Buckling and initial post-buckling behavior of thin-walled I columns

The behavior of axially compressed thin-walled I columns after torsional buckling is discussed. At first the state of strain in a column subjected to a large angle of twist is outlined. The considerations are based on the following assumptions: (1) cross section is non-deformable in its own plane, (2) shear deformation is negligible, and (3) strains are small and elastic. The initial post-buckling equilibrium paths are determined by utilizing a perturbation approach. The finite-element and analytical procedure is presented. It has been shown that the point of bifurcation for simply supported, or clamped I column with constant cross section is symmetric and stable. Two examples of I columns of variable cross section are also considered. It is worthwhile noticing that in these examples the critical loads are beyond the limits described by the critical force for column with the minimum and the maximum cross section. The points of bifurcation in these cases are also symmetric and stable. This property of the bifurcation point is very important with regard to the sensitivity to the initial geometrical imperfections. In the case of the unstable point of bifurcation a drastic decrease of the value of the critical loads is possible, which does not hold for stable point.     

Local bifurcations analysis of a state-dependent delay differential equation

In this paper, a first-order equation with state-dependent delay and a nonlinear right-hand side is considered. The conditions of the existence and uniqueness of the solution of the initial value problem are supposed to be executed.The task is to study the behavior of solutions of the considered equation in a small neighborhood of its zero equilibrium. The local dynamics depends on real parameters, which are coefficients of the right-hand side decomposition in a Taylor series.The parameter, which is a coefficient at the linear part of this decomposition, has two critical values that determine the stability domain of the zero equilibrium. We introduce a small positive parameter and use the asymptotic method of normal forms in order to investigate local dynamics modifications of the equation near each two critical values. We show that the stability exchange bifurcation occurs in the considered equation near the first of these critical values, and the supercritical Andronov–Hopf bifurcation occurs near the second of these values (provided the sufficient condition is executed). Asymptotic decompositions according to correspondent small parameters are obtained for each stable solution. Next, a logistic equation with state-dependent delay is considered to be an example. The bifurcation parameter of this equation has the only critical value. A simple sufficient condition of the occurrence of the supercritical Andronov–Hopf bifurcation in the considered equation near a critical value has been obtained as a result of applying the method of normal forms.     

A perturbation analysis of interactive static and dynamicbifurcations

The instability behavior of a nonlinear autonomous system in the vicinity of a coincident critical point, which leads to interactions between static and dynamic bifurcations, is studied. The critical point considered is characterized by a simple zero and a pair of pure imaginary eigenvalues of the Jacobian, and the system contains two independent parameters. The static and dynamic bifurcations and quasiperiodic motions resulting from the interaction of the bifurcation modes and the associated invariant tori are analyzed by a novel unification technique that is based on an intrinsic perturbation procedure. Divergence boundary, dynamic bifurcation boundary, secondary bifurcations, and invariant tori are determined explicitly. Two illustrative examples concerning control systems are presented     

On the finite element formulation of bifurcation mode of creep buckling of axisymmetric shells

In this paper, a finite element formulation is given in detail for the creep buckling of an axisymmetric shell. A special emphasis is placed on the bifurcation mode of creep buckling. A bifurcation point is determined by examining the shape of the potential energy in the vicinity of an axisymmetric equilibrium state obtained from a creep deformation analysis in the prebuckling stage. To illustrate the capability of the finite element formulation, a numerical example is presented for the creep buckling of a shallow spherical shell subjected to a uniform external pressure. In this analysis, not only the axisymmetric snap-through type but also the asymmetric bifurcation one are considered as buckling modes.     

Local feedback stabilization and bifurcation control, II. Stationary bifurcation

Local feedback stabilization and bifurcation control of nonlinear systems are studied for the case in which the critical linearized system possesses a simple zero eigenvalue. Sufficient conditions are obtained for local stabilizability of the equilibrium point at criticality and for local stabilizability of bifurcated equilibria. These conditions involve assumptions on the controllability of the critical mode for the linearized system. Explicit stabilizing feedback controls are constructed. The Projection Method of analysis of stationary bifurcations is employed. This work complements an earlier study by the same authors (Systems Control Lett.7 (1986) 11–17) of stabilization and bifurcation control in the (Hopf bifurcation) case of two pure imaginary eigenvalues of the linearized system at criticality.     

