Chaos in the softening duffing system under multi-frequency periodic forces

The chaotic dynamics of the softening-spring Duffing system with multi-frequency external periodic forces is studied. It is found that the mechanism for chaos is the transverse heteroclinic tori. The Poincare map, the stable and the unstable manifolds of the system under two incommensurate periodic forces were set up on a two-dimensional torus. Utilizing a global perturbation technique of Melnikov the criterion for the transverse interaction of the stable and the unstable manifolds was given. The system under more but finite incommensurate periodic forces was also studied. The Melnikov‘s global perturbation technique was therefore generalized to higher dimensional systems. The region in parameter space where chaotic dynamics may occur was given. It was also demonstrated that increasing the number of forcing frequencies will increase the area in parameter space where chaotic behavior can occur.     

Turnover of hysteresis determines novel bursting in Duffing system with multiple-frequency external forcings

This paper investigates bursting dynamics of the Duffing system with multiple-frequency external forcings, in which novel bursting patterns (i.e., the so-called turnover-of-hysteresis-induced bursting patterns) can be observed. Typically, distinct oscillations are observed in the quasi-static processes of these bursting patterns. We show that the oscillations appear in the quasi-static processes because the equilibrium hysteresis curve of the fast subsystem becomes the one with twists and turns, and this forms a new route to bursting, which we call turnover of hysteresis. Besides, we investigate the effects of forcing frequencies and amplitudes on the turnover-of-hysteresis-induced bursting. Our study shows that the three frequency components observed in the bursting are decided by the forcing frequencies, while the transition of the bursting depends on the forcing amplitudes.     

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。     

LOCAL BIFURCATION ANALYSIS OF STRONGLY NONLINEAR DUFFING SYSTEM

ＬＯＣＡＬＢＩＦＵＲＣＡＴＩＯＮＡＮＡＬＹＳＩＳＯＦＳＴＲＯＮＧＬＹＮＯＮＬＩＮＥＡＲＤＵＦＦＩＮＧＳＹＳＴＥＭＢｉＱｉｎｓｈｅｎｇ（毕勤胜）；ＣｈｅｎＹｕｓｈｕ（陈予恕）；ＷｕＺｈｉｑｉａｎｇ（吴志强）Ａｂｓｔｒａｃｔ：Ｂｙｕｓｉｎｇｃｏｏｒｄｉｎ...     

非线性电报方程组的3个双周期正解问题

This paper studies existence of at least three positive doubly periodic solutions of a coupled nonlinear telegraph system with doubly periodic boundary conditions. First, by using the Green function and maximum principle, existence of solutions of a nonlinear telegraph system is equivalent to existence of fixed points of an operator. By imposing growth conditions on the nonlinearities, existence of at least three fixed points in cone is obtained by using the Leggett-Williams fixed point theorem to cones in ordered Banach spaces. In other words, there exist at least three positive doubly periodic solutions of nonlinear telegraph system.     

RESEARCH OF THE PERIODIC SOLUTION FOR A CLASS OF NONLINEAR DIFFERENTIAL EQUATIONS

ＲＥＳＥＡＲＣＨＯＦＴＨＥＰＥＲＩＯＤＩＣＳＯＬＵＴＩＯＮＦＯＲＡＣＬＡＳＳＯＦ　ＮＯＮＬＩＮＥＡＲＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳＲＥＳＥＡＲＣＨＯＦＴＨＥＰＥＲＩＯＤＩＣＳＯＬＵＴＩＯＮＦＯＲＡＣＬＡＳＳＯＦＮＯＮＬＩＮＥＡＲＤＩＦＦＥ...     

Primary pulse transmission in a strongly nonlinear periodic system

The aim of this work is to study the transmission of stress waves in an impulsively forced semi-infinite repetitive system of linear layers which are coupled by means of strongly nonlinear coupling elements. Only primary pulse transmission and reflection at each nonlinear element is considered. This permits the reduction of the problem to an infinite set of first-order strongly nonlinear ordinary differential equations. A subset of these equations is solved both analytically and numerically. For a system possessing clearance nonlinearities it is found that the primary transmitted pulse propagates to only a finite number of layers, and that further transmission of energy to additional layers can occur only through time-delayed secondary pulses or does not occur at all. Hence, clearance nonlinearities in a periodic layered system can lead to energy entrapment in the leading layers. An alternative continuum approximation methodology is also outlined which reduces the problem of primary pulse transmission to the solution of a single strongly nonlinear partial differential equation. The use of the continuum approximation for studying maximum primary pulse penetration in the system with clearance nonlinearities is discussed.     

