Systems of differential equations that are competitive or cooperative. VI: A localCrClosing Lemma for 3-dimensional systems

1991 ◽  
Vol 11 (3) ◽  
pp. 443-454 ◽  
Author(s):  
Morris W. Hirsch
Keyword(s):  
Vector Field ◽  
Positive Integer ◽  
Differential Equations ◽  
Vector Fields ◽  
3 Dimensional ◽  
Planar Surfaces ◽  
Systems Of Differential Equations ◽  
Nonwandering Point

AbstractFor certainCr3-dimensional cooperative or competitive vector fieldsF, whereris any positive integer, it is shown that for any nonwandering pointp, every neighborhood ofFin theCrtopology contains a vector field for whichpis periodic, and which agrees withFoutside a given neighborhood ofp. The proof is based on the existence of invariant planar surfaces throughp.

scholarly journals Galoisian and qualitative approaches to linear Polyanin-Zaitsev vector fields

2018 ◽  
Vol 16 (1) ◽  
pp. 1204-1217
Author(s):  
Primitivo B. Acosta-Humánez ◽  
Alberto Reyes-Linero ◽  
Jorge Rodriguez-Contreras
Keyword(s):  
Vector Field ◽  
Differential Equations ◽  
Vector Fields ◽  
Qualitative Theory ◽  
Parametric Family ◽  
Liénard Equation ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Qualitative Approaches ◽  
Lienard Equation

AbstractIn this paper we study a particular parametric family of differential equations, the so-called Linear Polyanin-Zaitsev Vector Field, which has been introduced in a general case in [1] as a correction of a family presented in [2]. Linear Polyanin-Zaitsev Vector Field is transformed into a Liénard equation and, in particular, we obtain the Van Der Pol equation. We present some algebraic and qualitative results to illustrate some interactions between algebra and the qualitative theory of differential equations in this parametric family.

scholarly journals Complete lift of vector fields and sprays to T∞M

2015 ◽  
Vol 12 (10) ◽  
pp. 1550113 ◽  
Author(s):  
Ali Suri ◽  
Somaye Rastegarzadeh
Keyword(s):  
Vector Field ◽  
Differential Equations ◽  
Tangent Bundle ◽  
Vector Fields ◽  
Dimensional Manifold ◽  
Closed Curves ◽  
Infinite Dimensional ◽  
Integral Curves ◽  
Complete Lift ◽  
Infinite Dimensional Manifold

In this paper for a given Banach, possibly infinite dimensional, manifold M we focus on the geometry of its iterated tangent bundle TrM, r ∈ ℕ ∪ {∞}. First we endow TrM with a canonical atlas using that of M. Then the concepts of vertical and complete lifts for functions and vector fields on TrM are defined which they will play a pivotal role in our next studies i.e. complete lift of (semi)sprays. Afterward we supply T∞M with a generalized Fréchet manifold structure and we will show that any vector field or (semi)spray on M, can be lifted to a vector field or (semi)spray on T∞M. Then, despite of the natural difficulties with non-Banach modeled manifolds, we will discuss about the ordinary differential equations on T∞M including integral curves, flows and geodesics. Finally, as an example, we apply our results to the infinite-dimensional case of manifold of closed curves.

scholarly journals On constructing single-input non-autonomous systems of full rank

2018 ◽  
Keyword(s):  
Vector Field ◽  
Nonlinear System ◽  
Differential Equations ◽  
Autonomous System ◽  
Vector Fields ◽  
Autonomous Systems ◽  
Full Rank ◽  
System Of Differential Equations ◽  
Single Input ◽  
Class C

For a nonlinear system of differential equations $\dot x=f(x)$, a method of constructing a system of full rank $\dot x=f(x)+g(x)u$ is studied for vector fields of the class $C^k$, $1\le k<\infty$, in the case when $f(x)\not=0$. A method for constructing a non-autonomous system of full rank is proposed in the case when the vector field $f(x)$ can vanish.

scholarly journals Explicit Bracket in an Exceptional Simple Lie Superalgebra

1998 ◽  
Vol 08 (04) ◽  
pp. 479-495 ◽  
Author(s):  
Irina Shchepochkina ◽  
Gerhard Post
Keyword(s):  
Differential Equations ◽  
Generating Functions ◽  
Vector Fields ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Exact Form ◽  
3 Dimensional

This note is devoted to a more detailed description of one of the five simple exceptional Lie superalgebras of vector fields, [Formula: see text] a subalgebra of [Formula: see text]. We derive differential equations for its elements, and solve these equations. Hence we get an exact form for the elements of [Formula: see text]. Moreover we realize [Formula: see text] by "glued" pairs of generating functions on a (3∣3)-dimensional periplectic (odd symplectic) supermanifold and describe the bracket explicitly.

