Singular Perturbation Technique Research Articles

Use of memristive chaotic signal as a desired trajectory for a two-link flexible manipulator using contraction theory based on a composite control technique

Flexible manipulators are being considered as bench mark control problem in the field of nonlinear dynamics. Many of their inherent advantages create challenges while dealing with the dynamics. Tracking control and vibration suppression are two main control problems considered. In this paper a composite controller is designed for the memristive chaotic system signal as trajectory tracking control of a two-link flexible robot manipulator. The dynamics of the flexible manipulator is modelled by using assumed modes method and divided into two subsystems using the singular perturbation technique. The subsystems are called as the slow subsystem involving rigid dynamics of the manipulator and the fast sub-system which incorporates flexible dynamics of the manipulator. Separate control techniques are designed for each subsystem. Contraction theory based controllers are designed for the slow sub-system and fast subsystem for fast trajectory tracking of signal of a memristive chaotic system and quick suppression of the link deflections. The simulation results confirm the better performances of the proposed composite technique.

Postbuckling analysis of axially loaded nanoscaled shells embedded in elastic foundations based on Ru's surface elasticity theory

The nonlinear buckling and postbuckling characteristics of cylindrical nanoshells embedded in elastic foundations are investigated based on a strain-consistent elastic shell model including the both surface tension and the induced residual stress. For this purpose, in contrast to the previous models on the basis of the Gurtin-Murdoch elasticity theory, a new non-classical shell model based on Ru's surface elasticity theory is developed in which the non-strain displacement gradient terms are eliminated from the surface stress-strain relations. Using the virtual work's principle, the governing differential equations incorporating surface effects are derived. After that, the size-dependent governing equations are deduced to a boundary layer type problem which is subsequently solved through employing a two-stepped singular perturbation technique. It is revealed that because the edge supports of nanoshells are movable, before applying the axial compression, surface effects lead to an initial shortening due to induced residual strains, but the terms related to the residual strain and initial surface tension vanish in the size-dependent nonlinear governing equations. As a result, it is observed that before applying the axial compressive load, the surface effects cause an initial end-shortening for very thin nanoshells and these effects quickly diminish by increasing the shell thickness.Communicated by Krzysztof Kamil Żur

An asymptotic expansion method for geometric Asian options pricing under the double Heston model

Abstract The purpose of the paper is to provide an efficient method for the continuously monitored geometric Asian options under the double Heston model. By introducing two small parameters, we slightly modify the double Heston model. With singular and regular perturbation techniques, we derive the first-order asymptotic expansions for pricing geometric Asian options with fixed and floating strikes and provide the convergence of the asymptotic formulae. We also provide the Greeks of geometric Asian options. Numerical results verify the efficiency of the pricing method. We calibrate the modified model to real markets and examine the impacts of two-factor volatilities on geometric Asian option prices.

Radial postbuckling of nanoscaled shells embedded in elastic foundations based on Ru's surface stress elasticity theory

In this article, the nonlinear buckling and postbuckling response of cylindrical nanoshells under radial compressive load is investigated on the basis of a new surface elasticity theory. To this point, in contrast to the previous models based on the Gurtin-Murdoch elasticity theory, a new non-classical shell model based on the Ru's surface elasticity theory is developed in which the non-strain displacement gradient terms are eliminated from the surface stress-strain relations. The non-classical governing equations are derived using the principle of virtual work. Then they are deduced to boundary layer-type equations including simultaneously the nonlinear prebuckling deformations and the large postbuckling deflections. By using a singular perturbation technique, the new size-dependent problem is solved. It is indicated that due to the immovable edge supports and in the absence of the hydrostatic pressure, surface effects lead to an initial shortening caused by the induced residual strains. Additionally, the residual strain and initial surface tension terms vanish in the size-dependent nonlinear governing equations. Moreover, it is observed that this initial shortening diminishes quickly for thicker nanoshells.

Postbuckling of pressure-loaded FG-GRC laminated cylindrical panels resting on elastic foundations in thermal environments

This paper investigates the postbuckling behavior of graphene-reinforced composite (GRC) laminated cylindrical panels under lateral pressure loading. The panels are placed in a thermal environment and are supported by a Pasternak-type elastic foundation. The GRC layers are arranged in a piece-wise functionally graded (FG) pattern in the panel thickness direction. The anisotropic and temperature dependent material properties of GRC layers are predicted using the extended Halpin-Tsai micromechanical model. The higher order shear deformation theory and the von Karman strain-displacement relationships are employed to obtain the governing differential equations of the panels which also include the effects of temperature variation and panel-foundation interaction. The postbuckling of a cylindrical panel involves the boundary layer effect and the effects of the initial geometric imperfections. The nonlinear prebuckling deformations of the panel are also considered in the solution process. The postbuckling equilibrium path for the perfect and geometrically imperfect GRC laminated cylindrical panels are obtained by applying a singular perturbation technique along with a two-step perturbation approach. It is observed that, in the present examples, the considered FG-GRC laminated cylindrical panels subjected to lateral pressure experience pre-buckling deformation and the postbuckling equilibrium paths of the panels are no longer of the bifurcation type.

