Classification Of Such Systems Research Articles

On group classification of normal systems of linear second-order ordinary differential equations

In this paper we study the general group classification of systems of linear second-order ordinary differential equations inspired from earlier works and recent results on the group classification of such systems. Some interesting results and subsequent theorem arising from this particular study are discussed here. This paper considers the study of irreducible systems of second-order ordinary differential equations.

On three-dimensional quasi-Stäckel Hamiltonians

A three-dimensional integrable generalization of the Stäckel systems is proposed. A classification of such systems is obtained, which results in two families. The first family is the direct sum of the two-dimensional system which is equivalent to the representation of the Schottky–Manakov top in the quasi-Stäckel form and a Stäckel one-dimensional system. The second family is probably a new three-dimensional system. The system of hydrodynamic type, which we get from this family in the usual way, is a three-dimensional generalization of the Gibbons–Tsarev system. A generalization of the quasi-Stäckel systems to the case of any dimension is discussed.

Polynomial representations of piecewise-linear differential equations arising from gene regulatory networks

We describe generic sliding modes of piecewise-linear systems of differential equations arising in the theory of gene regulatory networks with Boolean interactions. We do not make any a priori assumptions on regulatory functions in the network and try to understand what mathematical consequences this may have in regard to the limit dynamics of the system. Further, we provide a complete classification of such systems in terms of polynomial representations for the cases where the discontinuity set of the right-hand side of the system has a codimension 1 in the phase space. In particular, we prove that the multilinear representation of the underlying Boolean structure of a continuous-time gene regulatory network is only generic in the absence of sliding trajectories. Our results also explain why the Boolean structure of interactions is too coarse and usually gives rise to several non-equivalent models with smooth interactions.

A connection between simulation relations and feedback transformations in nonlinear control systems

The recent introduction of simulation and bisimulation relations in the study of nonlinear control systems provides a broad potential framework for the study and classification of such systems. This paper reviews the definitions of simulation and bisimulation relations and develops connections between them and the well established notions of feedback transformations and feedback equivalences.

Lotka-Volterra equations in three dimensions satisfying the Kowalevski-Painlevé property

We examine a class of Lotka-Volterra equations in three dimensions which satisfy the Kowalevski-Painleve property. We restrict our attention to Lotka-Volterra systems defined by a skew symmetric matrix. We obtain a complete classification of such systems. The classification is obtained using Painleve analysis and more specifically by the use of Kowalevski exponents. The imposition of certain integrality conditions on the Kowalevski exponents gives necessary conditions. We also show that the conditions are sufficient.

Diffraction from quasi one‐dimensional crystals and nanotubes

AbstractIntensity distribution for diffraction from arbitrary quasi one‐dimensional crystal is calculated with the help of symmetry classification of such systems onto 15 conformation classes. Characteristic features of patterns are discussed. In particular, carbon nanotubes are single‐orbit systems generated by the fifth family line groups. It is shown how the identification of single‐wall tubes can be performed in two steps: at first we reconstruct the line group from the pattern, and afterwards the chirality indices are found easily due to the bi‐unique correspondence of the single‐wall carbon nanotubes and their symmetry groups. A comment on the diffraction from double‐ and multi‐wall nanotubes is made.

Self-Dual Transitive Spin 1/2 Models

Most general self-dual spin 1/2 models in any dimension, with interaction that is translation-invariant in a suitable sense (‘transitive models’), are determined. In the process of classification of such systems, a class of models which are self-dual in a particularly strong sense is introduced.

Geometric theory of differential systems: Linearization criterion for systems of second-order ordinary differential equations with a 4-dimensional solvable symmetry group of the Lie–Petrov type VI 1

In the framework of projective-geometric theory of systems of differential equations developed by the authors, this paper studies the group properties of systems of two (resolved with respect to the second derivatives) second-order ordinary differential equations whose right-hand sides are polynomials of the third degree with respect to the derivatives of the unknown functions. A classification of such systems admitting four-dimensional symmetry group of the Lie–Petrov type VI 1 is given. For each of the systems, a necessary and sufficient linearization criterion is obtained, i.e., the authors find the necessary and sufficient conditions under which, by a change of variables, the system can be reduced to a differential system whose integral curves are straight lines and are expressed by three linear parametric equations or two linear equations with constant coefficients. For all linearizable systems, the linearizing changes of variables are indicated.

