Controlling chaos in a nonlinear pendulum using an extended time-delayed feedback control method

Chaos control is employed for the stabilization of unstable periodic orbits (UPOs) embedded in chaotic attractors. The extended time-delayed feedback control uses a continuous feedback loop incorporating information from previous states of the system in order to stabilize unstable orbits. This article deals with the chaos control of a nonlinear pendulum employing the extended time-delayed feedback control method. The control law leads to delay-differential equations (DDEs) that contain derivatives that depend on the solution of previous time instants. A fourth-order Runge–Kutta method with linear interpolation on the delayed variables is employed for numerical simulations of the DDEs and its initial function is estimated by a Taylor series expansion. During the learning stage, the UPOs are identified by the close-return method and control parameters are chosen for each desired UPO by defining situations where the largest Lyapunov exponent becomes negative. Analyses of a nonlinear pendulum are carried out by considering signals that are generated by numerical integration of the mathematical model using experimentally identified parameters. Results show the capability of the control procedure to stabilize UPOs of the dynamical system, highlighting some difficulties to achieve the stabilization of the desired orbit.     

Hopf Bifurcation Calculations in Delayed Systems with Translational Symmetry

The Hopf bifurcation of an equilibrium in dynamical systems consisting of n equations with a single time delay and translational symmetry is investigated. The Jacobian belonging to the equilibrium of the corresponding delay-differential equations always has a zero eigenvalue due to the translational symmetry. This eigenvalue does not depend on the system parameters, while other characteristic roots may satisfy the conditions of Hopf bifurcation. An algorithm for this Hopf bifurcation calculation (including the center-manifold reduction) is presented. The closed form results are demonstrated for a simple model of cars following each other along a ring.     

Switching Time and Parameter Optimization in Nonlinear Switched Systems with Multiple Time-Delays

In this paper, we consider a dynamic optimization problem involving a general switched system that evolves by switching between several subsystems of nonlinear delay-differential equations. The optimization variables in this system consist of: (1) the times at which the subsystem switches occur; and (2) a set of system parameters that influence the subsystem dynamics. We first establish the existence of the partial derivatives of the system state with respect to both the switching times and the system parameters. Then, on the basis of this result, we show that the gradient of the cost function can be computed by solving the state system forward in time followed by a costate system backward in time. This gradient computation procedure can be combined with any gradient-based optimization method to determine the optimal switching times and parameters. We propose an effective optimization algorithm based on this idea. Finally, we consider three numerical examples, one involving the 1,3-propanediol fed-batch production process, to illustrate the effectiveness and applicability of the proposed algorithm.     

Some aspects of causal & neutral equations used in modelling

The problem that motivates the considerations here is the construction of mathematical models of natural phenomena that depend upon past states. The paper divides naturally into two parts: in the first, we expound the inter-connection between ordinary differential equations, delay-differential equations, neutral delay-differential equations and integral equations (with emphasis on certain linear cases). As we show, this leads to a natural hierarchy of model complexity when such equations are used in mathematical and computational modelling, and to the possibility of reformulating problems either to facilitate their numerical solution or to provide mathematical insight, or both. Volterra integral equations include as special cases the others we consider. In the second part, we develop some practical and theoretical consequences of results given in the first part. In particular, we consider various approaches to the definition of an adjoint, we establish (notably, in the context of sensitivity analysis for neutral delay-differential equations) rôles for well-defined adjoints and ‘quasi-adjoints’, and we explore relationships between sensitivity analysis, the variation of parameters formulae, the fundamental solution and adjoints.     

线性定常系统特征模型的特征参量辨识

说明线性定常系统特征模型的特征参量是一组由高阶线性定常系统的相关信息压缩而成,于是不能简单的作为与状态无关的慢时变参数来处理. 基于特征建模思想,建立了线性定常系统特征模型的特征参量与子空间方法之间的联系,给出了一种该特征模型的特征参量 的合成辨识算法.同时证明了在用于子空间辨识的样本量充分大和用于状态估计的时间充分长的情况下, 特征参量的估计值与真值之间的误差达到充分小. 最后,对于一个六阶的单输入单输出线性定常系统的仿真例子,对投影的带遗忘因子最小二乘算法和合成辨识算法进行了比较,验证了合成辨识算法的有效性.     

Consistent Numerical Schemes for Solving Nonlinear Inverse Source Problems with Gradient-Type Algorithms and Newton–Kantorovich Methods

In this paper, algorithms of solving an inverse source problem for systems of production–destruction equations are considered. Numerical schemes that are consistent to satisfy Lagrange’s identity for solving direct and adjoint problems are constructed. With the help of adjoint equations, a sensitivity operator with a discrete analog is constructed. It links perturbations of the measured values with those of the sought-for model parameters. This operator transforms the inverse problem to a quasilinear system of equations and allows applying Newton–Kantorovich methods to it. A numerical comparison of gradient algorithms based on consistent and inconsistent numerical schemes and a Newton–Kantorovich algorithm applied to solving an inverse source problem for a nonlinear Lorenz model is done.     

