Tikhonov Theorem Research Articles

A singularly perturbed SIS model with age structure

We present a preliminary study of an SIS model with a basic age structure and we focus on a disease with quick turnover, such as influenza or common cold. In such a case the difference between the characteristic demographic and epidemiological times naturally introduces two time scales in the model which makes it singularly perturbed. Using the Tikhonov theorem we prove that for certain classes of initial conditions the nonlinear structured SIS model can be approximated with very good accuracy by lower dimensional linear models.

Topological Methods in Analysis

Topological methods have played a seminal role in functional analysis since its birth in the early twentieth century. The Baire category theorem, for example, is the bedrock on which rest such basic principles of functional analysis as the open mapping theorem and the principle of uniform boundedness. Initial (weak) topologies, compactness, and the Tikhonov theorem drive such classical results of duality theory as the Banach-Alaoglu and Krein-Milman theorems. Topological methods also play a crucial role in Banach algebra theory (Gelfand topology), harmonic analysis (locally compact groups and function spaces), differential equations (fixed point theorems and Ascoli-Arzela theorem), and nonlinear analysis (fixed point existence theorems and topological degree theory) to mention just a few. The lofty goal of this special issue is to present some new outstanding contributions in this research area. They include results on fixed point theorems, metric spaces, C-algebras, modular spaces, algebraic hypersurfaces, and topological groups. Our hope is that this wide spectrum of topics will be of interest to many researchers in the area.

Differential Algebraic Equations of MOS Circuits and Jump Behavior

Abstract. Many nonlinear electronic circuits showing fast switching behavior exhibit jump effects which occurs when the state space of the electronic system contains a fold. This leads to difficulties during the simulation of these systems with standard circuit simulators. A method to overcome these problems is by regularization, where parasitic inductors and capacitors are added at the suitable locations. However, the transient solution will not be reliable if this regularization is not done in accordance with Tikhonov's Theorem. A geometric approach is taken to overcome these problems by explicitly computing the state space and jump points of the circuit. Until now, work has been done in analyzing example circuits exhibiting this behavior for BJT transistors. In this work we apply these methods to MOS circuits (Schmitt trigger, flip flop and multivibrator) and present the numerical results. To analyze the circuits we use the EKV drain current model as equivalent circuit model for the MOS transistors.

Neural firing rate model with a steep firing rate function

In this study we justify rigorously the approximation of the steep firing rate functions with a unit step function in a two-population neural firing rate model with steep firing rate functions. We do this justification by exploiting the theory of switching dynamical systems. It has been demonstrated that switching dynamics offer a possibility of simplifying the dynamical system and getting approximations of the solution of the system for any specific choice of parameters. In this approach the phase space of the system is divided into regular and singular domains, where the limit dynamics can be written down explicitly. The advantages of this method are illustrated by a number of numerical examples for different cases of the singular domains (i.e. for black, white and transparent walls) and for specific choices of the parameters involved. General conditions have been formulated on these parameters to give black, white and transparent walls. Further, the existence and stability of regular and singular stationary points have been investigated. It has been shown that the regular stationary points (i.e. stationary points inside the regular domains) are always stable and this property is preserved for smooth and sufficiently steep activation functions. In the most technical part of the paper we have provided conditions on the existence and stability of singular stationary points (i.e. those belonging to the singular domains). For the existence results, the implicit function theorem has been used, whereas the stability of singular stationary points is addressed by applying singular perturbation analysis and the Tikhonov theorem.

Local Cauchy problem for the MHD equations with mass diffusion

This paper studies the Cauchy problem for the MHD equations with mass diffusion in a bounded domain in $\mathbb{R}^{3}$. We use Tikhonov's fixed-point theorem to prove the existence and uniqueness of local solutions.

