电弹性体圆管的粘弹非线性动力学分析

研究了考虑粘弹性情形介电弹性体（DE）圆管的非线性动力学行为.利用弹簧-粘壶模型和不可压缩材料的neo-Hookean模型,建立了考虑粘弹性的介电弹性体圆管的动力学方程.采用小扰动方法,给出了瞬时固有频率的表达式,分析了特征时间、预拉伸以及圆管的厚度等参数对瞬时固有频率的影响规律.研究了受周期电压作用下的非线性振动特性,结果表明粘弹性会显著降低响应振幅,但不会影响准静态最终固有频率.在没有直流电压时,系统仅产生单一主共振,但当直流电压足够大时,会发生多频共振.     

Exact Solutions for Motion Equations of Symmetric Gyros System

A system of n heavy symmetric gyros coupled byideal spherical hinges is considered. The new class of nonstationaryexact solutions for the motion equations of this system is singled out.It contains majority of the known exact solutions of the problem. Themechanical system motions under which axes of symmetry of some bodiesare collinear to the vertical axis and axes of other bodies are invertical plane and they make equal angles with vertical correspond tothe constructed solutions. The necessary and sufficient conditions forthe existence of such motions are obtained. On the base of these resultsthe new dynamical properties which express the constancy of the totalenergy of each gyro and constancy of the vertical projections for theangular moments of the gyros subsystems are established.