scholarly journals LAGRANGIAN FLOWS FOR VECTOR FIELDS WITH GRADIENT GIVEN BY A SINGULAR INTEGRAL

2013 ◽  
Vol 10 (02) ◽  
pp. 235-282 ◽  
Author(s):  
FRANÇOIS BOUCHUT ◽  
GIANLUCA CRIPPA
Keyword(s):  
Vector Field ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Singular Integral ◽  
Vector Fields ◽  
Transport Equations ◽  
Well Posedness ◽  
Quantitative Stability ◽  
Quantitative Estimates

We prove quantitative estimates on flows of ordinary differential equations with vector field with gradient given by a singular integral of an L1 function. Such estimates allow to prove existence, uniqueness, quantitative stability and compactness for the flow, going beyond the BV theory. We illustrate the related well-posedness theory of Lagrangian solutions to the continuity and transport equations.

VECTOR FIELDS, VARIATIONAL EQUATIONS AND COMMUTATORS

2012 ◽  
Vol 22 (08) ◽  
pp. 1250190
Author(s):  
WILLI-HANS STEEB ◽  
YORICK HARDY ◽  
IGOR TANSKI
Keyword(s):  
Vector Field ◽  
Differential Equations ◽  
Fixed Points ◽  
Lie Algebra ◽  
Autonomous System ◽  
Vector Fields ◽  
Autonomous Systems ◽  
First Integrals ◽  
Variational Equations ◽  
First Order

We study autonomous systems of first order ordinary differential equations, their corresponding vector fields and the autonomous system corresponding to the vector field of the commutator of two such autonomous systems. These vector fields form a Lie algebra. From the variational equations of these autonomous systems, we form new vector fields consisting of the sum of the two vector fields. We show that these new vector fields also form a Lie algebra. Results about fixed points, first integrals and the divergence of the vector fields are also presented.

scholarly journals Construction of an autogenerator dynamic model applicable to nuclear processes

2011 ◽  
Vol 26 (1) ◽  
pp. 74-77
Author(s):  
Diana Dolicanin ◽  
Vladimir Amelkin ◽  
Milisav Stefanovic ◽  
Milos Vujisic
Keyword(s):  
Mathematical Model ◽  
Vector Field ◽  
Linear System ◽  
Differential Equations ◽  
Dynamic Model ◽  
New Method ◽  
Oscillation Regime ◽  
Systems Of Differential Equations ◽  
Non Linear System ◽  
Nuclear Processes

We propose a new method for constructing a mathematical model of a non-linear system in an auto-oscillation regime. The method is based on the divergence of a vector field having a constant value along the corresponding periodical motion. The variants of the obtained model could be used for describing nuclear processes that are represented by the systems of differential equations analogous to that of the presented model.

scholarly journals A HAIR AND HAIR VORTEX SIMULATION TECHNIQUE BASED ON VECTOR FIELDS ON A MANIFOLD

2016 ◽  
Vol 54 (1) ◽  
pp. 109 ◽  
Author(s):  
Nguyen Van Huan
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Dimensional Space ◽  
Simulation System ◽  
Physical Characteristics ◽  
Simulation Technique ◽  
Virtual Human ◽  
Surface Model ◽  
3 Dimensional ◽  
Time Required

Hair is an important component in the virtual human simulation system. During hair simulation, not only the time required to ensure that there are required to express the physical characteristics, chemical such as hair styles, colors, curves, twists, parting one’s hair and hair vortex (swirl), ... to represent the authenticity of the hair and improve hair simulation quality. While the studies has announced today that they mainly focus on simulating the styles, the motion of the hair that has not been expressed characteristics of hair as hair vortex,... The paper introduces the concept of scalp model as a manifold in 3-dimensional space. Based on the nature of the singularity of the vector field on the manifold, the paper proposes a hair simulation technique on the scalp surface model based on vector field on the manifold. Thus, we can simulate appropriately the hair vortex on the scalp model.

SOLUTION OF SOME SYSTEMS OF DIFFERENTIAL EQUATIONS IN MATHEMATICAL SYSTEM MAPLE

2013 ◽  
Vol 1 (05) ◽  
pp. 58-65
Author(s):  
Yunona Rinatovna Krakhmaleva ◽  
◽  
Gulzhan Kadyrkhanovna Dzhanabayeva ◽  
Keyword(s):  
Differential Equations ◽  
Systems Of Differential Equations ◽  
Mathematical System
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