Singularly perturbed reaction diffusion equations with time delay

Abstract A class of initial boundary value problems of differential-difference equations for reaction diffusion with a small time delay is considered. Under suitable conditions and by using the stretched variable method, a formal asymptotic solution is constructed. Then, by use of the theory of differential inequalities, the uniform validity of the solution is proved.     

Stability and bifurcation of a human respiratory system model with time delay

The stability and bifurcation of the trivial solution in the two-dimensional differential equation of a model describing human respiratory system with time delay were investigated. Formulas about the stability of bifurcating periodic solution and the directionof Hopf bifurcation were exhibited by applying the normal form theory and the center manifold theorem.Furthermore, numerical simulation was carried out.     

Singular perturbation of initial boundary value problems of reaction diffusion equation with delay

In this paper we consider the initial-boundary value problems for a class of general singularly perturbed delay reaction diffusion equations which are often met in applications, such as biomathematics and biochemistry. Applying the method of composite expansion we construct the formally asymptotic solution of the problem described. With the help of theory of upper and lower solutions we prove the uniformly validity of the formal solution and the existence of solution of the original problem. Project Supported by the Anhui Normal University Youth Natural Science Foundation of China     

Dynamic analysis of a flexible hub-beam system with tip mass

For a dynamic system of a rigid hub and a flexible cantilever beam, the traditional hybrid coordinate model assumes the small deformation in structural dynamics where axial and transverse displacements at any point in the beam are uncoupled. This traditional hybrid coordinate model is referred as the zeroth-order approximation coupling model in this paper, which may result in divergence to the dynamic problem of some rigid–flexible coupling systems with high rotational speed. In this paper, characteristics of a flexible hub-beam system with a tip mass is studied. Based on the Hamilton theory and the finite element discretization method, and in consideration of the second-order coupling quantity of the axial displacement caused by the transverse displacement of the beam, the rigid–flexible coupling dynamic model (referred as the first-order approximation coupling (FOAC) model in this paper) and the corresponding model in non-inertial system for the flexible hub-beam system with a tip mass are presented firstly, then the dynamic characteristics of the system are studied through numerical simulations under twos cases: the large motion of the system is known and is unknown. Simulation and comparison studies using both the traditional zeroth-order model and the proposed first-order model show that even small tip mass may affect dynamic characteristics of the system significantly, which may result in the largening of vibrating amplitude and the descending of vibrating frequency of the beam, and may affect end position of the hub-beam system as well. The effect of the tip mass becomes large along with the increasing of rotating speed of large motion of the system. When the large motion of the system is at low speed, the traditional ZOAC model may lead to a large error, whereas the proposed FOAC model is valid. When the large motion is at high speed, the ZOAC model may result in divergence to the dynamic problem of the flexible hub-beam system, while the proposed second model can still accurately describe the dynamic hub-beam system.     

Active control for a flexible beam with nonlinear hysteresis and time delay

A new active control method is presented to attenuate vibrations of a flexible beam with nonlinear hysteresis and time delay. The nonlinear and hysteretic behavior of the system is illustrated by the Bouc—Wen model. By specific transformation and augmentation of state parameters, we can convert the motion equation of the system with explicit time delay to the standard state space representation without any explicit time delay. Then the instantaneous optimal control method and Runge—Kutta method in fourth-order are applied to the controller design with time delay. Finally, in order to verify the effectivity of the time-delay controller proposed, numerical simulations are implemented. It is indicated by the simulation results that the control performance will deteriorate if neglect the time delay in process of the controller design and proposed time delay controller works well with both small and large time delay problems.     

Stability of a class of neural network models with delay

ＩｎｔｒｏｄｕｃｔｉｏｎＲｅｃｅｎｔｌｙ,ｔｈｅｏｒｅｔｉｃａｌａｎｄａｐｌｉｅｄｓｔｕｄｉｅｓｏｆｎｅｕｒａｌｎｅｔｗｏｒｋｍｏｄｅｌｈａｖｅｂｅｎｔｈｅｎｅｗｆｏｃｕｓｏｆｓｔｕｄｉｅｓｉｎｔｈｅｗｏｒｌｄ．Ｉｔｉｓｗｅｌ＿ｋｎｏｗｎｔｈａｔｑｕ...     

