Singularly Perturbed Systems with Diffusion and Time Delay

For a class of nonlinear oscillation problems described by a second order ordinary differential equation containing a small nonnegative perturbation parameter $\varepsilon $ we determine asymptotic solutions which are uniformly valid as $\varepsilon \downarrow 0$ on intervals of the order of magnitude $\varepsilon ^{ - 1} $. These asymptotic solutions are obtained from formal asymptotic solutions subject to a condition in order that they approximate the exact solution asymptotically. These formal asymptotic solutions are constructed in the form of a finite generalized asymptotic power series involving functions of the periodic two variable type.

In this paper, a new method is presented for identification of single-input single-output (SISO) nonlinear systems with long time delay. The proposed dynamic extreme learning machine (DELM) is different from extreme learning machine in that it adds adaptive delay parameters in the output layer of the network, and the delay parameter is optimized by use of particle swarm algorithm. The dynamic extreme learning machine has two main advantages: First, for any system, it needs only two input nodes. Second, it does not need to know the specific delay time. A pH neutralization process and a nonlinear dynamic system process are used to evaluate the performance of the proposed method. Simulation results demonstrate the effectiveness of the proposed identification algorithm for a class of nonlinear system with long time delay.

The asymptotic homogenization method is applied to obtain formal asymptotic solution and the homogenized solution of a Dirichlet boundary-value problem for an elliptic equation with rapidly os- cillating coefficients. The proximity of the formal asymptotic solution and the homogenized solution to the exact solution is proved, which provides the mathematical justification of the homogenization pro- cess. Preservation of the symmetry and positive-definiteness of the effective coefficient in the homogenized problem is also proved. An example is presented in order to illustrate the theoretical results.

The time delay in active and semi-active controls is an important research subject. Many researches on the time-delay control for deterministic systems have been made (Hu and Wang 2002, Yang et al. 1990, Abdel-Mooty and Roorda 1991, Pu 1998, Cai and Huang 2002), while the study on that for stochastic systems is very limited. The effects of the time delay on the control of nonlinear systems under Gaussian white noise excitations have been studied by Bilello et al. (2002). The controlled linear systems with deterministic and random time delay subjected to Gaussian white noise excitations have been treated by Grigoriu (1997). Recently, a stochastic averaging method for quasi-integrable Hamiltonian systems with time delay has been proposed (Liu and Zhu 2007). In the present paper, a stochastic optimal time-delay control method for stochastically excited nonlinear structural systems is proposed based on the stochastic averaging method for quasi Hamiltonian systems with time delay and the stochastic dynamical programming principle. An example of stochastically excited and controlled hysteretic column is given to illustrate the proposed control method.

Linear stability methods were applied to a biomechanical model of the human musculoskeletal spine to investigate effects of reflex gain and reflex delay on stability. Equations of motion represented a dynamic 18 degrees-of-freedom rigid-body model with time-delayed reflexes. Optimal muscle activation levels were identified by minimizing metabolic power with the constraints of equilibrium and stability with zero reflex time delay. Muscle activation levels and associated muscle forces were used to find the delay margin, i.e., the maximum reflex delay for which the system was stable. Results demonstrated that stiffness due to antagonistic co-contraction necessary for stability declined with increased proportional reflex gain. Reflex delay limited the maximum acceptable proportional reflex gain, i.e., long reflex delay required smaller maximum reflex gain to avoid instability. As differential reflex gain increased, there was a small increase in acceptable reflex delay. However, differential reflex gain with values near intrinsic damping caused the delay margin to approach zero. Forward-dynamic simulations of the fully nonlinear time-delayed system verified the linear results. The linear methods accurately found the delay margin below which the nonlinear system was asymptotically stable. These methods may aid future investigations in the role of reflexes in musculoskeletal stability.

This paper deals with an adaptive robust tracking control using a multilayer neural network (NN) for a class of nonlinear dynamic systems with unknown time varying state delays. Typical adaptive NN backstepping controllers for uncertain nonlinear systems with time-delay give rise to computation complexity caused by the the repeated derivatives of virtual controllers and nonlinear functions. Moreover, the combined techniques usually result in only uniformly ultimately bounded (UUB) stability caused by the inherent NN approximation error. This paper presents a control scheme that uses an integral sliding mode control as a feedback term and an adaptive neural controller as a feedforward term based on the desired compensation adaptive law (DCAL) technique. First, we develop a new DCAL formulation which avoids the explosion of complexity caused by the general NN backstepping scheme to compensate for nonlinear system uncertainties, bounded system disturbances, and unknown state time delays. Then, using a Lyapunov-Krasovskii (LK) functional, it is shown that the proposed controller renders the class of uncertain nonlinear time-delay systems asymptotically stable.

