Singular bifurcations in higher index differential-algebraic equations

The present paper extends the singularity induced bifurcation theorem (SIBT) to higher index differential-algebraic equations (DAEs) in Hessenberg form. The SIBT arises in power system theory, and is also significant within the context of electrical circuits. This phenomenon is due to the presence of singularities in parameter-dependent problems, and it was originally proved for semiexplicit index-1 DAEs. We introduce a new proof of a matrix pencil-based version of this result, relying on the spectral features of the linearization of the underlying ordinary differential equation. This approach is then applied to higher index DAEs in Hessenberg form. For these structures, the SIBT is shown to follow from a minimal index change at the singularity. Additionally, we show that a Kronecker index change is also necessary for the existence of a singularity induced bifurcation point in semi-explicit index-1 and Hessenberg DAEs.     

非线性指标-3微分-代数系统的波形松弛算法的实现

本文探讨非线性指标-3微分-代数系统的波形松弛算法所涉及的理论模型和具体算例的求解。对于收敛性问题,我们利用谱半径条件证明该理论模型的收敛性,所得结论的收敛性条件相对较弱,并利用算例测试了收敛定理的正确性。结果显示:在求解非线性微分-代数系统时,波形松弛类算法是具有内在并行性的有效算法。     

关于微分-代数系统的Runge-Kutta迭代方法的研究

为了求解非自制指标-1的微分-代数系统,我们研究基于Runge-Kutta方法的动力学迭代过程,得到相关的非线性微分-代数方程的收敛理论,这类迭代过程具有一般性和灵活性,且沿着时间域网格点可以选取不同的插值函数.     

Differential-algebraic equations and singular perturbation methods in recurrent neural learning

This paper introduces a mathematical framework based on dynamical system theory, differential-algebraic equations (DAEs) and singularly perturbed (SP) systems, oriented to the analysis and design of on-line schemes for fixed point recurrent neural learning. New schemes proposed in this framework make it possible to relax some common assumptions in usual recurrent backpropagation (RBP) implementations. The scope of the work is not necessarily restricted to gradient-based adaptation methods, the results being applicable to more general learning strategies. The presented models clarify the relative timescaling between the network dynamics and the adaptation process in on-line techniques, including adjoint-based approaches. Certain restrictions on the learning speed are formalized through a 'rate of learning' limit appearing in the DAE/SP setting. Local convergence is rigorously stated, and certain Newton-based stabilization techniques are proposed regarding global issues in the presence of bifurcations, which typically introduce severe difficulties in common RBP methods. Some simulation examples concerning the synthesis of associative memories illustrate the applicability of the proposed techniques.     

多体系统动力学微分-代数方程广义-α 投影法

高效、稳定的微分-代数方程数值求解方法是多体系统动力学领域的关键问题之一。该文针对多体系统动力学指标3微分-代数方程,对目前多体系统动力学中引入的隐式时域逐步积分方法进行了深入研究,提出了适用于一般质量矩阵的广义-α -S法,并结合约束投影方法,构造了广义-α -S投影法。该方法既能较好地保持系统总能量,又能较高程度地同时满足位移约束、速度级约束和加速度级约束,并且在步长较大时可稳定求解,计算效率较高。     

A review of stochastic resonance: circuits and measurement

Noise in dynamical systems is usually considered a nuisance. However, in certain nonlinear systems, including electronic circuits and biological sensory systems, the presence of noise can enhance the detection of weak signals. The phenomenon is termed stochastic resonance and is of great interest for electronic instrumentation. We review and investigate the stochastic resonance of several bistable circuits. A new type of S characteristic circuit is demonstrated using simple nonlinear elements with an operational amplifier. Using this circuit, the effects on stochastic resonance were determined as the slope of the S shaped characteristic curve was varied     

An unsymmetrical mode of ferroresonance

Earlier studies of ferroresonance note the existence of unsymmetrical modes of operation of series circuits with square-loop reactors. Here the observation of stable second harmonics in a series ferroresonant circuit with a reactor having very nonsquare-loop core material is noted; these harmonics produce unsymmetrical circuit waveforms. Circuit behavior is described, and a qualitative explanation is offered. An approximate solution of the nonlinear circuit equations using the method of reversion of series shows good correlation with the experimental waveforms. From the analysis, the necessary condition for this mode of circuit operation is shown to be the presence of a constant component of flux in the circuit. Potential application of the circuit as a logic element is suggested.     