Development of a doubly periodic system of cavities in viscous bodies

The plane unknown-boundary problem of the development of a doubly periodic system of cavities in viscous media in the presence of finite strains is considered. Under conditions of timedependent slow Newtonian viscous fluid flow the solution of the doubly periodic problem of the development of a system of identical cavities whose centers are arranged in square and triangular grids is obtained.     

Nonlinear dynamic analysis of a quarter vehicle system with external periodic excitation

In this paper, a nonlinear dynamic model of a quarter vehicle with nonlinear spring and damping is established. The dynamic characteristics of the vehicle system with external periodic excitation are theoretically investigated by the incremental harmonic balance method and Newmark method, and the accuracy of the incremental harmonic balance method is verified by comparing with the result of Newmark method. The influences of the damping coefficient, excitation amplitude and excitation frequency on the dynamic responses are analyzed. The results show that the vibration behaviors of the vehicle system can be control by adjusting appropriately system parameters with the damping coefficient, excitation amplitude and excitation frequency. The multi-valued properties, spur-harmonic response and hardening type nonlinear behavior are revealed in the presented amplitude-frequency curves. With the changing parameters, the transformation of chaotic motion, quasi-periodic motion and periodic motion is also observed. The conclusions can provide some available evidences for the design and improvement of the vehicle system.     

The necessary condition for the appearance of chaos in two classes of forced vibration system with both square and cubic nonlinear terms

Two classes of forced vibration system with both square and cubic nonlinear terms are treated as Hamiltonian integrable systems under small perturbation. The Melnikov's functions are given. Thus the necessary condition for induction of chaos in the systems is obtained.     

Oscillatory modes in a nonlinear second-order differential equation with delay

Solution properties of the nonlinear second-order delay-differential equation x(t)=–ax(t)+f[x(t–)] are studied wheref is a piecewise constant function which mimics negative feedback. We show that the solutions can be obtained by a simple geometrical construction which, in principle, can be implemented using a ruler and a compass. Analytical results guarantee the existence and stability properties of limit cycle solutions. Computer-aided constructions reveal a remarkable richness of different types of dynamical behaviors including a variety of unconventional bifurcation schemes.     

Limiting state of an elastoplastic orthotropic plate with a periodic system of collinear cracks

A modified Dugdale model is used to study the fracture of an orthotropic elastoplastic plate with a periodic system of rectilinear cracks. The material of the plate obeys a general yield criterion. The general form of solution is obtained in terms of Kolosov-Muskhelishvili potentials. The size of the plastic zone is expressed in terms of the external load and geometrical parameters. The equations for the determination of the stresses in the plastic zone and the crack opening displacement are derived. The effect of anisotropy on the formation of the plastic zones at the crack tip is examined __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 5, pp. 80–88, May 2007.     

Surface waves induced by external periodic pressure in a fluid with an uneven bottom

The behavior of waves generated by periodic pressure on the free surface is considered within the linear shallow-water theory. The fluid depth is a piecewise-constant function, which implies the presence of a finite-size bottom trench or elevation. For an arbitrary shape of bottom unevenness, the solution of the problem reduces to a system of integral boundary equations. Manifestation of wave-guiding properties of bottom unevenness is illustrated by an example of an extended rectangular elevation.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 1, pp. 70–77, January–February, 2005.     

The study on a kind of control system with nonlinear parabolic distributed parameters

The modelling of one kind of nonlinear parabolic distributed parameter control system with moving boundary, which had extensive applications was presented. Two methods were used to investigate the basic characteristics of the system: 1) transforming the system in the variable domain into that in the fixed domain; 2) transforming the distributed parameter system into the lumped parameter system. It is found that there are two critical values for the control variable: the larger one determines whether or not the boundary would move, while the smaller one determines whether or not the boundary would stop automatically. For one-dimensional system of planar, cylindrical and spherical cases the definite solution problem can be expressed as a unified form. By means of the computer simulation the open-loop control system and close-cycle feedback control system have been investigated. Numerical results agree well with theoretical results. The computer simulation shows that the system is well posed, stable, measurable and controllable. Foundation item: the National Natural Science Foundation of China (59936140) Biographies:ZHOU Jian-jun (1953-); XU Yan-hou (1933-)     

