Bifurcations and Attractor-Repeller Splittings of Non-Saddle Sets

This paper is devoted to the study of some aspects of the stability theory of flows. In particular, we study Morse decompositions induced by non-saddle sets, including their corresponding Morse equations, attractor-repeller splittings of non-saddle sets and bifurcations originated by implosions of the basin of attraction of asymptotically stable fixed points. We also characterize the non-saddle sets of the plane in terms of the Euler characteristic of their region of influence.     

强非线性动力系统的频率增量法

提出一类强非线性动力系统的暧时频率增量法,将描述动力系统的二阶常微分方程,化为以相位为自变量、瞬廛频率为未知函数的积分方程;用谐波平衡原理,将求解瞬时频率的积分问题,归结为求解以频率增量的Fourier系数为独立变量的线性代数方程组;给出了若干例子。     

Nonlinear Dynamics and Stability of a Machine Tool Traveling Joint

In this work, we investigate the nonlinear dynamics and stability of a machine tool traveling joint. The dynamical system considered includes contacting elements of a lathe joint and the cutting process where the onset of instability is governed by mode coupling. The equilibrium equations of the dynamical system yield a unique fixed point that can change its stability via a Hopf bifurcation. The unstable domain is primarily governed by the cutting tool location, the contact stiffness of the joint and the depth of material to be removed. Self excited vibrations due to a mode coupling instability evolve around the unstable fixed point and one or more limit cycles may coexist in the statically unstable domain. Stability and accuracy of the approximate analytical solutions are analyzed by applying Floquet analysis. Perturbation of the dynamical system with weak periodic excitation results with periodic and aperiodic solutions.     

Nonlinear travelling waves in a hyperelastic rod composed of a compressible Mooney-Rivlin material

In this paper, we study strongly nonlinear axisymmetric waves in a circular cylindrical rod composed of a compressible Mooney-Rivlin material. To consider the travelling wave solutions for the governing partial differential system, we first reduce it to a nonlinear ordinary differential equation. By using the bifurcation theory of planar dynamical systems, we show that the reduced system has seven periodic annuluses with different boundaries which depend on four parameters. We further consider the bifurcation behavior of the phase portraits for the reduced one-parameter vector fields when other three parameters are fixed. Corresponding to seven different periodic annuluses, we obtain seven types of travelling wave solutions, including solitary waves of radial contraction, solitary waves of radial expansion, solitary shock waves of radial contraction, solitary shock waves of radial expansion, periodic waves and two types of periodic shock waves. These are physically acceptable solutions by the governing partial differential system. The rigorous parameter conditions for the existence of these waves are given.     

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.     

求解非线性动力系统周期解的改进打靶法

针对有周期解的动力系统边值问题可以转化为初值问题这一特点,改进了周期解的打靶 法数值求解. 在计算边界条件代数方程关于待定初值参数导数的过程中利用前一次 Runge-Kutta方法计算得到的节点函数值并通过再次利用Runge-Kutta方法获得了该导数值. 用此方法求解了Duffing方程及非线性转子---轴承系统的周期解,用Floquet理论判断了 周期解的稳定性,与普通打靶法作了比较,验证了方法的有效性.     

Asymptotic Nonlinear Behaviors in Transverse Vibration of an Axially Accelerating Viscoelastic String

This paper investigates longtime dynamical behaviors of an axially accelerating viscoelastic string with geometric nonlinearity. Application of Newton's second law leads to a nonlinear partial-differential equation governing transverse motion of the string. The Galerkin method is applied to truncate the partial-differential equation into a set of ordinary differential equations. By use of the Poincare maps, the dynamical behaviors are presented based on the numerical solutions of the ordinary differential equations. The bifurcation diagrams are presented for varying one of the following parameter: the mean transport speed, the amplitude and the frequency of transport speed fluctuation, the string stiffness or the string dynamic viscosity, while other parameters are fixed.     

NONLINEAR DYNAMICS OF AXIALLY ACCELERATING VISCOELASTIC BEAMS BASED ON DIFFERENTIAL QUADRATURE

This paper investigates nonlinear dynamical behaviors in transverse motion of an axially accelerating viscoelastic beam via the differential quadrature method. The governing equation, a nonlinear partial-differential equation, is derived from the viscoelastic constitution relation using the material derivative. The differential quadrature scheme is developed to solve numerically the governing equation. Based on the numerical solutions, the nonlinear dynamical behaviors are identified by use of the Poincare map and the phase portrait. The bifurcation diagrams are presented in the case that the mean axial speed and the amplitude of the speed fluctuation are respectively varied while other parameters are fixed. The Lyapunov exponent and the initial value sensitivity of the different points of the beam, calculated from the time series based on the numerical solutions, are used to indicate periodic motions or chaotic motions occurring in the transverse motion of the axially accelerating viscoelastic beam.     