Diffusion approximation of near critical branching processes in fixed and random environment

We consider Bienaymé-Galton-Watson and continuous-time Markov branching processes and prove diffusion approximation results in the near critical case, in fixed and random environment. In one hand, in the fixed environment case, we give new proofs and derive necessary and sufficient conditions for diffusion approximation to get hold of Feller-Jiřina and Jagers theorems. In the other hand, we propose a continuous-time Markov branching process with random environments and obtain diffusion approximation results. An averaging result is also presented. Proofs here are new, where weak convergence in the Skorohod space is proved via singular perturbation technique for convergence of generators and tightness of the distributions of the considered families of stochastic processes.

An analytical approximation method for pricing barrier options under the double Heston model

The purpose of the paper is to provide an efficient pricing method for single barrier options under the double Heston model. By rewriting the model as a singular and regular perturbed BS model, the double Heston model can separately mimic a fast time-scale and a slow time-scale. With the singular and regular perturbation techniques, we analytically derive the first-order asymptotic expansion of the solution to a barrier option pricing partial differential equation. The convergence and efficiency of the approximate method is verified by Monte Carlo simulation. Numerical results show that the presented asymptotic pricing method is fast and accurate.

Some results on the p(u)-Laplacian problem

The p-Laplacian problem with the exponent of nonlinearity p depending on the solution u itself is considered in this work. Both situations when p(u) is a local quantity or when p(u) is nonlocal are studied here. For the associated boundary-value local problem, we prove the existence of weak solutions by using a singular perturbation technique. We also prove the existence of weak solutions to the nonlocal version of the associated boundary-value problem. The issue of uniqueness for these problems is addressed in this work as well, in particular by working out the uniqueness for a one dimensional local problem and by showing that the uniqueness is easily lost in the nonlocal problem.

Postbuckling of doubly curved FG-GRC laminated panels subjected to lateral pressure in thermal environments

This article presents an investigation on the postbuckling behavior of doubly curved graphene-reinforced composite (GRC) laminated panels supported by an elastic foundation and subjected to lateral pressure and in thermal environments. The piece-wise GRC layers are arranged in a functionally graded (FG) pattern along the thickness direction of the panels. The overall mechanical properties of the FG-GRCs are assumed to be temperature dependent and are estimated through the extended Halpin-Tsai micromechanical model. The governing differential equations for the doubly curved panels are based on a higher order shear deformation shell theory with von Kármán strain-displacement relationships and the panel-foundation interaction. The initial deflections caused by lateral pressure and thermal bending stresses are both taken into account. The governing equations are first deduced to a boundary layer type that includes nonlinear prebuckling deformations and initial geometric imperfections of the panel. The postbuckling equilibrium path for the perfect and geometrically imperfect GRC laminated doubly curved panels are obtained by applying a singular perturbation technique along with a two-step perturbation approach. The impacts of material property gradient, temperature variation, panel curvature ratio, as well as foundation stiffness on the postbuckling behavior of FG-GRC laminated doubly curved panels are investigated.

Analytical and Numerical Investigation of Fe3O4–Water Nanofluid Flow over a Moveable Plane in a Parallel Stream with High Suction

In the current framework, a model is constituted to explore the impacts of high suction and partial slip on Fe3O4–water nanoliquid flow over a porous moveable surface in a parallel free stream. The mechanisms of heat transfer are also modeled in the existence of Newtonian heating effect. The obtaining PDEs are transformed into a non-linear ODE system employing appropriate boundary conditions to diverse physical parameters. The governing ODE system is solved using a singular perturbation technique that results in an analytical asymptotic solution as a function of the physical parameters. The obtained solution allows us to carry out an analytical parametric study to investigate the impact of the physical parameters on the nonlinear attitude of the system. The precision of the proposed method is verified by comparisons between the numerical and analytical results. The results confirm that the proposed technique yields a good approximation to the solution as well as the solution calculation has no CPU time-consuming or round off error. Numerical solutions are computed and clarified in graphs for the model embedded parameters. Moreover, profiles of the skin friction coefficient and the heat transfer rate are also portrayed and deliberated. The data manifests that both solid volume fraction and slip impact significantly alter the flow profiles. Moreover, an upward trend in temperature is anticipated for enhancing Newtonian heating strength. Additionally, it was found that both the nanofluid velocity and temperature distributions are decelerated when the solid volume fraction and suction parameters increase. Furthermore, a rise in slip parameter causes an increment in velocity profiles, and a rise in Biot number causes an increment in the temperature profiles.

