Normal Form for a Class of Three-Dimensional Systems with Free-Divergence Principal Part

2018 ◽  
pp. 37-65
Author(s):  
Antonio Algaba ◽  
Natalia Fuentes ◽  
Estanislao Gamero ◽  
Cristóbal García
Keyword(s):  
Normal Form ◽  
Principal Part ◽  
Three Dimensional

On the Boundary Value Problem in A Dihedral Angle for Normally Hyperbolic Systems of First Order

1998 ◽  
Vol 5 (2) ◽  
pp. 121-138
Author(s):  
O. Jokhadze
Keyword(s):  
Partial Differential Equations ◽  
Boundary Value Problem ◽  
Principal Part ◽  
Hyperbolic Systems ◽  
Boundary Value ◽  
Three Dimensional ◽  
Order System ◽  
First Order ◽  
Dimensional Boundary ◽  
Class Of Functions

Abstract Some structural properties as well as a general three-dimensional boundary value problem for normally hyperbolic systems of partial differential equations of first order are studied. A condition is given which enables one to reduce the system under consideration to a first-order system with the spliced principal part. It is shown that the initial problem is correct in a certain class of functions if some conditions are fulfilled.

Unique Normal Form and the Associated Coefficients for a Class of Three-Dimensional Nilpotent Vector Fields

2017 ◽  
Vol 27 (14) ◽  
pp. 1750224
Author(s):  
Jing Li ◽  
Liying Kou ◽  
Duo Wang ◽  
Wei Zhang
Keyword(s):  
Normal Form ◽  
Vector Fields ◽  
Three Dimensional ◽  
Recursive Formula ◽  
Higher Order ◽  
Dimensional Vector ◽  
Symbolic Calculations

In this paper, we mainly focus on the unique normal form for a class of three-dimensional vector fields via the method of transformation with parameters. A general explicit recursive formula is derived to compute the higher order normal form and the associated coefficients, which can be achieved easily by symbolic calculations. To illustrate the efficiency of the approach, a comparison of our result with others is also presented.

Quadratic Differential Systems with a Finite Saddle-Node and an Infinite Saddle-Node (1, 1)SN - (B)

2021 ◽  
Vol 31 (09) ◽  
pp. 2130026
Author(s):  
Joan C. Artés ◽  
Marcos C. Mota ◽  
Alex C. Rezende
Keyword(s):  
Normal Form ◽  
Bifurcation Diagram ◽  
Three Dimensional ◽  
Phase Portraits ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Saddle Node ◽  
Complete Study ◽  
Real Quadratic ◽  
Finite Singularity

This paper presents a global study of the class [Formula: see text] of all real quadratic polynomial differential systems which have a finite semi-elemental saddle-node and an infinite saddle-node formed by the coalescence of a finite singularity and an infinite singularity. This class can be divided into two different families, namely, [Formula: see text] phase portraits possessing a finite saddle-node as the only finite singularity and [Formula: see text] phase portraits possessing a finite saddle-node and also a simple finite elemental singularity. Each one of these two families is given by a specific normal form. The study of family [Formula: see text] was reported in [Artés et al., 2020b] where the authors obtained [Formula: see text] topologically distinct phase portraits for systems in the closure [Formula: see text]. In this paper, we provide the complete study of the geometry of family [Formula: see text]. This family which modulo the action of the affine group and time homotheties is three-dimensional and we give the bifurcation diagram of its closure with respect to a specific normal form, in the three-dimensional real projective space. The respective bifurcation diagram yields 631 subsets with 226 topologically distinct phase portraits for systems in the closure [Formula: see text] within the representatives of [Formula: see text] given by a specific normal form. Some of these phase portraits are proven to have at least three limit cycles.

scholarly journals Topological Classification of Quadratic Polynomial Differential Systems with a Finite Semi-Elemental Triple Saddle

2016 ◽  
Vol 26 (11) ◽  
pp. 1650188 ◽  
Author(s):  
Joan C. Artés ◽  
Regilene D. S. Oliveira ◽  
Alex C. Rezende
Keyword(s):  
Normal Form ◽  
Bifurcation Diagram ◽  
Limit Cycles ◽  
Three Dimensional ◽  
Quadratic Polynomial ◽  
Phase Portraits ◽  
Structure Stability ◽  
Differential Systems ◽  
Bifurcation Set ◽  
Polynomial Differential Systems

The study of planar quadratic differential systems is very important not only because they appear in many areas of applied mathematics but due to their richness in structure, stability and questions concerning limit cycles, for example. Even though many papers have been written on this class of systems, a complete understanding of this family is still missing. Classical problems, and in particular Hilbert’s 16th problem [Hilbert, 1900, 1902], are still open for this family. In this article, we make a global study of the family [Formula: see text] of all real quadratic polynomial differential systems which have a finite semi-elemental triple saddle (triple saddle with exactly one zero eigenvalue). This family modulo the action of the affine group and time homotheties is three-dimensional and we give its bifurcation diagram with respect to a normal form, in the three-dimensional real space of the parameters of this normal form. This bifurcation diagram yields 27 phase portraits for systems in [Formula: see text] counting phase portraits with and without limit cycles. Algebraic invariants are used to construct the bifurcation set and we present the phase portraits on the Poincaré disk. The bifurcation set is not just algebraic due to the presence of a surface found numerically, whose points correspond to connections of separatrices.

scholarly journals Integrable zero-Hopf singularities and three-dimensional centres