We address the homogenisation of a linear viscoelastic and hierarchical composite material in a one-dimensional (1D) framework via a three-scale asymptotic homogenisation method. We consider a family of heterogeneous problems with periodic, rapidly-oscillating and piece-wise coefficients that model a structure with two hierarchical levels of organisation. Here, we assume continuity contact conditions at the interfaces among the constituents and set a straightforward geometrical configuration in order to gain a better insight of the multiscale problem. The main goal is to provide a general overview of the procedure and validate the approach by means of a comparison between the solution of the original heterogeneous problem, the homogenised problem and the formal asymptotic solution. In addition, we show that the three-scale approach presents a clear improvement over the recursive two-scale one, and we illustrate the convergence of the solutions towards the solution of the homogenised problem when the asymptotic parameters approach.

Microfiltration of particles is modelled by the motion of particles embeddedin a Stokes flow near a porous membrane in which Darcy equations apply.Stokes flow also applies on the other side of the membrane. A pressure gradient isapplied across the membrane. Beavers and Joseph slip boundary condition applies along the membrane surfaces.This coupled Stokes-Darcy problem is solved by a perturbation method, considering that the particle size is muchlarger than the pores of the membrane. The formal asymptotic solution is developed in detail up to 3rd order.The method is applied to the example case of a spherical particle moving normal to a membrane. The solution,limited here to an impermeable slip surface (described from 3rd order expansion), uses as an intermediate step theboundary integral technique for Stokes flow near an impermeable surface with a no-slip boundary condition. Results of the perturbation solution are in good agreement with O'Neill and Bhatt analytical solution for this case.

Singular perturbation of boundary value problem for quasilinear third-order ordinary differential equations involving two small parameters

Averaging of the system of equations of motion of a viscous fluid in a porous medium

Tracking control for multiple-integrator systems is regarded as a fundamental problem associated with nonlinear dynamic systems in the physical and mathematical sciences, with many applications in engineering fields. In this paper, we adopt the Zhang neural network method to solve this nonlinear dynamic problem. In addition, in order to adapt to the requirements of real-world hardware implementations with higher-order precision for this problem, the multiple-order derivatives in the Zhang neural network method are estimated using backward finite-divided difference formulas with quadratic-order precision, thus producing time delays. As such, we name the proposed method the Zhang neural network method with time delay. Moreover, we present five theorems to describe the convergence property of the Zhang neural network method without time delay and the quadratic-order error pattern of the Zhang neural network method with time delay derived from the backward finite-divided difference formulas with quadratic-order precision, which specifically demonstrate the effect of the time delay. Finally, tracking controllers with quadratic-order precision for multiple-integrator systems are constructed using the Zhang neural network method with time delay, and two numerical experiments are presented to substantiate the theoretical results for the Zhang neural network methods with and without time delay.

The Gevrey asymptotics in the initial value problem for singularly perturbed nonlinear differential equations

Time delays are frequently appearing in many real-life phenomena and the presence of time delays in chaotic systems enriches its complexities. The analysis of fractional-order chaotic real nonlinear systems with time delays has a plenty of interesting results but the research on fractional-order chaotic complex nonlinear systems with time delays is in the primary stage. This paper studies the problem of hybrid projective synchronization (HPS) of fractional-order chaotic complex nonlinear systems with time delays. HPS is one of the extensions of projective synchronization, in which different state vectors can be synchronized up to different scaling factors. Based on Laplace transformation and the stability theory of linear fractional-order systems, a suitable nonlinear controller is designed to achieve synchronization between the master and slave fractional-order chaotic complex nonlinear systems with time delays in the sense of HPS with different scaling factors. Finally, the HPS between fractional-order delayed complex Lorenz system and fractional-order delayed complex Chen system and that of fractional-order delayed complex Lorenz system and fractional-order delayed complex Lu system are taken into account to demonstrate the effectiveness and feasibility of the proposed HPS techniques in the numerical example section.

This paper studies subharmonic responses of a parametrically excited asymmetric nonlinear system with time delay. It is known that it is difficult to investigate the nonlinear system with time delay by analytical methods because the system is described by a difference-differential equation hard to be solved. To analyze this kind of a system, we have already introduced an averaging method for the functional differential equation. The previous work showed that this averaging method is effective for self excited vibrations and forced vibrations arising in a Duffing type vibration system with time delay. However, the previous work dealt only with a fundamental harmonic responses and the target system has a relatively simple structure. To investigate a more practical example, this paper studies not only fundamental but also higher harmonic responses of a more practical vibration system with time delay such as a one dimensional rotor system of asymmetric rigidity supported by plain bearings. The result showed that stability conditions estimated by the averaging method is deeply connected with behaviors obtained from a direct numerical simulation of the target system.

Finite difference solutions of reaction diffusion equations with continuous time delays