A study case for the analysis of asymptotic expansions beyond the tQSSA for inhibitory mechanisms in enzyme kinetics.

AbstractIn this paper we study the model of the chemical reaction of fully competitive inhibition and determine the appropriate parameter∊(related to the chemical constants of the model), for the application of singular perturbation techniques. We determine the inner and the outer solutions up to the first perturbation order and the uniform expansions. Some numerical results are discussed.

Full-State Tracking Control for Flexible Joint Robots With Singular Perturbation Techniques

This paper proposes a practical method to realize multivariable full-state tracking control for industrial robots with elastic joints. Unlike existing methods, the proposed method does not require high-order derivatives of the link states such as acceleration and jerk. Therefore, the proposed method does not suffer from chatter related to inaccurate estimation of high-order derivatives. The method is derived by adopting a singular perturbation technique. A decoupled error dynamics is achieved by two decoupling control loops: a fast loop that controls the deflection error and a slow loop for tracking control on the link side. Our stability analysis based on a linear system shows that the proposed control system is stable as long as the fast system is at least twice as fast as the slow system. A practical method to select the gain is also presented such that the closed-loop poles are placed at the desired locations. In simulation, we compare the proposed method with feedback linearization. The results indicate that in an ideal scenario the proposed method can obtain a similar performance as feedback linearization. However, the proposed method obtains a superior performance in a realistic scenario. A real-world experiment with a six degree-of-freedom commercial industrial robot is carried out to further validate our approach.

Torsional postbuckling behavior of FG-GRC laminated cylindrical shells in thermal environments

This paper presents the buckling and postbuckling behaviors of graphene-reinforced composite (GRC) laminated cylindrical shells subjected to torsion in thermal environments. The GRC layers of the shell are arranged in a piece-wise functionally graded (FG) distribution pattern in the thickness direction and each layer of the shell contains different volume fraction of graphene reinforcement. The extended Halpin-Tsai micromechanical model is employed to determine the temperature dependent material properties of GRC layers. The governing equations of the GRC laminated cylindrical shells under torsion are derived based on a higher-order shear deformation shell theory with the geometric nonlinearity being defined by the von Kármán strain-displacement relationship. A singular perturbation technique along with a two-step perturbation approach is employed to determine the buckling torques and the torsional postbuckling equilibrium paths of the FG-GRC laminated cylindrical shells in thermal environments. The numerical results obtained reveal that the piece-wise FG distribution of graphene volume fraction can enhance the buckling torque and the torsional postbuckling strength while the rise of temperature may lead to the reduction of the torsional buckling torques and torsional postbuckling strength of the GRC laminated cylindrical shell.

Towards sensorless observers for sinusoidal electric machines with variable speed and no mechanical model: A promising approach for PMSMs

One of the major issues in developing sensorless observers for AC Sinusoidal Machines, and for all kinds of Electric Machines, is to deal with variable speed relying only on the electromagnetic model, without using any mechanical information. In this paper, the case of Permanent Magnet Synchronous Machines is considered as first benchmark. A novel and promising design strategy is presented to develop a simple sixth-order observer for estimating rotor speed/position and magnetic flux amplitude, in the context of non-zero variable speed, with unknown constant sign and bounded derivative. This framework does not cover yet any arbitrarily-varying mechanical speed, but it goes far beyond the typical simplifying assumption of “slowly-varying speed”, which is actually meant as “constant speed” in the common theoretical analysis. In the proposed method, the rotation dynamics of the machine back-electromotive force is represented by means of the Lie Groups formalism, and no open-loop integration of stator voltages and currents is adopted. Lyapunov-like and Singular Perturbations techniques are then exploited to achieve regional practical asymptotic stability, with a very wide region of attraction. The mild limitations on such stability domain are carefully analyzed and discussed. Numerical simulations are provided to show the effectiveness of the proposed observer, under heavily variable mechanical speed. Finally, taking cue from the features of the presented approach, future steps are outlined in order to further weaken the restrictions on the speed variations and to extend the results to other electric machines.