Nonlinear formulation for flexible multibody system with large deformation

In this paper, nonlinear modeling for flexible multibody system with large deformation is investigated. Absolute nodal coordinates are employed to describe the displacement, and variational motion equations of a flexible body are derived on the basis of the geometric nonlinear theory, in which both the shear strain and the transverse normal strain are taken into account. By separating the inner and the boundary nodal coordinates, the motion equations of a flexible multibody system are assembled. The advantage of such formulation is that the constraint equations and the forward recursive equations become linear because the absolute nodal coordinates are used. A spatial double pendulum connected to the ground with a spherical joint is simulated to investigate the dynamic performance of flexible beams with large deformation. Finally, the resultant constant total energy validates the present formulation. The project supported by the National Natural Science Foundation of China (10472066, 10372057). The English text was polished by Yunming Chen.     

Boundary value problem for a singularly perturbed nonlinear system

I.Intr0ducti0n'Manyresultshavebeeng0tinstudyingthesingu1arperturbation0fboundaryvalueproblemfornonlinearsystemeg#=j(t,g,y',8)(l.l)bythemethodandthetechniqueofdiagonalizationl'~'l.Thepapersarebasedonsucha.conditionthattheeigenvaluesofJacobimatrixlu'offwit…     

Initial layer phenomena for a class of singular perturbed nonlinear system with slow variables

The initial layer phenomena for a class of singular perturbed nonlinear system with slow variables are studied. By introducing stretchy variables with different quantity levels and constructing the correction term of initial layer with different “ thickness“, the Norder approximate expansion of perturbed solution concerning small parameter is obtained, and the “ multiple layer“ phenomena of perturbed solutions are revealed. Using the fixed point theorem, the existence of perturbed solution is proved, and the uniformly valid asymptotic expansion of the solutions is given as well.     

Active control and experiment study of a flexible hub-beam system

The first-order approximation coupling （FOAC） model was proposed recently for dynamics and control of flexible hub-beam systems. This model may deal with system dynamics for both low and high rotation speed, while the classical zeroth-order approximation coupling （ZOAC） model is only available for low rotation speed. This paper assumes the FOAC model to present experimental study of active positioning control of a flexible hub-beam system. Linearization and nonlinear control strategies are both considered. An experiment system based on a DSP TMS320F2812 board is introduced. The difference between linearization and nonlinear control strategies are studied both numerically and experimentally. Simulation and experimental results indicate that, linearized controller can make the system reach an expected position with suppressed vibration of flexible beam, but the time taken to position is longer than expected, whereas nonlinear controller works well with precise positioning, suppression of vibration and time control.     

Stability and bifurcation analysis in tri-neuron model with time delay

A simple delayed neural network model with three neurons is considered. By constructing suitable Lyapunov functions, we obtain sufficient delay-dependent criteria to ensure global asymptotical stability of the equilibrium of a tri-neuron network with single time delay. Local stability of the model is investigated by analyzing the associated characteristic equation. It is found that Hopf bifurcation occurs when the time delay varies and passes a sequence of critical values. The stability and direction of bifurcating periodic solution are determined by applying the normal form theory and the center manifold theorem. If the associated characteristic equation of linearized system evaluated at a critical point involves a repeated pair of pure imaginary eigenvalues, then the double Hopf bifurcation is also found to occur in this model. Our main attention will be paid to the double Hopf bifurcation associated with resonance. Some Numerical examples are finally given for justifying the theoretical results.     

漂浮基柔性两杆空间机械臂的关节运动鲁棒控制与柔性振动最优控制Robust joint motion control and vibration optimal control for a free-floating two-flexible-link space manipulator

讨论了载体位置、姿态均不受控情况下,具有有界干扰及有界未知参数的漂浮基柔性两杆空间机械臂的具有鲁棒性的关节运动控制与柔性振动最优控制算法设计问题。首先选择合理的联体坐标系,利用拉格朗日方程并结合动量守恒原理得到漂浮基柔性两杆空间机械臂系统的动力学方程。通过合理选择联体坐标系与利用奇异摄动理论,实现了两个柔性杆柔性振动之间、关节运动与两柔性杆柔性振动的解耦,得到了柔性两杆空间机械臂的慢变子系统与柔性臂快变子系统。针对两个子系统设计相应的控制规律,即增广鲁棒慢变子系统控制律与柔性臂快变子系统最优控制律,这两个相应的子系统控制规律综合到一起构成飘浮基柔性两杆空间机械臂总的关节运动与臂柔性振动控制的组合控制律。系统的数值仿真证实了方法的有效性。该控制方案不需要直接测量漂浮基的位置、移动速度和移动加速度。     

