Effect of bounded noise on chaotic motion of duffing oscillator under parametric excitation

A harmonic function with constant amplitude and random frequency and phase is called bounded noise. In this paper, the effect of bounded noise on the chaotic behavior of the Duffing oscillator under parametric excitation is studied in detail. The random Melnikov process is derived and a mean-square criterion is used to detect the chaotic dynamics in the system. It is found that the threshold of bounded noise amplitude for the onset of chaos in the system increases as the intensity of the noise in frequency increases. The threshold of bounded noise amplitude for the onset of chaos is also determined by the numerical calculation of the largest Lyapunov exponents. The effect of bounded noise on the Poincaré map and power spectra is also investigated. The numerical results qualitatively confirm the conclusion drawn by using the random Melnikov process with mean-square criterion for larger noise intensity.     

The differential equations for the feynman amplitude of a single-loop graph with four vertices

A system of second-order partial differential equations for the Feynman amplitude of a single-loop graph with four vertices is obtained. It is proved that the symbol of differential operators of this system is singular (in the sense of I. N. Bernshtein) on the Landau manifold of the Feynman amplitude under consideration. The derived system of differential equations is a multidimensional generalization of the system of differential equations for the hypergeometric function of two variables of Appell and Kampé de Fériet.Translated from Matematicheskie Zametki, Vol. 23, No. 1, pp. 113–119, January, 1978.     

Asymptotic Solution of the Autoresonance Problem

We study the problem of forced oscillations near a stable equilibrium of a two-dimensional nonlinear Hamiltonian system of equations. A given exciting force is represented as rapid oscillations with a small amplitude and a slowly varying frequency. We study the conditions under which such a perturbation makes the phase trajectory of the system recede from the original equilibrium point to a distance of the order of unity. To study the problem, we construct asymptotic solutions using a small amplitude parameter. We present the solution for not-too-small values of time outside the original boundary layer.     

Direct and Inverse Problems of Electromagnetic Control

A mathematic model is suggested of the direct and universe problems of flaw detection. The direct problem consists in determining the complex amplitude of EMF in the through-type-induction- transducer–air gap–ferromagnetic cylinder system. In the direct problem, the distribution laws of magnetic permeability and electrical conduction are assumed to be known. A formula is obtained for calculating the complex amplitude of EMF in the class of piecewise-constant dependences of electromagnetic parameters. The inverse problem of flaw detection for a ferromagnetic cylinder consists in determining the magnetic permeability based on the measured values of the moduli of the EMF amplitudes in the system on a fixed frequency grid. An approximate solution of the inverse problem is searched in the class of piecewise-constant functions. Tikhonov’s method of regularization is used to solve the inverse problem. The results of numerical and physical modeling are presented.     

Reconstruction of random-disturbance amplitude in linear stochastic equations from measurements of some of the coordinates

The problem of reconstructing the unknown amplitude of a random disturbance in a linear stochastic differential equation is studied in a fairly general formulation by applying dynamic inversion theory. The amplitude is reconstructed using discrete information on several realizations of some of the coordinates of the stochastic process. The problem is reduced to an inverse one for a system of ordinary differential equations satisfied by the elements of the covariance matrix of the original process. Constructive solvability conditions in the form of relations on the parameters of the system are discussed. A finite-step software implementable solving algorithm based on the method of auxiliary controlled models is tested using a numerical example. The accuracy of the algorithm is estimated with respect to the number of measured realizations.     

Ray Method Expansions for Surface and Internal Waves in Inhomogeneous Oceans of Variable Depth

A ray method formalism is developed for the analysis of surface and internal waves in an inhomogeneous ocean of variable depth. In this method, we deduce from the governing system of equations a system of first order ordinary differential equations, for the group lines (rays of the ray method) and the propagation of phase and amplitude on them. The dispersion relation for these waves arises as an eigen-condition on an eigen-value problem involving an ordinary differential equation in the depth variable. The deduced equation for amplitude propagation has the interpretation of a statement of conservation of action.     

一类带有功能性反应和脉冲效应的捕食系统的动力学性质

本文讨论了一类带有HollingⅡ类功能性反应和脉冲投放的一食饵两捕食者系统.运用Floquet和小振幅扰动理论,证明了当投放周期小于某个临界值时,系统食饵绝灭的周期解是全局渐近稳定的,同时研究了系统的持续生存.     

Nonlinear stability and chaos in electrohydrodynamics

Using the method of multiple scales, the nonlinear instability problem of two superposed dielectric fluids is studied. The applied electric filed is taken into account under the influence of external modulations near a point of bifurcation. A time varying electric field is superimposed on the system. In addition, the viscosity and variable gravity force are considered. A generalized equation governing the evolution of the amplitude is derived in marginally unstable regions of parameter space. A bifurcation analysis of the amplitude equation is carried out when the dissipation due to viscosity and the control parameter are both assumed to be small. The solution of a nonlinear equation in which parametric and external excitations are obtained analytically and numerically. The method of generalized synchronization is applied to determine the equations that describe the modulation of the amplitude and phase. These equations are used to determine the steady state equations. Frequency response curves are presented graphically. The stability of the proposed solution is determined applying Liapunov's first method. Numerical solutions are presented graphically for the effects of the different equation parameters on the system stability, response and chaos.     