Multiscale analysis of defective multiple-Hopf bifurcations

An adapted version of the Multiple Scale Method is formulated to analyze 1:1 resonant multiple Hopf bifurcations of discrete autonomous dynamical systems, in which, for quasi-static variations of the parameters, an arbitrary number m of critical eigenvalues simultaneously crosses the imaginary axis. The algorithm therefore requires discretizing continuous systems in advance. The method employs fractional power expansion of a perturbation parameter, both in the state variables and in time, as suggested by a formal analogy with the eigenvalue sensitivity analysis of nilpotent (defective) matrices, also illustrated in detail. The procedure leads to an order-m differential bifurcation equation in the complex amplitude of the unique critical eigenvector, which is able to capture the dynamics of the system around the bifurcation point. The procedure is then adapted to the specific case of a double Hopf bifurcation (m = 2), for which a step-by-step, computationally-oriented version of the method is furnished that is directly applicable to solve practical problems. To illustrate the algorithm, a family of mechanical systems, subjected to aerodynamic forces triggering 1:1 resonant double Hopf bifurcations is considered. By analyzing the relevant bifurcation equation, the whole scenario is described in a three-dimensional parameter space, displaying rich dynamics.     

Bifurcation analysis of PID-controlled neuromuscular blockade in closed-loop anesthesia

The problem of PID-controlled neuromuscular blockade (NMB) in closed-loop anesthesia is considered. Contrary to the usual practice of designing PID-controllers for nonlinear systems on the basis of a linearized model and online tests, bifurcation analysis is utilized in this paper for that purpose. Two nonlinear Wiener models for the NMB are considered: a conventional pharmacokinetic/pharmacodynamic (PK/PD) model and a parsimonious model suitable for online parameter estimation. The models under a PID feedback are analyzed in order to discern the safe intervals of the controller parameters that are not subject to complex dynamical phenomena. The parsimony of the mathematical model is instrumental in minimizing the number of bifurcation parameters. The analyses show that the closed-loop systems undergo Andronov–Hopf bifurcation at a point in the model parameter space giving rise to nonlinear oscillations. For steeper, but still feasible slopes of the nonlinear function parameterizing the static nonlinearity of the Wiener models, deterministic chaos can arise in the closed loop for lower concentrations of the anesthetic drug. A model-based PID-controller tuning procedure is suggested that guarantees a certain settling time and robustness margin of the resulting loop with respect to the bifurcation. The tuning procedure is illustrated on mathematical models identified from patient data and the corresponding PID controllers.     

分数阶随机Duffing系统的Hopf分岔

运用正交多项式逼近原理，研究了分数阶随机Duffing系统在零平衡点的Hopf分岔．首先，运用Laguerre正交多项式逼近法将含有随机参数的分数阶Duffing系统转化为等价的确定性系统，然后通过数值计算求得其响应．最后，利用两个引理求得等价系统发生Hopf分岔行为的临界值，并通过数值模拟验证了理论分析结果．     

分数阶计算机病毒模型的稳定性与分岔分析

随着有线和无线通信网络的普及,计算机病毒已经成为当代信息社会的一大威胁,单纯依靠杀毒软件已经无法彻底清除病毒,而通过对其在互联网上的传播机制的分析,以及对其模型的研究,可以找到有效的防范计算机病毒的对策。因此,基于非线性动力学与分数阶系统理论,建立了一类具有饱和发生率的分数阶时滞SIQR计算机病毒模型。计算出模型的平衡点,并通过分析相应的特征方程研究了时滞对平衡点稳定性的影响。选择时滞作为分岔参数,得到了发生Hopf分岔的时滞临界值。研究发现,系统的动力学行为依赖于分岔的临界值,同时给出了系统局部稳定和产生Hopf分岔的条件。在此基础上,研究了分数阶阶次的变化对分岔阈值的影响。最后,通过数值模拟验证了理论分析的正确性。     

Stability and Hopf bifurcation of a general delayed recurrent neural network.