Wave train selection by invasion fronts in the FitzHugh–Nagumo equation

We study invasion fronts in the FitzHugh–Nagumo equation in the oscillatory regime using singular perturbation techniques. Phenomenologically, localized perturbations of the unstable steady-state grow and spread, creating temporal oscillations whose phase is modulated spatially. The phase modulation appears to be selected by an invasion front that describes the behavior in the leading edge of the spreading process. We construct these invasion fronts for large regions of parameter space using singular perturbation techniques. Key ingredients are the construction of periodic orbits, their unstable manifolds, and the analysis of pushed and pulled fronts in the fast system. Our results predict the wavenumbers and frequencies of oscillations in the wake of the front through a phase locking mechanism. We also identify a parameter regime where nonlinear phase locked fronts are inaccessible in the singularly perturbed geometry of the traveling-wave equation. Direct simulations confirm our predictions and point to interesting phase slip dynamics.

Pinned Solutions in a Heterogeneous Three-Component FitzHugh\u2013Nagumo Model

We analyse pinned front and pulse solutions in a singularly perturbed three-component FitzHugh–Nagumo model with a small jump-type heterogeneity. We derive explicit conditions for the existence and stability of these type of pinned solutions by combining geometric singular perturbation techniques and an action functional approach. Most notably, in certain parameter regimes we can explicitly compute the pinning distance of a localised solution to the defect.

On some stationary Navier–Stokes type problems

The goal of this note is to present a simple proof of existence of solution to the stationary Navier–Stokes problem using a singular perturbation technique and to address some nonlocal issues related to it.

Effect of Hyper-Resistivity on Nonlinear Tearing Modes**Supported by the National Natural Science Foundation of China under Grant No 11675257, the Strategic Priority Research Program of Chinese Academy of Sciences under Grant No XDB16010300, the Key Research Program of Frontier Science of Chinese Academy of Sciences under Grant No QYZDJ-SSW-SYS016, and the External Cooperation Program of Chinese Academy of Sciences under

We analytically investigate nonlinear tearing modes with the anomalous electron viscosity or, as it is normally called, hyper-resistivity. In contrast to the flux average method used by previous work, we employ the standard singular perturbation technique and a quasilinear method to obtain the time evolution equation of tearing modes. The result that the magnetic flux grows with time in a scaling as t2/3 demonstrates that nonlinear tearing modes with the hyper-resistivity effect alone have a weaker dependence on time than that of the corresponding resistive case.

Surface energy effect on nonlinear buckling and postbuckling behavior of functionally graded piezoelectric cylindrical nanoshells under lateral pressure

In this paper, the surface energy effect on the nonlinear buckling and postbuckling behavior of functionally graded piezoelectric (FGP) cylindrical nanoshells subjected to lateral pressure is studied based on the electro-elastic surface/interface theory together with von-Kármán-Donnell-type kinematics of nonlinearity. The total strain energy of the FGP nanoshell, including surface energy, is derived by considering the constitutive formulations of surface phase. The principle of minimum potential energy is utilized to establish the nonlinear governing differential equations, and the singular perturbation technique is employed to obtain the asymptotic solutions. Then, two sets of comparison are conducted to validate the present work, and some numerical examples are given to study the effects of surface parameters, power law index and aspect ratio on the buckling and postbuckling behavior of FGP nanoshells. The results show that the critical buckling load and postbuckling path of FGP nanoshell are significantly size-dependent.

Stability analysis of a general class of singularly perturbed linear hybrid systems

We introduce and analyze a general class of singularly perturbed linear hybrid systems with both switches and impulses, in which the slow or fast nature of the variables can be mode-dependent. This means that, at switching instants, some of the slow variables can become fast and vice-versa. Firstly, we show that using a mode-dependent variable reordering we can rewrite this class of systems in a form in which the variables preserve their slow or fast nature over time. Secondly, we establish, through singular perturbation techniques, an upper bound on the minimum dwell-time ensuring the overall system’s stability. Remarkably, this bound is the sum of two terms. The first term, which can be equal to zero, only depends on the matrices of the reduced order linear hybrid system describing the slow dynamics and corresponds to an upper bound on the minimum dwell time ensuring the stability of that system. The order of magnitude of the second term is determined by that of the parameter defining the ratio between the two time-scales of the singularly perturbed system. We show that the proposed framework can also take into account the change of dimension of the state vector at switching instants. Numerical illustrations complete our study.