Iterative parameter identification methods for nonlinear functions

This paper considers identification problems of nonlinear functions fitting or nonlinear systems modelling. A gradient based iterative algorithm and a Newton iterative algorithm are presented to determine the parameters of a nonlinear system by using the negative gradient search method and Newton method. Furthermore, two model transformation based iterative methods are proposed in order to enhance computational efficiencies. By means of the model transformation, a simpler nonlinear model is achieved to simplify the computation. Finally, the proposed approaches are analyzed using a numerical example.     

Symbolic and numerical analysis for studying complex nonlinear behavior

Carleman linearization and symbolic compution are used in order to derive explicit solutions in terms of exponential polynomials depending on the parameters and initial conditions. This new method is combined with a numerical algorithm in order to compute the Lyapunov exponents associated with the system. The aim of such an approach is to propose efficient tools in order to determine the intervals of the parameters where chaotic behavior exists.     

A fractional order HIV/AIDS model based on the effect of screening of unaware infectives

In this paper, a fractional order model for the spread of human immunodeficiency virus (HIV) infection is proposed to study the effect of screening of unaware infected individuals on the spread of the HIV virus. For this purpose, local asymptotic stability analysis of the disease‐free equilibrium is investigated. In addition, the model is studied for different values of the fractional order to show the relation between the variations of the reproduction number and the order of the proposed model. Finally, numerical solutions are simulated by using a predictor‐corrector method to illustrate the dynamics between susceptible individuals and unaware infected individuals.     

The impact of Wolbachia on dengue transmission dynamics in an SEI–SIS model

Dengue has grown dramatically in recent decades globally. In order to investigate the spread of dengue with vector control, especially, the impact of Wolbachia on dengue transmission, a mathematical model is established and analyzed to study dengue transmission between humans and mosquitoes. Firstly, model qualitative analysis including the existence and local asymptotic stability of dengue-free equilibria and endemic equilibria is done. It is found that dengue will disappear when the basic reproduction number is less than one, and dengue will prevail when the basic reproduction number is larger than one. More important finding is that the persistence of Wolbachia is determined by its fitness effect on mosquitoes, and Wolbachia can drastically reduce dengue fever transmission. All the results are verified by numerical simulation. Secondly, sensitivity analysis is done to explore the relative importance of different parameters on the system. It is obtained that parameters with strong sensitivity and controllability are the biting rate, the probability of dengue infection between mosquitoes and humans and the recovery rate of infectious humans. Finally, the control methods are discussed.     

Liquidity Costs: A New Numerical Methodology and an Empirical Study

We consider rate swaps which pay a fixed rate against a floating rate in the presence of bid-ask spread costs. Even for simple models of bid-ask spread costs, there is no explicit strategy optimizing an expected function of the hedging error. We here propose an efficient algorithm based on the stochastic gradient method to compute an approximate optimal strategy without solving a stochastic control problem. We validate our algorithm by numerical experiments. We also develop several variants of the algorithm and discuss their performances in terms of the numerical parameters and the liquidity cost.     

一类非线性动力系统的数值分析及控制策略研究

根据艾滋病在新疆的流行特点,建立了一个非线性动力系统的数学模型来研究艾滋病在新疆高危人群中传播的规律.通过查阅大量的统计数据和文献资料,确定了模型中部分参数的具体数值,然后通过数据拟合的方法得到了各个高危人群中的HIV病毒的传染性系数.在模型中,选择2004年底(2005年初)作为系统的初始点,预测了艾滋病未来几年内在新疆的流行趋势.最后,提出并比较遏止艾滋病传播的各项干预措施.     

An inventory model for deteriorating items with stock-dependent consumption rate and shortages under inflation and time discounting

This paper derives an inventory model for deteriorating items with stock-dependent consumption rate and shortages under inflation and time discounting over a finite planning horizon. We show that the total cost function is convex. With the convexity, a simple solution algorithm is presented to determine the optimal order quantity and the optimal interval of the total cost function. The results are discussed with a numerical example and particular cases of the model are discussed in brief. A sensitivity analysis of the optimal solution with respect to the parameters of the system is carried out.     

Cholera dynamics with Bacteriophage infection: A mathematical study

Mathematical modeling of waterborne diseases, such as cholera, including a biological control using Bacteriophage viruses in the aquatic reservoirs is of great relevance in epidemiology. In this paper, our aim is twofold: at first, to understand the cholera dynamics in the region around a water body; secondly, to understand how the spread of Bacteriophage infection in the cholera bacterium V. cholerae controls the disease in the human population. For this purpose, we modify the model proposed by Codeço, for the spread of cholera infection in human population and the one proposed by Beretta and Kuang, for the spread of Bacteriophage infection in the bacteria population [1, 2]. We first discuss the feasibility and local asymptotic stability of all the possible equilibria of the proposed model. Further, in the numerical investigation, we have found that the parameter ϕ, called the phage adsorption rate, plays an important role. There is a critical value, ϕ_c, at which the model possess Hopf-bifurcation. For lower values than ϕ_c, the equilibrium E^* is unstable and periodic solutions are observed, while above ϕ_c, the equilibrium E^* is locally asymptotically stable, and further shown to be also globally asymptotically stable. We investigate the effect of the various parameters on the dynamics of the infected humans by means of numerical simulations.     