Submarine springs and coastal karst aquifers: A review

This article reports on current knowledge of coastal karst aquifers, in which conduit flow is dominant, and its aim is to characterise the functioning of these systems which are closely linked to the sea. First, earlier and recent studies of these aquifers are discussed. On the basis of their findings, it can be shown that two essential mechanisms are involved in the functioning of these systems, i.e., aquifer discharge through submarine springs and saline intrusion through conduits open to the sea. Then, the conditions that give rise to these aquifers are described and particular emphasis is placed on the influence of deep karstification when the sea level falls. The base-level variations are attributed to the glaciations or, in the specific case of the Mediterranean, to the salinity crisis in the Messinian period. It is this inherited structure, sometimes containing very deep conduits below sea level, that today conditions the aquifer flow. The flow in the conduits open to the sea depends on the hydraulic head gradient between the aquifer and the sea and is therefore a function of the water density and head losses in the aquifer. This survey of coastal karst aquifers has revealed some common characteristics that show the development and/or functional capacity of their karstic drainage networks. A classification of such systems into three categories is proposed with the aim of assisting in the decision-making concerning potential exploitation of water resources in coastal regions.

On the robust stability of implicit linear systems containing a small parameter in the leading term

This paper deals with the robust stability of implicit linear systems containing a small parameter in the leading term. Based on possible changes in the algebraic structure of the matrix pencils, a classification of such systems is given. The main attention is paid to the cases when the appearance of the small parameter causes some structure change in the matrix pencil. First, we give sufficient conditions providing the asymptotic stability of the parameterized system. Then, we give a formula for the complex stability radius and characterize its asymptotic behaviour as the parameter tends to zero. The structure-invariant cases are discussed, too. A conclusion concerning the parameter dependence of the robust stability is obtained.

Computer Algebra Systems and Symbolic Computations

General principles of the construction and architecture of computer algebra systems are considered. The range of tasks that can be solved with their help is indicated. A classification of such systems is presented, and the most popular and significant packages for symbolic computing are listed. Fundamental characteristics of computer algebra systems, as well as data types, are mentioned. A difference between symbolic computations and numerical methods is emphasized. Two examples of algebraic calculations in the packages Maxima and GAP 4.2 are given. They are concerned with the solution of nonlinear algebraic systems and computations with subgroup lattices, respectively. Bibliography: 7 titles.

The Geometry of Quadratic Differential Systems with a Weak Focus of Third Order

AbstractIn this article we determine the global geometry of the planar quadratic differential systems with a weak focus of third order. This class plays a significant role in the context of Hilbert's 16-th problem. Indeed, all examples of quadratic differential systems with at least four limit cycles, were obtained by perturbing a system in this family. We use the algebro-geometric concepts of divisor and zero-cycle to encode global properties of the systems and to give structure to this class. We give a theorem of topological classification of such systems in terms of integer-valued affine invariants. According to the possible values taken by them in this family we obtain a total of 18 topologically distinct phase portraits. We show that inside the class of all quadratic systems with the topology of the coefficients, there exists a neighborhood of the family of quadratic systems with a weak focus of third order and which may have graphics but no polycycle in the sense of [15] and no limit cycle, such that any quadratic system in this neighborhood has at most four limit cycles.

Smart surgical tools and augmenting devices

In this survey paper, the authors analyze the general structure of robotic systems for computer-assisted surgery, present a classification of such systems based on the degree of intelligence of the tools, and discuss some examples of different classes of devices. Computer-assisted surgery accelerated progress is related, on the one hand, to the improvement of medical imaging techniques and, on the other hand, to the evolution of surgical instrumentation. The integration of these two factors has determined an extraordinary progress that is not just a linear temporal development, but it is a discontinuity as regards traditional surgical procedures. Specifically, the authors consider the following classes of robotic-derived surgical devices/systems: a) handheld tools augmenting the capabilities of the surgeon; b) teleoperated surgical tools; and c) autonomous surgical robots. The paper will focus essentially on the analysis of systems and components of robots and tools designed for minimally invasive surgery. Although different classification methods exist on the basis of the clinical needs and/or on the design approach, the devices which will be illustrated in this paper are classified on the basis of their scale, degrees of freedom, autonomy, embedded intelligence, and features of the interface between the surgeon and the patient.