In this paper, stability and bifurcation of a general recurrent neural network with multiple time delays is considered, where all the variables of the network can be regarded as bifurcation parameters. It is found that Hopf bifurcation occurs when these parameters pass through some critical values where the conditions for local asymptotical stability of the equilibrium are not satisfied. By analyzing the characteristic equation and using the frequency domain method, the existence of Hopf bifurcation is proved. The stability of bifurcating periodic solutions is determined by the harmonic balance approach, Nyquist criterion, and graphic Hopf bifurcation theorem. Moreover, a critical condition is derived under which the stability is not guaranteed, thus a necessary and sufficient condition for ensuring the local asymptotical stability is well understood, and from which the essential dynamics of the delayed neural network are revealed. Finally, numerical results are given to verify the theoretical analysis, and some interesting phenomena are observed and reported.     

Hopf bifurcation and oscillations in a communication network with heterogeneous delays

In this paper, we investigate stability, bifurcation and oscillations arising in a single-link communication network model with a large number of heterogeneous users adopting a Transmission Control Protocol (TCP)-like rate control scheme with an Active Queue Management (AQM) router. In the system considered, different user delays are known and fixed but taken from a given distribution. It is shown that for any given distribution of delays, there exists a critical amount of feedback (due to AQM) at which the equilibrium loses stability and a limit cycling solution develops via a Hopf bifurcation. The nature (criticality) of the bifurcation is investigated with the aid of Lyapunov-Schmidt perturbation method. The results of the analysis are numerically verified and provide valuable insights into dynamics of the AQM control system.     

超声速流中二元机翼的颤振与极限环

以沉浮和俯仰自由度上具有间隙立方结构非线性的二元机翼模型为例,考虑系统的结构阻尼,建立了系统的非线性动力学方程.通过修正的三阶活塞理论模拟了超声速流中机翼的非定常气动力和气动力矩.引入无量纲参数将系统动力学方程无量纲化,通过数值模拟得到了二元机翼的时域响应和系统的相轨迹变化规律.通过系统的分岔图得到了无量纲参数和系统周...     

Bifurcation and post-buckling analysis of laminated composite plates via reduced basis technique

A reduced basis technique and a problem-adaptive computational algorithm are presented for the bifurcation and post-buckling analysis of laminated anisotropic plates. The computational algorithm can be conveniently divided into three distinct stages. The first stage is that of determining the bifurcation point. The plate is discretized by using displacement finite element (or finite difference) models. The special symmetries exhibited by the response of the anisotropic plate are used to reduce the size of the analysis region. The vector of unknown nodal parameters is expressed as a linear combination of a small number of basis vectors, and a Rayleigh-Ritz technique is used to approximate the finite element equations by a small system of algebraic equations. The reduced equations are used to determine the bifurcation point and the associated eigen mode of the panel.In the second stage of the bifurcation buckling mode is used to obtain a nonlinear solution in the vicinity of the bifurcation point and new (updated) sets of basis vectors and reduced equations are generated. In the third stage the reduced equations are used to trace the post-buckling paths.The effectiveness of the proposed technique for predicting the bifurcation and post-buckling behavior of plates is demonstrated by means of numerical examples for plates loaded by means of prescribed edge displacements.     

Oscillation control for n-dimensional congestion control model via time-varying delay

Time delay or round trip time (RTT) is an important parameter in the model of Internet congestion control. On the one hand, the delay may induce oscillation via the Hopf bifurcation. In the present paper, a congestion control model of n dimensions is considered to study the delay-induced oscillation. By linear analysis of the n-dimensional system, the critical delay for the Hopf bifurcation is obtained. To describe the relation between the delay and oscillation analytically, the method of multiple scales (M...     

