Limit cycles and homoclinic orbits and their bifurcation of Bogdanov-Takens system

A quantitative analysis of limit cycles and homoclinic orbits, and the bifurcation curve for the Bogdanov-Takens system are discussed. The parameter incremental method for approximate analytical-expressions of these problems is given. These analytical-expressions of the limit cycle and homoclinic orbit are shown as the generalized harmonic functions by employing a time transformation. Curves of the parameters and the stability characteristic exponent of the limit cycle versus amplitude are drawn. Some of the limit cycles and homoclinic orbits phase portraits are plotted. The relationship curves of parameters μ and A with amplitude a and the bifurcation diagrams about the parameter are also given. The numerical accuracy of the calculation results is good.     

On homoclinic bifurcation system in the presence of stochastic perturbation

By virture of the singular point theory for one-dimension diffusion process and the stochastic averaging approach of energy envelop, the bifurcation behavior of a homoclinic bifurcation system, which is in the presence of parametric white noise and is concealed behind a codimension two bifurcation point, is investigated in this paper. Supported by the National Science Foundation of China under Grant No. 19602016.     

A Collocation-Based Algorithm for Computing the Pull-In Range of a Class of Phase-Locked Loops

The pull-in range (ω_p) of a phase-locked loop (PLL) is defined as the maximum value of loop detuning ω_0s for which pull-in occurs from anywhere on the PLL's phase plane. That is, pull-in is guaranteed from anywhere on the phase plane if ω_0s < ω_p. Simple approximation is available for computing ω_p for the high gain PLL where saddle-node bifurcation occurs at ω_0s = ω_p. Unlike the high gain case, a simple approximation for ω_p is not available for the low gain case where bifurcation from a separatrix cycle occurs at ω_0s = ω_p. The vector field model for a class of second-order PLLs is shown to have rotational properties, which imply the existence of a separatrix cycle. The external stability of this separatrix cycle is an indicator of the type of bifurcation (saddle-node or separatrix cycle) which terminates the limit cycle associated with the PLL's stable false lock state and the PLL pulls-in (i.e. achieve phase lock). A formula is given for determining the separatrix cycle's stability, which indicates that these paratrix cycle is externally stable for small values of closed loop gain. A collocation-based algorithm is presented for computing the PLL's separatrix cycle and the value of pull-in range frequency ω_0s = ω_p at which a stable separatrix cycle exists.     

Supercritical as well as subcritical Hopf bifurcation in nonlinear flutter systems

The Hopf bifurcations of an airfoil flutter system with a cubic nonlinearity are investigated, with the flow speed as the bifurcation parameter. The center manifold theory and complex normal form method are Used to obtain the bifurcation equation. Interestingly, for a certain linear pitching stiffness the Hopf bifurcation is both supercritical and subcritical. It is found, mathematically, this is caused by the fact that one coefficient in the bifurcation equation does not contain the first power of the bifurcation parameter. The solutions of the bifurcation equation are validated by the equivalent linearization method and incremental harmonic balance method.     

A Note on Bifurcations of Motions in the Lorentz System

The limit-cycle phenomenon in the Lorenz system is studied with considering bifurcation slates of a dynamic system. It is established that the trajectory has a complex structure and includes intervals of periodic solutions of different kinematics and an interval of saddle-node solution     

Control of the homoclinic bifurcation in buckled beams: Infinite dimensional vs reduced order modeling

A method for controlling non-linear dynamics and chaos is applied to the infinite dimensional dynamics of a buckled beam subjected to a generic space varying time-periodic transversal excitation. The homoclinic bifurcation of the hilltop saddle is identified as the undesired dynamical event, because it triggers, e.g., cross-well scattered (possibly chaotic) dynamics. Its elimination is then pursued by a control strategy which consists in choosing the best spatial and temporal shape of the excitation permitting the maximum shift of the homoclinic bifurcation threshold in the excitation amplitude-frequency parameters space.The homoclinic bifurcation is detected by the Holmes and Marsden's theorem [A partial differential equation with infinitely many periodic orbits: chaotic oscillations of a forced beam, Arch. Ration. Mech. Anal. 76 (1981) 135-165] constituting a generalization of the classical Melnikov's theory. Two classes of boundary conditions (b.c.) are identified: for the first, the Melnikov function is exactly the same as obtained with the reduced order models, while for the second, which is more general, this is no longer true, and the non-linear normal modes theory is used. Based on this distinction, the control method is then separately applied to the two cases, and the optimal spatial and temporal shapes of the excitation are determined.A detailed comparison of the infinite vs finite dimensional models is performed with respect to the control features, and it is shown that, depending on the b.c., the control based on the reduced order model provides either exact or engineering acceptable results, although more systematic investigations are required to generalize the last conclusion.     