THE STUDY ON A KIND OF CONTROL SYSTEM WITH NONLINEAR PARABOLIC DISTRIBUTED PARAMETERS

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｅａｃｔｉｖｉｔｉｅｓｓｕｃｈａｓｔｈｅａｅｒｏｓｐａｃｅｅｎｇｉｎｅｅｒｉｎｇｈｅａｔｐｒｏｔｅｃｔｉｏｎ[1,2 ],ｔｈｅｌａｒｇｅ＿ｓｃａｌｅｃｏｎｃｒｅｔｅｐｏｕｒｉｎｇ ,ｔｈｅｃｈｅｍｉｃａｌｐｒｏｄｕｃｔｉｏｎ ,ｔｈｅｐｏｗｄｅｒｍｅｔａｌｌｕｒｇｙａｎｄｔｈｅｆｉｒｅｓａｆｅｔｙｅｎｇｉｎｅｅｒｉｎｇ[3,4],ｔｈｅｒｅｉｓｏｎｅｋｉｎｄｏｆｎｏｎｌｉｎｅａｒｐａｒａｂｏｌｉｃｄｉｓｔｒｉｂｕｔｅｄｐａｒａｍｅｔｅｒｃｏｎｔｒｏｌｓｙｓｔｅｍｗｉｔｈｍｏｖｉ…     

Hamiltonian formulation of nonlinear water waves in a two-fluid system

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｇｅｏｍｅｔｒｉｚａｔｉｏｎｏｆｍｅｃｈａｎｉｃｓｉｓａｔｅｎｄｅｎｃｙｏｆｔｈｅｄｅｖｅｌｏｐｍｅｎｔｏｆｃｏｎｔｉｎｕｕｍｍｅｃｈａｎｉｃｓａｎｄｄｒａｗｓｅｘｔｅｎｓｉｖｅａｔｅｎｔｉｏｎｏｆｒｅｓｅａｒｃｈｅｒｓ...     

一类平面自治非线性时滞控制系统的多稳态和混沌

众所周知,平面自治系统即使具有光滑非线性存在,系统也不会出现复杂的动力学行为。本文研究这样的系统存在时滞时,时滞量对系统的动力学行为的影响。通过对一个平面自治非线性系统引入时滞反馈,得到数学模型。利用泛函分析和平均法建立系统平衡态随时滞量变化的失稳机理,研究表明：时滞量平面自治系统动力学行为的影响是本质的．时滞量不但可以使系统出现Hopf分岔,产生周期振动。而且还可以使系统出现多稳态的周期运动或周期吸引子,这些共存的吸引子相碰是导致系统复杂的动力学行为,包括概周期和混沌运动。     

A very complicated structure of resonances in a nonlinear system with cyclic symmetry: Nonlinear forced localization

The fundamental and subharmonic resonances of a nonlinear cyclic assembly are examined using the asymptotic method of multiple-scales. The system consists of a number of identical cantilever beams coupled by means of weak linear stiffnesses. Assuming beam inextensionality, geometric nonlinearities arise due to longitudinal inertia and the nonlinear relation between beam curvature and transverse displacement. The governing nonlinear partial differential equations are discretized by a Galerkin procedure and the resulting set of coupled ordinary differential equations is solved using an asymptotic analysis. The unforced assembly is known to possess localized nonlinear normal modes, which give rise to a very complicated topological structure of fundamental and subharmonic response curves. In contrast to the linear system which exhibits as many forced resonances as its number of degrees of freedom, the nonlinear system is found to possess a number of additional resonance branches which have no counterparts in linear theory. Some of the additional resonances are spatially localized, corresponding to motions of only a small subset of periodic elements. The analytical results are verified by numerical Poincaré maps, and the forced localization features of the nonlinear assembly are demonstrated by considering its response to impulsive excitations.     

Stability and Oscillations in a Time-Delayed Vehicle System with Driver Control

In this paper, linear stability and chaotic motion of a time-delayednonlinear vehicle system are studied. The stability is determined bycomputing the spectrum associated with a system of linear retardedfunctional differential equations, which reveals that a loss ofstability occurs following a Hopf bifurcation. Beyond the critical valuefor linear stability, the system exhibits limit cycle motions.Subharmonic, quasi-periodic and chaotic motions are observed for asystem excited by a periodic disturbance.     

Computer computation of the method of multiple scales—dirichlet problem for a class of system of nonlinear differential equations

The method of boundary layer with multiple scales and computer algebra were applied to study the asymptotic behavior of solution of boundary value problems for a class of system of nonlinear differential equations. The asymptotic expansions of solution were constructed. The remainders were estimated. And an example was analysed. It provides a new foreground for the application of the method of boundary layer with multiple scales. Contributed by Jiang Fu-ru, Original Member of Editorial Committe, AMM Biography: Xie La-bing (1976∼); Jiang Fu-ru(1927∼)