Delayed Hopf bifurcation in time-delayed slow-fast systems

This paper presents an investigation on the phenomenon of delayed bifurcation in time-delayed slow-fast differential systems.Here the two delayed's have different meanings.The delayed bifurcation means that the bifurcation does not happen immediately at the bifurcation point as the bifurcation parameter passes through some bifurcation point,but at some other point which is above the bifurcation point by an obvious distance.In a time-delayed system,the evolution of the system depends not only on the present ...     

一类具有饱和发生率的时滞恶意病毒传播模型的分岔控制策略

考虑非线性的饱和发生率,建立一种刻画信息物理融合系统(cyber-physical systems, CPS)中恶意病毒传播的SIRS(susceptible-infected-recovered-susceptible)模型.为了避免因Hopf分岔的产生致使恶意病毒传播扩散,采用参数调节法和状态反馈法相结合的混合分岔控制策略,研究信息物理融合系统的Hopf分岔控制问题,建立受控系统的稳定性条件和分岔判据,探明控制增益参数对Hopf分岔点和分岔极限环幅值的影响规律,并给出分岔阈值与增益参数间的关系图.数值仿真结果表明,所提出的混合分岔控制策略不仅能够改变Hopf分岔点的位置,而且可以有效调节极限环幅值的大小,使得信息物理融合系统产生预期的动力学行为,有效降低恶意病毒传播的危害.     

Static,eigenvalue problem and bifurcation analysis of MEMS arches actuated by electrostatic fringing-fields

In this work, we investigate the structural behavior of a micro-electromechanical system arch microbeam actuated by electric fringing-fields where the electrodes are located at both side of the microbeam. In this particular configuration, the electrostatic actuating force is caused by the asymmetry of the fringing electric fields acting in a direction opposite to the relative deflection of the microbeam. A reduced-order model is derived for the considered system using the so-called Galerkin decomposition and assuming linear undamped mode shapes of a straight beam as basis functions in the decomposition process. A static analysis is performed to investigate the occurrence of any structural instability. The eigenvalue problem is then investigated to calculate the fundamental as well as higher natural frequencies variation of the microbeam with the applied DC load. A bifurcation analysis is then implemented to derive a criterion for whether symmetric or asymmetric bifurcation is occurring during the static structural instability. The results show elimination of the so-called pull-in instability in this kind of systems as compared to the regular case of parallel-plates electrostatic actuation. The bifurcation analysis shows that the arch goes for asymmetric bifurcation (symmetry breaking) with increase in initial elevation without the occurrence of symmetric bifurcation (snap-through) for any initial elevation.     

On the rotating beam: Some theoretical and numerical results

A bifurcation analysis for a thin beam rotating about an axis is carried out by the Golubitsky-Schaeffer theory of singularities. The rotation velocity is the bifurcation parameter, while the constants describing the position of the beam are considered as small perturbation parameters. A finite difference approximation of the problem is also introduced, and an error analysis is given.     

The nonlinear behaviour of a slender flexible cylinder pinned or clamped at both ends and subjected to axial flow

The nonlinear dynamics of a slender flexible cylinder subjected to axial flow is studied when both its ends are either pinned or clamped, particularly focusing on the post-critical behaviour. In both cases, the system is stable at low flow velocities until the critical flow for divergence, at which point the initial equilibrium position of the cylinder becomes unstable and a new stable buckled solution arises (together with its symmetric counterpart). The amplitude of the buckled solution increases with the flow velocity. At higher flow, the buckled stationary cylinder loses stability by a Hopf bifurcation, after which a periodic solution arises. The frequency of oscillation after this Hopf bifurcation, in the case of a pinned–pinned cylinder, is almost twice as high as that in the clamped–clamped case, due to the dynamic loss of stability in a higher mode in the former case. The periodic solution is followed by a period-doubling bifurcation, giving rise to period-2 oscillations. The system undergoes a torus bifurcation afterward, followed by quasiperiodic and chaotic oscillations at higher flow velocities. All the critical flow velocities for the pinned–pinned cylinder are smaller than those for the clamped–clamped one. In the case of a pinned–pinned cylinder, at still higher flow velocities, there exists a range of flow velocities in which chaotic and static solutions co-exist; this has not been observed in the clamped–clamped case.