The global bifurcation and chaos for the coupling between longitudinal and transverse vibration of a thin elastic plate in large overall motion

This paper presents the global bifurcation and chaotic behavior for the coupling of longitudinal and transverse vibration of a thin elastic plate in large overall motion. First the parametric equations of the homoclinic orbits of such a system is obtained. Then, by using the Melnikov method and digital computer simulation. the behavior of bifurcation and chaos of this vibration system is investigated in the cases of different resonances. The obvious difference between the transverse vibration and the coupling of transverse and longitudinal vibration is also shown.The project supported by the National Natural Science Foundation of China.     

Coexistence of the chaos and the periodic solutions in planar fluid flows

This paper discusses the dynamic behavior of the Kelvin-Stuart cat’s eye flow underperiodic perturbations.By means of the Melnikov method the conditions to havebifurcations to subharmonics of even order for the oscillating orbits and to have bifurcationsto subharmonics of any order for the rotating orbits are given,and further,the coexistencephenomena of the chaotic motions and periodic solutions are presented.     

Separatrices and limit cycles of strongly nonlinear oscillators by the perturbation-incremental method

The perturbation-incremental method is applied to determine the separatrices and limit cycles of strongly nonlinear oscillators. Conditions are derived under which a limit cycle is created or destroyed. The latter case may give rise to a homoclinic orbit or a pair of heteroclinic orbits. The limit cycles and the separatrices can be calculated to any desired degree of accuracy. Stability and bifurcations of limit cycles will also be discussed.     

Buckling of a beam subjected to electromagnetic force and its stabilization by controlling the perturbation of the bifurcation

For a beam subjected to electromagnetic force, magnetoelastic buckling due to the increase of such force is theoretically investigated by taking account of the nonlinearity of the electromagnetic force and the elastic force of the beam. Using Liapunov-Schmidt method and center manifold theory, the equilibrium space, the bifurcation set and the bifurcation diagram are theoretically derived. Also, the effect of the higher modes other than the buckling mode on the mode shape of the postbuckling state is discussed. Furthermore, a control method to stabilize the magnetoelastic buckling is proposed, and the unstable equilibrium state of the beam in the postbuckling state, i.e., the straight position of the beam, is stabilized by controlling the perturbation of the bifurcation.     

Theoretical study on the bifurcation of vortexes structure for flow in curved tube

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Axisymmetric bifurcation analysis in soils by the tangential-subloading surface model

This article discusses localized bifurcation modes corresponding to shear band formation and diffuse bifurcation modes corresponding to bulge formation for cylindrical soil specimen subjected to an axisymmetric load under undrained conditions. We employ the tangential-subloading surface model, which exhibits the characteristic regimes of the governing equations: elliptic, hyperbolic and parabolic. Also, conditions for shear band formation, shear band inclination, diffuse bulging formation, and the long and short wavelength limits of diffuse bulging modes are discussed in relation to material properties and their state of stress, i.e. the stress ratio and the normal-yield ratio. Tangential-plastic strain rate term is required for the analyses of shear band and diffuse bulging. The shear band and the diffuse bulging are generated in not only normal-yield but also subyield states and they are severely affected by the normal-yield ratio describing the degree of approach to the normal-yield state.     