The Schauder-Tikhonov theorem in countably normed spaces

For the particular case in which the spaceE is normed, this theorem was proved earlier by Schauder. A proof of the Schauder theorem can be found in Nirenberg’s monograph [3]. In that book, the theorem is proved under the assumption that E is a Banach space, but that proof can be repeated literally for a normed space E. There are numerous versions of the Schauder–Tikhonov theorem, see, e.g., [4]. In [5], this theorem is given in most complete form. In that paper, Theorem 1.1 was proved without the assumption that E is a locally convex space. A distinguishing feature of the locally convex space E is that its geometry can well be described in terms of the topologically dual space E′. This is related to the fact that topologically dual spaces are sufficiently close to locally convex spaces. So, for example, ifA,B ⊂ E are nonintersecting convex sets andA is open, then there is a functional f ∈ E′ that separates these sets, i.e., f(A) ∩ f(B) = ∅ [6]. If a topological vector space is not locally convex, then its dual can consists only of the zero element. A standard example of such a kind is given by the space Lp(0, 1), 0 < p < 1 [7]. The topology in this space (after the corresponding factorization) is induced by the metric

The asymptotic structure of the Morris–Lecar model

In this paper, the asymptotic structure of the Morris–Lecar (ML) model in the concepts of slow manifold and fast foliation is analyzed based on the Tikhonov's theorem. The rationality of the well-known V, N reduced ML system is discussed, and a new V, N reduced ML system is derived from a more reasonable asymptotic structures of the original ML system, which we propose.

A new development for the Tikhonov Theorem in nonlinear singular perturbation systems

This paper deals with the exponential stability of nonlinear perturbation systems under a new condition. A novel criterion of exponential stability of nonlinear systems is firstly given in a general form. In this criterion, a new kind of characteristic value is introduced, which makes the exponential stability measurable. Based on this criterion, a new development for the Tikhonov Theorem in nonlinear singular perturbation systems is presented.

Tikhonov's theorem and quasi-steady state

There exists a systematic approach to asymptotic properties for quasi-steady state phenomena via the classical theory of Tikhonov and Fenichel. This observation allows, on the one hand, to settle convergence issues, which are far from trivial in asymptotic expansions. On the other hand, even if one takes convergence for granted, the approach yields a natural way to compute a reduced system on the slow manifold, with a reduced equation that is frequently simpler than the one obtained by the ad hoc approach. In particular, the reduced system is always rational. The paper includes a discussion of necessary and sufficient conditions for applicability of Tikhonov's and Fenichel's theorems, computational issues and a direct determination of the reduced system. The results are applied to several relevant examples.

Order reduction of a non-Lipschitzian model of monodisperse spray ignition

It is pointed out that the order reduction of singular perturbed systems with non-Lipschitzian nonlinearities can be performed, using the new concept of positively (negatively) invariant manifolds, if the five assumptions of the Tikhonov theorem are satisfied. Examples when these assumptions are satisfied and not satisfied are presented and discussed. The previously derived singularly perturbed system of three equations (droplet radii, gas temperature and fuel vapour concentration), describing evaporation and ignition of monodisperse fuel sprays, are shown to satisfy all five assumptions of the Tikhonov theorem. This allows us to reduce this system to a system of two equations describing the gas temperature and fuel concentration when the droplet evaporation process has been completed.

A boundary value problem on the half-line for higher-order nonlinear differential equations

This article is devoted to the study of the existence of solutions as well as the existence and uniqueness of solutions to a boundary-value problem on the half-line for higher-order nonlinear ordinary differential equations. An existence result is obtained by the use of the Schauder–Tikhonov theorem. Furthermore, an existence and uniqueness criterion is established using the Banach contraction principle. These two results are applied, in particular, to the specific class of higher-order nonlinear ordinary differential equations of Emden–Fowler type and to the special case of higher-order linear ordinary differential equations, respectively. Moreover, some (general or specific) examples demonstrating the applicability of our results are given.

Command Filtered Backstepping

Implementation of backstepping becomes increasingly complex as the order of the system increases. This increasing complexity is mainly driven by the need to compute command derivatives at each step of the design, with the ultimate step requiring derivatives of the same order as the plant. This article addresses a modification that obviates the need to compute analytic derivatives by introducing command filters in the backstepping design. While the concept of the command filter has previously been introduced in the literature, the main contribution of this technical note is the rigorous analysis of the effect of the command filter on closed-loop stability and performance, and a proof of stability based on Tikhonov's theorem. The implementation approach includes a compensated tracking error that retains the standard stability properties of backstepping approaches.