Global stability of autonomous and nonautonomous hepatitis B virus models in patchy environment

Autonomous and nonautonomous hepatitis B virus infection models in patchy environment are investigated respectively to illustrate the influences of population migration and almost periodicity for infection rate on the spread of hepatitis B virus. The basic reproduction number is determined and asymptotic stabilities of disease-free and endemic equilibria are established in case of autonomous system. Moreover, in the nonautonomous system case, existence and global attractivity of almost periodic solution for this system are studied. Finally, feasibility of main theoretical results is showed with the aid of numerical examples for model with two patches.     

Numerical optimization methods for controlled systems with parameters

First- and second-order numerical methods for optimizing controlled dynamical systems with parameters are discussed. In unconstrained-parameter problems, the control parameters are optimized by applying the conjugate gradient method. A more accurate numerical solution in these problems is produced by Newton’s method based on a second-order functional increment formula. Next, a general optimal control problem with state constraints and parameters involved on the righthand sides of the controlled system and in the initial conditions is considered. This complicated problem is reduced to a mathematical programming one, followed by the search for optimal parameter values and control functions by applying a multimethod algorithm. The performance of the proposed technique is demonstrated by solving application problems.     

On the system governed by parabolic partial delay-differential equations with first boundary conditions

Summary In this paper, we consider a system governed by second order parabolic partial delay-differential equations with first boundary conditions. All the coefficients of the system are assumed bounded measurable and contain delays in their arguments. The second order parabolic partial delay-differential equation is in ? divergence form ?. In Theorem4.1, we present results on the existence and uniqueness of weak solution in the sense of Aronson for this class of systems. In Theorem4.2, we prove that whenever the coefficients and forcing terms converge in the almost everywhere topology the corresponding solutions converge weakly in an appropriate Sobolev space. Entrata in Redazione il 9 novembre 1977.     

Restrictions and unfolding of double Hopf bifurcation in functional differential equations

The normal form of a vector field generated by scalar delay-differential equations at nonresonant double Hopf bifurcation points is investigated. Using the methods developed by Faria and Magalhães (J. Differential Equations 122 (1995) 181) we show that (1) there exists linearly independent unfolding parameters of classes of delay-differential equations for a double Hopf point which generically map to linearly independent unfolding parameters of the normal form equations (ordinary differential equations), (2) there are generically no restrictions on the possible flows near a double Hopf point for both general and -symmetric first-order scalar equations with two delays in the nonlinearity, and (3) there always are restrictions on the possible flows near a double Hopf point for first-order scalar delay-differential equations with one delay in the nonlinearity, and in nth-order scalar delay-differential equations (n?2) with one delay feedback.     

Adjoint Sensitivity Analysis of a Fluid-Structure Interaction Problem

A fluid-structure mathematical model usually includes parameters whose actual values are known only approximately or can vary around some reference values. The objective of the sensitivity analysis is to determine quantitatively the behavior of the responses of a fluid-structure system locally around a chosen point of the trajectory in the phase-space of parameters and dependent variables. In this work, the response considered is the total mechanical energy of the structure. The sensitivities with respect to all the parameters the fluid-structure system depends on are useful in many situations as well as for optimization purposes. We present the theoretical developments necessary for the application of the adjoint sensitivity analysis methods (ASAM) for the fully coupled governing equations of an aeroelastic system. The algorithm is general and can be applied for any kind of fluid-structure interaction problems. Illustrative numerical examples are presented for the case of typical section with two degrees of freedom. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Detailed simulation of the pulsating detonation wave in the shock-attached frame

The paper is devoted to the numerical investigation of the stability of propagation of pulsating gas detonation waves. For various values of the mixture activation energy, detailed propagation patterns of the stable, weakly unstable, irregular, and strongly unstable detonation are obtained. The mathematical model is based on the Euler system of equations and the one-stage model of chemical reaction kinetics. The distinctive feature of the paper is the use of a specially developed computational algorithm of the second approximation order for simulating detonation wave in the shock-attached frame. In distinction from shock capturing schemes, the statement used in the paper is free of computational artifacts caused by the numerical smearing of the leading wave front. The key point of the computational algorithm is the solution of the equation for the evolution of the leading wave velocity using the second-order grid-characteristic method. The regimes of the pulsating detonation wave propagation thus obtained qualitatively match the computational data obtained in other studies and their numerical quality is superior when compared with known analytical solutions due to the use of a highly accurate computational algorithm.