The Periodic System and the Metrology of Mesoscopic Quantum Systems

The current view is presented on the quantitative description of the periodic system, which is a multiparticle quantum system. Principles are proposed for extending the classification of such systems.

Special Control Features of Manipulative Systems with Unsteady Base

The main technical characteristics and quality performances of the manipulative systems under control with unsteady base are considered in the paper. Brief classification of such systems and the analysis of problems related with control systems synthesis are presented. The results of analysis allow us to outline the ways for possible solving the problems as follows: real time control implementation, insuring the needed accuracy of reference trajectory reproduction and reliability of information from sensors, efficiency increasing of noisy signals processing etc.

On integrable coupled KdV-type systems

In this paper we describe a new method for constructing integrable systems of differential equations. We are looking for systems in two variables in such forms that the reduction v = u leads us to a single equation in u. We give a complete classification of such systems that reduce to Korteweg-de Vries-type equations. Furthermore, we present an extensive (and complete for the systems of the Sawada-Kotera and Kaup-Kupershmidt types) classification of fifth-order equations in the same weighting. We show that the scalar integrable equations give rise to large classes of integrable systems. Moreover, we present a previously unknown example of a system that can be written in biHamiltonian form in infinitely many different ways, thereby solving the problem of the number of biHamiltonian forms that can have a differential equation. Finally, we present examples of nondegenerate systems possessing degenerate symmetries, which is impossible in the scalar case.

A multi-dimensional classification of production systems for the design and selection of production planning and control systems

A primary requirement for improved understanding of the management of production systems is an appropriate classification of such systems. This paper proposes a classification that facilitates a better understanding of real production systems. It combines all the essential features, e.g. the flow of materials with new classification perspectives with respect to response time, repetitiveness and work organization. As much previous work has influenced the approach proposed here, the paper also presents a review of relevant literature. The classification has four groups of characteristics, comprising eight dimensions of descriptors, encompassing 12 variables. Choosing or designing an appropriate production planning and control system (PPCS) is a difficult task due to the integrative character of the production planning and control area: it tends to interface with all functional areas of the enterprise. An important objective of the classification system proposed here is to provide a tool to assist in undertaking this difficult task. Real production systems are becoming more hybrid in order to be able to cope with change. We show that our classification can successfully deal with such systems.

Analysis of implicit hyperbolic multivariable systems

This paper deals with the coupled distributed parameter systems described by the system of partial differential equations (PDEs) with singular matrix coefficients. A basic classification of such systems is provided, and some properties of the systems are analyzed using the spectral theory of matrix polynomial pencils. This theory is used next for analysis of the transient behavior of superconducting energy storage coils. Several illustrative examples are included, and some numerical results showing the distribution of voltages and electric field stresses are presented.

Quadratic isoparametric systems in ℝn+mp

The notions of isoparametric maps and submanifolds in semi-Riemannian spaces are the generalizations of such notions in Riemannian spaces. The generalizations are different according to the purposes. We take the definitions as in the Riemannian case. Quadratic isoparametric maps and submanifolds are interesting examples which can be studied in detail. In this paper we study what we call quadratic isoparametric systems. In fact we give a classification of such systems of codimension 2. We use three different methods to show that quadratic isoparametric submanifolds of codimension 2 are homogeneous. The classification of quadratic isoparametric systems is done algebraically. By this we have changed the geometric problem of classifying quadratic submanifolds of codimension 2 into the algebraic problem of classifying quadratic isoparametric systems of codimension 2. The classification of such systems with arbitrary codimension is still open.

Conformal invariance and critical behavior

A review is given of recent work on the application of conformal invariance to systems at critical points, with emphasis on the classification of such systems in two dimensions using the constraints of positivity and modular invariance on a torus.