Side‐wall effects on the bifurcation of the flow through a sudden expansion

Three‐dimensional computations have been performed to study the flow through a symmetric sudden expansion with an expansion ratio of 3 at low Reynolds numbers. The aspect ratio of the flow channel is allowed to vary within a wide range to examine its influence on the flow which bifurcates from a symmetric state to an asymmetric state. The results reveal that the critical Reynolds number of the symmetry‐breaking bifurcation increases while the aspect ratio is reduced. The flow behaviour near the side walls is illustrated by using limiting streamlines. The origin of the singular points identifiable on the side wall can be traced back to the recirculating flows and the relevant reattachment/separation points in the core of the channel. It is seen that the determination of the exact critical Reynolds number is not trivial because it depends on how to define asymmetric flow. Computations have also been conducted to show that a slight asymmetry in the channel geometry causes a smooth transition from symmetric to non‐symmetric states. Copyright © 2007 John Wiley & Sons, Ltd.     

Improvements on the arc-length-type method

Arc-length-type and energy-type methods are two main strategies used in structural nonlinear tracing analysis, but the former is widely used due to the explicitness and clarity in conception, as well as the convenience and reliability in calculation. It is very important to trace the complete load-deflection path in order to know comprehensively the characteristics of structures subjected to loads. Unfortunately, the nonlinear analysis techniques are only workable for tracing the limit-point-type equilibrium path. For the bifurcation-point-type path, most of them cannot secure a satisfactory result. In this paper, main arc-length-type methods are reviewed and compared, and the possible reasons of failures in tracing analysis are briefly discussed. Some improvements are proposed, a displacement perturbation method and a force perturbation method are presented for tracing the bifurcation-point-type paths. Finally, two examples are analyzed to verify the ideas, and some conclusions are drawn with respect to the arc-length-type methods. The project supported by the Special Research Fund for Doctor Program of Universities (9424702)     

Chaotic Saddles in Wada Basin Boundaries and Their Bifurcations by the Generalized Cell-Mapping Digraph (GCMD) Method

By means of the generalized cell-mapping digraph (GCMD) method, we studybifurcations governing the escape of periodically forced oscillatorsfrom a potential well, in which a chaotic saddle plays an extremelyimportant role. In this paper, we find the chaotic saddle anddemonstrate that it is embedded in a strange fractalbasin boundary which has the Wada property that any point that is on theboundary of that basin is also simultaneously on the boundary of atleast two other basins. The chaotic saddle in the Wada basin boundary,by colliding with a chaotic attractor, leads to a chaotic boundarycrisis with indeterminate outcome. A local saddle-node fold bifurcation,if the saddle of the saddle-node fold is located in tangency with thechaotic saddle in the Wada basin boundary, also results in a strangeglobal phenomenon, namely that the local saddle-node fold bifurcation hasglobally indeterminate outcome. We also investigate the origin andevolution of the chaotic saddle in the Wada basin boundary, particularlyconcentrating on its discontinuous bifurcations (metamorphoses). Wedemonstrate that the chaotic saddle in the Wada basin boundary iscreated by a collision between two chaotic saddles in differentfractal basin boundaries. After a final escape bifurcation, there onlyexists the attractor at infinity and a chaotic saddle with a beautifulpattern is left behind in the phase space.     

Selecting the Coefficients of Sliding Friction in the Problem on Frictional Interaction of Three Solids

The choice of the coefficients of sliding friction in the static equilibrium problem for a system of three solids with friction at two points is discussed. This system simulates the mechanism of gravitational seismic isolation of a solid. It is shown that the coefficients of friction must be identical at both points of frictional contact irrespective of the type of the material and surface finish. A formula for the balanced coefficient of friction is derived based on experimental data     

用拉格朗日函数分析不同模量静不定桁架内力

在外载荷作用下的不同模量静不定桁架平衡问题,是任意有限多个自变量的多元函数在任意有限多个约束条件下的极值问题,对采用拉格朗日乘数法求解此类极值问题进行了数学证明．通过求解不同模量静不定桁架极限载荷的几个算例,阐述拉格朗日乘数法在计算不同模量静不定桁架极限载荷中的应用．研究结果表明：采用拉格朗日乘数法求解不同模量静不定桁架极限载荷的通用性较强,用拉格朗日乘数法求解不同模量静不定桁架极限载荷的方法不但克服了常规方法需利用几何关系建立协调方程的缺陷,且具有力学概念清晰直观、计算过程简便、便于工程设计人员在实际中掌握和应用．