Numerical stability analysis of numerical methods for volterra integral equations with delay argument

The present paper deals with the stability properties of numerical methods for Volterra integral equations with delay argument. We assess the numerical stability of numerical methods with respect to the following test equations (0.1a) $$y\left( t \right) = \psi \left( 0 \right) + \int_0^t {\left( {py\left( s \right) + q\left( {s - \tau } \right)} \right)ds (0 \leqslant t \leqslant X)}$$ (0.1b) $$y\left( t \right) = \psi \left( t \right) \left( {t \in [ - \tau ,0)} \right)$$ where τ is a positive constant, and P and q are complex valued. We investigate the stability properties of reducible quadrature methods and θ-methods in the case of the above test equations     

In-plane mechanical behaviors of 2D repetitive frameworks with four-coordinate flexible joints and elbowed beam members

The mechanical properties of an open-framework structure constructed of joints and beam members are strongly influenced by both its geometrical configuration and joint flexibility. This paper clarifies the relationship between joint flexibility and Poisson's ratio, which is a mechanical criterion for solid deformation, and discusses two types of in-plane anisotropic structures made up of four-coordinate flexible joints and elbowed beam members. Uniaxial tensile analyses estimate the linear and nonlinear elastic properties of these frameworks by applying straightforward joint modeling with multi-rotational degrees of freedom. The numerical results show that these proposed frameworks produce a variety of deformability dependent on the joint flexibility in auxetic deformation with a negative Poisson's ratio, the folding mechanics under kinematic indeterminacy, and the transition of Poisson's ratio between the positive and negative values. The geometrical and topological aspects of the obtained mechanical behaviors are discussed.     

Stability analysis of a noise control system in a duct by using delay differential equation

The paper deals with the criteria for the closed- loop stability of a noise control system in a duct. To study the stability of the system, the model of delay differential equation is derived from the propagation of acoustic wave governed by a partial differential equation of hyperbolic type. Then, a simple feedback controller is designed, and its closed- loop stability is analyzed on the basis of the derived model of delay differential equation. The obtained criteria reveal the influence of the controller gain and the positions of a sensor and an actuator on the closed-loop stability. Finally, numerical simulations are presented to support the theoretical results.     

Chaotic oscillations in a piecewise linear spring-mass system

Chaotic oscillations are useful in assessing the health of a structure. Hence, simple chaotic systems which can easily be realized mechanically or electro-mechanically are highly desired. We study a new piecewise linear spring-mass system. The chaotic behaviour in this system is characterized using bifurcation diagrams and the invariant parameters of the dynamics. We also show that there exists a stochastic analogue of this system, which mimics the dynamical features of its deterministic counterpart. This allows a greater flexibility in practical designs as the chaotic oscillations are obtained either deterministically or stochastically. Also, the oscillations are low dimensional, which reduces the computational resources needed for obtaining the invariant parameters of this system.     

Stability analysis of the active control system with time delay using IHB method

In this paper, the incremental harmonic balance method is employed to solve the periodic solution that a vibration active control system with double time delays generates, and the stability analysis of which is achieved by the Poincare theorem. The system stability regions can be obtained in view of time delay and feedback gain, the variation of which is also studied. It turns out that along with the increase of time delay, the active control system is not always from stable to unstable, and the system can be from stable to unstable state, whereas the system can be from unstable to stable state. The extent that the two times delays impact on the system stability region is mainly related to the relative magnitude of the two feedback gains. The system can maintain the stable state under the condition of the well-matched feedback gains. The results can provide evidence to design the control strategy of time-delayed feedback.     

Stability of vertical oscillations in an electrodynamic suspension system with a discrete guideway structure

Conclusions 1. The coupling of the vehicle oscillation equation and the equations for the currents in the guideway structures must be taken into account in the stability analysis of vertical oscillations in an electrodynamic suspension system with a discrete guideway structure.2. The reported calculations indicate that vertical oscillations are unstable over a wide range of travel velocities. Consequently, the execution of steady-state vehicle motion requires vertical stabilization of the system.3. Effective stabilization can be achieved by means of a dynamic vibration damper. Even for a small relative mass =0.025 of the optimally tuned damper the perturbed motion decays rapidly.Physicotechnical Problems Branch of the Institute of Geotechnical Mechanics, National Academy of Sciences of Ukraine, Dnepropetrovsk. Translated from Prikladnaya Mekhanika, Vol. 31, No. 7, pp. 88–93, July, 1995.