Analysis of Elastic-PlasticWaves in a Thin-Walled Tube By a Novel Lie-Group Differential Algebraic Equations Method

In this paper, we adopt the viewpoint of a nonlinear complementarity problem (NCP) to derive an index-one differential algebraic equations (DAEs) system for the problem of elastic-plastic wave propagation in an elastic-plastic solid undergoing small deformations. This is achieved by recasting the pointwise complementary trio in the elastic-plastic constitutive equations into an algebraic equation through the Fischer-Burmeister NCP-function. Then, for an isotropicallyhardening/ softening material under prescribed impulse loadings on a thin-walled tube with combined axial-torsional stresses, we can develop a novel algorithm based on the Lie-group differential algebraic equations (LGDAE) method to iteratively solve the resultant DAEs at each time marching step, which converges very fast. The one-dimensional axial-torsional wave propagation problems under different imposed dynamical loading conditions and initial conditions are solved, to assess the performance of the LGDAE.     

Parallel sensitivity analysis for efficient large-scale dynamic optimization

An efficient parallel algorithm for the computation of parametric sensitivities for differential-algebraic equations (DAEs) with a focus on dynamic optimization problems is presented. A speedup of about 4 can be obtained for process models of more than 13500 DAEs and 75 parameters employing 8 processor cores in parallel using a Windows based system. The algorithm obtains its efficiency by decoupling the sensitivity equations from the state equations of the DAE. Furthermore, the costly Jacobian matrices are computed separately by other processes. The computational effort for a combined state and sensitivity integration can almost be reduced to the computational effort of the pure state integration, which is the theoretical limit of the suggested approach.     

On the implementation of an analog ATPG: the nonlinear case

In order to reduce the complexity of the fault diagnosis equations and still retain computational simplicity, a self-testing algorithm has been proposed and implemented on a VMS VAX 11/780 for linear circuits. A prototype implementation of such an algorithm for nonlinear circuits and systems is presented. The proposed analog automatic test program generator (AATPG) for nonlinear circuits and systems is divided into offline and online processes. Unlike the simulation of the pseudocircuits in the linear case, which can be achieved by a matrix/vector multiplication, the circuit simulator SPICE is used to simulate the nonlinear pseudocircuits. The automatic SPICE code generator required for this simulation is presented. The proposed AATPG for nonlinear circuits has been implemented on a VMS VAX 11/780. The actual test can be run in either a fully automatic mode or interactively     

Dynamics of a Modified Predator-Prey System to Allow for a Functional Response and Time Delay

A modified predator-prey system described by two differential equations and an algebraic equation is discussed. Formulae for determining the direction of a Hopf bifurcation and the stability of the bifurcating periodic solutions are derived differential-algebraic system theory, bifurcation theory and centre manifold theory. Numerical simulations illustrate the results, which includes quite complex dynamical behaviour.     

The variational solution of electric and magnetic circuits

Voltage and current distributions in electric circuits comply with a stationary power condition. An alternative circuit analysis technique can be derived from this property. The unknowns can be varied until the power consumption of the circuit is calculated to be unchanging. At this point the voltages and currents are at their true solution. This variational approach to circuit analysis has been generally overlooked. No doubt this is due to the sufficiency of conventional circuit analysis techniques. However, a closer examination of the variational approach reveals a straightforward analytic procedure with unifying properties and graphically illustrated solutions. These features provide new educational opportunities and insight. As a research tool, the variational approach offers a unified method of solving problems which include fields and devices. The application to nonlinear circuits is also of current interest. An introductory step-by-step guide to the variational procedure for electric and magnetic networks is described in this article     

Double singularity-induced bifurcation points and singular Hopf bifurcations

The singularity-induced bifurcation and singular Hopf bifurcation theorems and the degeneracies that arise when Newton's laws are coupled to Kirchhoff's laws are explored. Such models are used in the electrical engineering literature to describe electrical power systems and they can take the form of either an index-1 differential-algebraic equation (DAE) or a singularly perturbed ordinary differential equation (ODE). As a consequence of the debate in the engineering literature as to which class of system is the 'true' representation of power systems, a discussion is included of the consequences of the power engineer's 'load-flow singularity' for both ODE and DAE.     

Deconstructing the core dynamics from a complex time-lagged regulatory biological circuit

Complex regulatory dynamics is ubiquitous in molecular networks composed of genes and proteins. Recent progress in computational biology and its application to molecular data generate a growing number of complex networks. Yet, it has been difficult to understand the governing principles of these networks beyond graphical analysis or extensive numerical simulations. Here the authors exploit several simplifying biological circumstances which thereby enable to directly detect the underlying dynamical regularities driving periodic oscillations in a dynamical nonlinear computational model of a protein?protein network. System analysis is performed using the cell cycle, a mathematically well-described complex regulatory circuit driven by external signals. By introducing an explicit time delay and using a `tearing-and-zooming? approach the authors reduce the system to a piecewise linear system with two variables that capture the dynamics of this complex network. A key step in the analysis is the identification of functional subsystems by identifying the relations between statevariables within the model. These functional subsystems are referred to as dynamical modules operating as sensitive switches in the original complex model. By using reduced mathematical representations of the subsystems the authors derive explicit conditions on how the cell cycle dynamics depends on system parameters, and can, for the first time, analyse and prove global conditions for system stability. The approach which includes utilising biological simplifying conditions, identification of dynamical modules and mathematical reduction of the model complexity may be applicable to other well-characterised biological regulatory circuits.     