Arbitrary amplitude periodic solutions for parametrically excited systems with time delay

Periodic solutions for parametrically excited system under state feedback control with a time delay are investigated. Using the asymptotic perturbation method, two slow-flow equations for the amplitude and phase of the parametric resonance response are derived. Their fixed points correspond to limit cycles (phase-locked periodic solutions) for the starting system. In the system without control, periodic solutions (if any) exist only for fixed values of amplitude and phase and depend on the system parameters and excitation amplitude. In many cases, the amplitudes of periodic solutions do not correspond to the technical requirements. On the contrary, it is demonstrated that, if the vibration control terms are added, stable periodic solutions with arbitrarily chosen amplitude and phase can be accomplished. Therefore, an effective vibration control is possible if appropriate time delay and feedback gains are chosen.     

Low-Dimensional Description of Oscillatory Thermal Convection: The Small Prandtl Number Limit

 Low-dimensional models are derived for transitional, buoyancy-driven flow in a vertical channel with prescribed spatially periodic heating. Stationary characteristic structures (empirical eigenfunctions) are identified by applying proper orthogonal decomposition to numerical solutions of the governing partial differential equations. A Galerkin procedure is then employed to obtain suitable low-order dynamical models. Stability analysis of the fixed points of the low-order systems predicts conditions at the primary flow instability that are in very good agreement with direct numerical solutions of the full model. This agreement is found to hold as long as the low-order system possesses a possible Hopf bifurcation (minimum two-equation) mechanism. The effect of the number of retained eigenmodes on amplitude predictions is examined. Received 25 January 1996 accepted 19 July 1996     

On relative equilibria and integrable dynamics of point vortices in periodic domains

The motion of two point vortices defines an integrable Hamiltonian dynamical system in either singly or doubly periodic domains. The motion of three point vortices in these domains is also integrable when the net circulation is zero. The relative vortex motion in both domains can be reduced to advection of a passive particle by fixed vortices in an equivalent Hamiltonian system. A survey of the solutions for vortex motion in these systems is discussed. Some initial conditions lead to relative equilibria, or vortex configurations that move without change of shape or size. These configurations can be determined as stagnation points in the reduced problem or through explicit solution of the governing equations. These periodic point-vortex systems present a rich collection of interesting solutions despite the few degrees of freedom, and several questions on this subject remain open.     

An Estimate for the Morse Index of a Stokes Wave

Stokes waves are steady periodic water waves on the free surface of an infinitely deep irrotational two-dimensional flow under gravity without surface tension. They can be described in terms of solutions of the Euler–Lagrange equation of a certain functional; this allows one to define the Morse index of a Stokes wave. It is well known that if the Morse indices of the elements of a set of non-singular Stokes waves are bounded, then none of them is close to a singular one. The paper presents a quantitative variant of this result.     

Bifurcations in a parametrically excited non-linear oscillator

A non-linear parametrically excited oscillator, that includes van der Pol as well as Duffing type non-linearities, is studied for its small non-linear motions using the method of averaging. The averaged equations, which form a dynamical system on the plane and depend on the linear damping and the detuning, are analyzed for their constant and periodic solutions. Bendixon's criterion is used to deduce the existence and the non-existence of limit cycle solutions for various values of the parameters. Then, using local bifurcation theory for “saddle-node”, pitchfork and “Hopf” bifurcations and some results from one and two parameter unfoldings of degenerate singularities, a partial bifurcation set is constructed. Since constant and periodic solutions of the averaged system correspond, respectively, to the periodic solutions and almost periodic or amplitude modulated motions of the original oscillator, the bifurcation set indicates some ways in which periodic solutions can become “entrained” or can break the entrainment for almost periodic oscillations.     