2017 ◽  
Vol 148 (2) ◽  
pp. 327-340
Author(s):  
Isaac A. García
Keyword(s):  
Phase Space ◽  
Normal Form ◽  
Periodic Orbits ◽  
Parameter Space ◽  
Three Dimensional ◽  
Quadratic Polynomial ◽  
Affine Variety ◽  
Higher Dimensional ◽  
Dimensional Version ◽  
Poincaré Return Map

In this paper we show that the well-known Poincaré–Lyapunov non-degenerate analytic centre problem in the plane and its higher-dimensional version, expressed as the three-dimensional centre problem at the zero-Hopf singularity, have a lot of common properties. In both cases the existence of a neighbourhood of the singularity in the phase space completely foliated by periodic orbits (including equilibria) is characterized by the fact that the system is analytically completely integrable. Hence its Poincaré–Dulac normal form is analytically orbitally linearizable. There also exists an analytic Poincaré return map and, when the system is polynomial and parametrized by its coefficients, the set of systems with centres corresponds to an affine variety in the parameter space of coefficients. Some quadratic polynomial families are considered.

MATHEMATICAL ANALYSIS OF A SAINT-VENANT MODEL WITH VARIABLE TEMPERATURE

2010 ◽  
Vol 20 (08) ◽  
pp. 1251-1297 ◽  
Author(s):  
VINCENT GIOVANGIGLI ◽  
BINH TRAN
Keyword(s):  
Partial Differential Equations ◽  
Normal Form ◽  
Differential Equations ◽  
Energy Equation ◽  
Variable Temperature ◽  
Transport Coefficients ◽  
Three Dimensional ◽  
Invariance Property ◽  
Three Dimensional Model ◽  
Partial Differential

We investigate the derivation and the mathematical properties of a Saint-Venant model with an energy equation and with temperature-dependent transport coefficients. These equations model shallow water flows as well as thin viscous sheets over fluid substrates like oil slicks, atlantic waters in the Strait of Gilbraltar or float glasses. We exhibit an entropy function for the system of partial differential equations and by using the corresponding entropic variable, we derive a symmetric conservative formulation of the system. The symmetrized Saint-Venant quasilinear system of partial differential equations is then shown to satisfy the nullspace invariance property and is recast into a normal form. Upon establishing the local dissipative structure of the linearized normal form, global existence results and asymptotic stability of equilibrium states are obtained. We finally derive the Saint-Venant equations with an energy equation taking into account the temperature-dependence of transport coefficients from an asymptotic limit of a three-dimensional model.

scholarly journals Applications of Game Theory in Project Management: A Structured Review and Analysis

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 858 ◽  
Author(s):  
Mahendra Piraveenan
Keyword(s):  
Game Theory ◽  
Project Management ◽  
Normal Form ◽  
Cooperative Games ◽  
Three Dimensional ◽  
Citation Network ◽  
Extensive Form ◽  
Management Scenarios ◽  
Dimensional Classification ◽  
Non Cooperative Games

This paper provides a structured literature review and analysis of using game theory to model project management scenarios. We select and review thirty-two papers from Scopus, present a complex three-dimensional classification of the selected papers, and analyse the resultant citation network. According to the industry-based classification, the surveyed literature can be classified in terms of construction industry, ICT industry or unspecified industry. Based on the types of players, the literature can be classified into papers that use government-contractor games, contractor–contractor games, contractor-subcontractor games, subcontractor–subcontractor games or games involving other types of players. Based on the type of games used, papers using normal-form non-cooperative games, normal-form cooperative games, extensive-form non-cooperative games or extensive-form cooperative games are present. Also, we show that each of the above classifications plays a role in influencing which papers are likely to cite a particular paper, though the strongest influence is exerted by the type-of-game classification. Overall, the citation network in this field is sparse, implying that the awareness of authors in this field about studies by other academics is suboptimal. Our review suggests that game theory is a very useful tool for modelling project management scenarios, and that more work needs to be done focusing on project management in ICT domain, as well as by using extensive-form cooperative games where relevant.

Equivariant N-Dof Hamiltonians Via Generalized Normal Forms

2003 ◽  
Vol 05 (03) ◽  
pp. 449-480 ◽  
Author(s):  
Jesús Palacián ◽  
Patricia Yanguas
Keyword(s):  
Normal Form ◽  
Hamiltonian Systems ◽  
Principal Part ◽  
Degrees Of Freedom ◽  
Qualitative Description ◽  
Hamiltonian Vector Field ◽  
Perturbed System ◽  
Formal Integral ◽  
Truncated Normal ◽  
Lie Transformations

In the present paper we study polynomial Hamiltonian systems depending on one or various real parameters. We determine the values that these parameters should take in order to be able to construct formal (asymptotic) integrals of the system. In this respect, a method to calculate the formal integrals of a polynomial Hamiltonian vector field is presented. The original Hamilton function represents a family of dynamical systems composed by a principal part (quadratic terms) plus the perturbation (terms of degree three or bigger). We extend an integral of the principal part to the perturbed system by means of Lie transformations for autonomous Hamiltonian systems. Thus, the procedure is carried out order by order starting with polynomials of degree three. We obtain the conditions that the external parameters have to satisfy so that the integral of the quadratic terms persists for the whole system up to a certain order of approximation. Once the formal integral is computed the departure system has been transformed into a generalized normal form, i.e. a system which is equivalent to the initial one but easier to be analysed by making use of reduction theory. The truncated normal form defines a system with less degrees of freedom than the original Hamiltonian and is written exactly in terms of the polynomial first integrals associated to the quadratic part of the new integral and it contains the qualitative description of the initial system. The theory is illustrated with two examples borrowed from Physics.

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