On a driven nonlinear circuit and chaotic behaviour

Abstract

The chaotic behaviour of the sinusoidally driven nonlinear circuit proposed by Hunt is investigated. The results are compared with those from other driven nonlinear circuits.     

A combined finite element and loop analysis for nonlinearly interacting magnetic fields and circuits

A general technique is described for combining a finite element representation of a quasi-stationary nonlinear magnetic field problem and the loop equations for all circuits, moving or stationary, which interact with the field. Various power quantities associated with the field and with each independent circuit loop can be estimated directly. Methods are presented for time integration of the first-order matrix differential equation describing both field and circuits. Saturating nonlinearities are treated iteratively. In particular, a transient problem in an idealized induction motor is solved. When permitted to reach steady state this solution yields points on the torque versus slip curve for the motor.     

Modified nodal analysis: an essential addition to electricalcircuit theory and analysis

The known limitations of classical mesh and nodal methods of analysing linear electrical circuits are described before considering an established modification of the nodal approach. The method, known as 'modified nodal analysis', has none of the limitations of the basic nodal technique and is well suited both to symbolic and numeric analysis of complex circuits using modern matrix-based software. The simplicity of incorporating into the matrix equations all types of passive and active circuit elements is demonstrated and examples are used to illustrate further the efficacy of the method. It is emphasised that the absence of this circuit analysis technique from many academic engineering courses is totally at variance with its widespread application in modern circuit simulation packages     

Frequency control technique applicable to either voltage- or current-feedback magnetically-coupled switching-transistor oscillators

Through the use of nonlinear circuit techniques similar to those normally associated with series-connected magnetic amplifiers, it is possible to modify the basic circuits of almost all of the numerous types of magnetically-coupled switching-transistor multivibrators so as to provide for frequency control of these circuits by means of a low-level dc control signal. Using two well-known types of such inverter circuits as examples, this paper describes the basic principles involved in this frequency control technique and discusses the operation of two test circuits. A primary advantage of this frequency control method is its physical simplicity and low cost.     

热冲击下的复合材料壳刚柔耦合动力学研究

研究了在热冲击下任意形状(仅一个方向有曲率)复合材料壳的非线性刚柔耦合动力学响应。根据Mindlin理论,建立了任意形状的复合材料壳的非线性应变-位移关系。借助于数学理论以及几何关系,描述了壳上任意点的变曲率。用虚功原理建立了动力学变分方程,并采用等参单元对壳的连续动力学方程进行离散,建立了中心刚体-复合材料壳的刚-柔耦合动力学方程。用高斯积分计算常值阵,为了提高计算效率,采用广义-α法结合Newton-Raph-son迭代法对动力学方程进行积分。将采用该方法计算得到的频率与ANSYS软件计算得到的作对比,验证了模型的正确性。通过算例分析了在热冲击作用下复合材料壳的线性、非线性的动力学特性,以及曲率、材料特性对动力学响应的影响。     

General-purpose analysis of nonlinear circuits containingsaturating/hysteretic inductors by the harmonic-balance technique

The systematic application of the harmonic-balance (HB) technique to nonlinear circuits containing saturating/hysteretic inductors subjected to high driving levels is discussed. Ferromagnetic hysteresis is described by the Jiles-Atherton (JA) model. The traditional HB approach is extended to accept a generalized parametric formulation of the nonlinearities allowing the JA model to be dealt with in a straightforward way. The nonlinear HB system is solved by Newton iteration coupled with the exact calculation of the Jacobian matrix and an advanced norm-reduction scheme. This results in excellent robustness and fast convergence of the analysis algorithm even at very large signal levels, in spite of the high nonlinearity of the JA differential equation. Two techniques for finding the initial magnetization curve are also discussed. The proposed approach is validated by comparing the hysteresis cycles of several materials obtained from HB calculations with time-domain and experimental results. An example of a general nonlinear circuit analysis problem is also provided. The analysis technique is complemented by a systematic algorithm for the local and global stability analysis of the large-signal steady state