Forced vibration of plastic structures taking into account the elastic and damping properties of the material (systems with one degree of freedom)

A method, which takes into account the elastic properties and hysteresis losses of plastic materials, is proposed for calculating the forced vibrations of plastic structures with one degree of freedom. A solution of the equation of motion of the system is found in trigonometric form, and an algebraic equation for determining the vibration amplitude is obtained. In its general form the latter equation can be solved for special cases only; when it is required to determine the vibration amplitude in the general case, the successive approximations method is recommended, or rigorous solution of the algebraic equation for each specific case.Mekhanika Polimerov, Vol. 1, No. 3, pp. 101–106, 1965     

随机能源Logistic反馈控制系统的分岔研究

建立了非线性随机动力模型—带噪声的能源Logistic反馈控制模型,应用随机平均法对随机动力模型进行了简化,得到了一个二维的扩散过程.二维过程满足Ito型随机微分方程,应用不变测度理论研究了该模型的随机分岔.最后,给出了数值实验验证了相应的结论.     

Novel digitally controlled drive for the vibratory conveyors

The vibratory conveyors are used in the industry for continuous transportation of loose substances over short distances (2–10 m). The presented conveyor is designed for transportation in aseptic conditions. To obtain high efficiency for granulated mass with different size of grain and different physical properties it is necessary to set the appropriate vibration amplitude independently of vibration frequency. This is not possible with the conveyors currently being used. Therefore it was proposed to apply the novel drive made up of rotary exciter with digital controller. Moreover some methods for dynamic stiffening of the panels were developed and their effectiveness was ascertained. A digitally controlled exciter allowed dynamic panel stiffening by using light closed panels in construction of the conveyor. The control system with encoders for measuring the vibration amplitude of the panel was also established. A study of the conveyor's prototype shows the effectiveness of the drive system. The possibility of digitally setting the vibration amplitude of the light panel to obtain the maximum productivity for particular materials was also shown in results of the research. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Ivlev型捕食系统的全局动力学分析

研究捕食者与食饵均具有线性密度制约的Ivlev型捕食动力系统.应用常微分方程定性方法,得到了正平衡点的全局稳定性和非小振幅极限环的存在唯一性的充分条件.特别地,在一定条件下,证明了极限环的存在唯一性与正平衡点的局部不稳定性是等价的,正平衡点的局部稳定性隐含它的全局稳定性,因此,系统的全局动力学性质完全由正平衡点的局部性质所决定.     

一类带有Holling Ⅱ类功能性反应和相互干扰及脉冲效应的捕食系统

A kind of predator-prey system of Holling typeⅡand interaction perturbation with impulsive effect is presented.By using Floquet theory and small amplitude perturbations skills,the locally asymptotical stability of prey-eradication periodic solution and the permanence of the system are discussed and the corresponding threshold conditions are given respectively.Finally,the existence of positive periodic solution is investigated by the bifurcation theory.     

A Kind of Predator-prey System of Holling Type Ⅱ and Interaction Perturbation with Impulsive Effect

A kind of predator-prey system of Holling type Ⅱ and interaction perturbation with impulsive effect is presented.By using Floquet theory and small amplitude perturbations skills,the locally asymptotical stability of prey-eradication periodic solution and the permanence of the system are discussed and the corresponding threshold conditions are given respectively.Finally,the existence of positive periodic solution is investigated by the bifurcation theory.     

Almost periodic solutions of neutral impulsive systems with periodic time-dependent perturbed delays

A neutral impulsive system with a small delay of the argument of the derivative and another delay which differs from a constant by a periodic perturbation of a small amplitude is considered. If the corresponding system with constant delay has an isolated ω-periodic solution and the period of the delay is not rationally dependent on ω, then under a nondegeneracy assumption it is proved that in any sufficiently small neighbourhood of this orbit the perturbed system has a unique almost periodic solution.     

Synchronization of two coupled multimode oscillators with time-delayed feedback

Effects of synchronization in a system of two coupled oscillators with time-delayed feedback are investigated. Phase space of a system with time delay is infinite-dimensional. Thus, the picture of synchronization in such systems acquires many new features not inherent to finite-dimensional ones. A picture of oscillation modes in cases of identical and non-identical coupled oscillators is studied in detail. Periodical structure of amplitude death and “broadband synchronization” zones is investigated. Such a behavior occurs due to the resonances between different modes of the infinite-dimensional system with time delay.     

Stability of equilibrium with respect to white noise

A system of ordinary differential equations with a local asymptotically stable equilibrium is considered. The problem of stability with respect to a persistent perturbation of the white noise type is discussed. Stability with given bounds is proved on a large time interval with length of the order of the squared inverse perturbation amplitude. The proof is based on the construction of a barrier function for the parabolic Kolmogorov equation associated with the perturbed dynamical system.     

Dynamic responses to sinusoidal excitations of beams with frictional joints

We consider the dynamic responses of a beam with a frictional joint. The frictional force at the joint is modeled using the Coulomb friction model. The frictional force at the joint makes the nature of the boundary conditions at the joint uncertain. Therefore, this problem represents a type of nonlinear problems where the boundary conditions are coupled to the solutions. Using numerical integration of the resulting differential equations obtained by combining the finite element method and the Lagrange equations, we study the steady-state solutions of the system to sinusoidal excitations. We explore the dependence of the system responses to various parameters including the frictional force, the forcing frequency and the forcing amplitude. A result of special interest is the existence of an optimum friction force if the frictional joint is used to control the system response amplitude. We also examine the ways that friction affects the resonance frequency of the structure. Experiments are carried out, which agree qualitatively with the numerical results.     

具有脉冲扰动和非单调功能反应的三种群捕食系统的分析

研究一类具有脉冲效应和非单调功能反应的两个捕食者一个食饵害虫控制系统．通过脉冲微分方程的Floquet理论和小幅扰动方法,证明了当脉冲周期小于某个临界值时,系统存在一个渐近稳定的害虫根除周期解,否则系统是持续生存的．最后,通过数值实例,给出了一简单讨论．