Asymptotic study of the Rayleigh equation

This paper provides an asymptotic treatment of the Cauchy problem for the Rayleigh equation µy + ?3/3 + ay – ? = 0 modelling a harmonic oscillator. The method of boundary layer functions and our version of the Tikhonov theorem are used. The models of asymptotic approximations are derived

Dynamic Inversion for Nonaffine-in-Control Systems via Time-Scale Separation. Part I

This paper presents a new method for approximate dynamic inversion for nonaffine-in-control systems via time-scale separation. The control signal is sought as a solution of "fast" dynamics and is shown to asymptotically stabilize the original nonaffine system. Sufficient conditions are formulated, which are consistent with the assumptions of Tikhonov's theorem in singular perturbations theory. Several examples illustrate the theoretical results.

Asymptotic behavior of Morris–Lecar system

In this paper, the asymptotic behavior of the Morris–Lecar model is studied. Firstly, the phase plane characteristic of the model is investigated, secondly, in terms of the concepts of slow manifold and fast foliation, the asymptotic structure of the Morris–Lecar model is discussed based on Tikhonov's theorem. Finally, computation and analysis results show that the spiking of action potential of Morris–Lecar neuron actually has the jump onset–jump return structure.

COMPUTATIONAL DESIGN OF THE TWO-LEVEL CONTROL FOR THE SINGULARLY PERTURBED SYSTEM

We present a new algorithm of the two-level control for a singularly perturbed system. If the mathematical model of the controllable dynamic system has a singular perturbation, it can be reduce with the help of hierarchical controls that can be obtained with Tikhonov's theorem. This is the stabilization of the adjoining system around an equilibrium point and to stabilize the desired movement through the use of a simplified system. In this paper, we present the application of the stabilization and the control algorithm of an aeronautics problem.

Tracking and Vibration Control of Flexible Robots Using Shape Memory Alloys

This paper considers the tracking control problem for flexible robots with active vibration suppression by shape memory alloys (SMAs). With a singular perturbation technique, the flexible modes and joint angles are modeled as fast and slow variables, respectively. The fast subsystem comprising the nonlinear SMA dynamics is obtained in a strict feedback form feasible for the backstepping design. The SMA actuators are configured in redundant pairs for an active suppression of vibrations. For the slow subsystem, asymptotically stable tracking is ensured, while for the fast subsystem, uniform exponential stability is guaranteed. By invoking Tikhonov's theorem, it is shown that the error arising from the nonideal time-scale separation in the singular perturbation model is small. Simulation demonstrates the effectiveness of the proposed control

A Stochastic Tikhonov Theorem in Infinite Dimensions

The present paper studies the problem of singular perturbation in the infinite-dimensional framework and gives a Hilbert-space-valued stochastic version of the Tikhonov theorem. We consider a nonlinear system of Hilbert-space-valued equations for a "slow" and a "fast" variable; the system is strongly coupled and driven by linear unbounded operators generating a C0-semigroup and independent cylindrical Brownian motions. Under well-established assumptions to guarantee the existence and uniqueness of mild solutions, we deduce the required stability of the system from a dissipativity condition on the drift of the fast variable. We avoid differentiability assumptions on the coefficients which would be unnatural in the infinite-dimensional framework.

THE ASYMPTOTIC STRUCTURE OF THE HODGKIN–HUXLEY EQUATIONS

We analyze the asymptotic structure of the Hodgkin–Huxley system of equations, in terms of the concepts of slow manifold and fast foliation, based on Tikhonov's theorem on asymptotics of solutions of slow–fast systems of differential equations. We test Zeeman's conjecture that the jump onset–slow return structure of the action potential in realistic equations of biological excitability may be due to a cusp singularity of the slow manifold with respect to the fast foliation. We find that although the cusp singularity can appear in such equations, the characteristic features in question cannot be reproduced within the Tikhonov scheme and require development of different asymptotic approaches.

Nonstandard singular perturbation systems and higher index differential-algebraic systems

In order to obtain trajectory approximation results for a given singular perturbation system (SPS), two systems are derived from it: the slow and the fast one. Tikhonov's theorem gives sufficient conditions on them to ensure a good approximation for a standard SPS, i.e., its corresponding slow system is a differential-algebraic system (DAS) of index 1. In this paper it is shown that a nonstandard SPS with the parameter set to zero can be seen as a DAS of higher index. This connection allows us to obtain a Tikhonov's theorem when this DAS is of index 2.