Heteroclinic Connection of Periodic Solutions of Delay Differential Equations

For a certain class of delay equations with piecewise constant nonlinearities we prove the existence of a rapidly oscillating stable periodic solution and a rapidly oscillating unstable periodic solution. Introducing an appropriate Poincaré map, the dynamics of the system may essentially be reduced to a two dimensional map, the periodic solutions being represented by a stable and a hyperbolic fixed point. We show that the two dimensional map admits a one dimensional invariant manifold containing the two fixed points. It follows that the delay equations under consideration admit a one parameter family of rapidly oscillating heteroclinic solutions connecting the rapidly oscillating unstable periodic solution with the rapidly oscillating stable periodic solution.      

Characteristic solutions of scalar newton equations in one space dimension

Newton equations are dynamical systems on the space of fields. The solutions of a given equation which are curves of characteristic fields for its force are planar and have constant angular momentum. Separable solutions are characteristic with angular momentum equal to zero. A Newton equation is separable if and only if its characteristic equation is homogeneous. Separable equations correspond to invariants of homogeneous ordinary differential equations, and those associated with a given homogenous equation correspond to its generalized dilation symmetries. A Newton equation is compatible with the characteristic condition if and only if its characteristic equation is linear. Such equations correspond to invariants of linear ordinary differential equations. Those associated with a given linear equation correspond to the central force problems on its solution space. Regardless of compatibility, any Newton equation with a plane of characteristic fields has non-separable characteristic 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.     

Travelling wave solutions for a second order wave equation of KdV type

The theory of planar dynamical systems is used to study the dynamical behaviours of travelling wave solutions of a nonlinear wave equations of KdV type.In different regions of the parametric space,sufficient conditions to guarantee the existence of solitary wave solutions,periodic wave solutions,kink and anti-kink wave solutions are given.All possible exact explicit parametric representations are obtained for these waves.     

The Sub-Harmonic Bifurcation of Stokes Waves

Steady periodic water waves on the free surface of an infinitely deep irrotational flow under gravity without surface tension (Stokes waves) can be described in terms of solutions of a quasi-linear equation which involves the Hilbert transform and which is the Euler-Lagrange equation of a simple functional. The unknowns are a 2π-periodic function w which gives the wave profile and the Froude number, a dimensionless parameter reflecting the wavelength when the wave speed is fixed (and vice versa). Although this equation is exact, it is quadratic (with no higher order terms) and the global structure of its solution set can be studied using elements of the theory of real analytic varieties and variational techniques. In this paper it is shown that there bifurcates from the first eigenvalue of the linearised problem a uniquely defined arc-wise connected set of solutions with prescribed minimal period which, although it is not necessarily maximal as a connected set of solutions and may possibly self-intersect, has a local real analytic parametrisation and contains a wave of greatest height in its closure (suitably defined). Moreover it contains infinitely many points which are either turning points or points where solutions with the prescribed minimal period bifurcate. (The numerical evidence is that only the former occurs, and this remains an open question.) It is also shown that there are infinitely many values of the Froude number at which Stokes waves, having a minimal wavelength that is an arbitrarily large integer multiple of the basic wavelength, bifurcate from the primary branch. These are the sub-harmonic bifurcations in the paper's title. (In 1925 Levi-Civita speculated that the minimal wavelength of a Stokes wave propagating with speed c did not exceed 2πc ²/g. This is disproved by our result on sub-harmonic bifurcation, since it shows that there are Stokes waves with bounded propagation speeds but arbitrarily large minimal wavelengths.) Although the work of Benjamin & Feir} and others [9, 10] has shown Stokes waves on deep water to be unstable, they retain a central place in theoretical hydrodynamics. The mathematical tools used to study them here are real analytic-function theory, spectral theory of periodic linear pseudo-differential operators and Morse theory, all combined with the deep influence of a paper by Plotnikov [36]. Accepted: December 6, 1999     

Bifurcations of travelling wave solutions for a coupled nonlinear wave system

Introduction FangShaomeiandGuoBoling[1]consideredthefollowingtimeperiodicproblemof dampedcouplednonlinearwaveequations:ut f(u)x-αuxx βuxxx 2vvx=G1(u,v) h1(x),vt-γvxx 2(uv)x g(v)x=G2(u,v) h2(x),(1)whereα,β,γareconstants,andγ>0,β≠0.Undertheperiodicboundaryconditions,the authorsobtainedtheuniqueexistenceofstrongsolutionsfortheabovesystem.InthispaperweshallconsiderbifurcationbehaviorofthetravellingwavesolutionsofEq.(1)inthecaseGi(u,v)≡0,hi(u,v)≡0(i=1,2).Letξ=x-ct,u=u(x-ct),where cis…     

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。