Relevant bibliographies by topics / Planar vector field

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Academic literature on the topic 'Planar vector field'

Author: Grafiati
Published: 4 June 2021
Last updated: 4 February 2022

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Journal articles on the topic "Planar vector field"

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Roussarie, Robert. "A topological study of planar vector field singularities." Discrete & Continuous Dynamical Systems - A 40, no. 9 (2020): 5217–45. http://dx.doi.org/10.3934/dcds.2020226.

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Algaba, Antonio, Cristóbal García, and Jaume Giné. "Orbital Reversibility of Planar Vector Fields." Mathematics 9, no. 1 (December 23, 2020): 14. http://dx.doi.org/10.3390/math9010014.

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In this work we use the normal form theory to establish an algorithm to determine if a planar vector field is orbitally reversible. In previous works only algorithms to determine the reversibility and conjugate reversibility have been given. The procedure is useful in the center problem because any nondegenerate and nilpotent center is orbitally reversible. Moreover, using this algorithm is possible to find degenerate centers which are orbitally reversible.
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Yoldas, Halil. "Some Results on Cosymplectic Manifolds Admitting Certain Vector Fields." Journal of Geometry and Symmetry in Physics 60 (2021): 83–94. http://dx.doi.org/10.7546/jgsp-60-2021-83-94.

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The purpose of present paper is to study cosymplectic manifolds admitting certain special vector fields such as holomorphically planar conformal (in short HPC) vector field. First, we prove that an HPC vector field on a cosymplectic manifold is also a Jacobi-type vector field. Then, we obtain the necessary conditions for such a vector field to be Killing. Finally, we give an important characterization for a torse-forming vector field on such a manifold given as to be recurrent.
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Nagloo, Joel, Alexey Ovchinnikov, and Peter Thompson. "Commuting planar polynomial vector fields for conservative Newton systems." Communications in Contemporary Mathematics 22, no. 04 (April 3, 2019): 1950025. http://dx.doi.org/10.1142/s0219199719500251.

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We study the problem of characterizing polynomial vector fields that commute with a given polynomial vector field on a plane. It is a classical result that one can write down solution formulas for an ODE that corresponds to a planar vector field that possesses a linearly independent commuting vector field. This problem is also central to the question of linearizability of vector fields. Let [Formula: see text], where [Formula: see text] is a field of characteristic zero, and [Formula: see text] the derivation that corresponds to the differential equation [Formula: see text] in a standard way. Let also [Formula: see text] be the Hamiltonian polynomial for [Formula: see text], that is [Formula: see text]. It is known that the set of all polynomial derivations that commute with [Formula: see text] forms a [Formula: see text]-module [Formula: see text]. In this paper, we show that, for every such [Formula: see text], the module [Formula: see text] is of rank [Formula: see text] if and only if [Formula: see text]. For example, the classical elliptic equation [Formula: see text], where [Formula: see text], falls into this category.
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LI, JIBIN, MINGJI ZHANG, and SHUMIN LI. "BIFURCATIONS OF LIMIT CYCLES IN A Z2-EQUIVARIANT PLANAR POLYNOMIAL VECTOR FIELD OF DEGREE 7." International Journal of Bifurcation and Chaos 16, no. 04 (April 2006): 925–43. http://dx.doi.org/10.1142/s0218127406015210.

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By using the bifurcation theory of planar dynamical systems and the method of detection functions, the bifurcations of limit cycles in a Z2-equivariant planar perturbed Hamiltonian polynomial vector fields of degree 7 are studied. An example of a special Z2-equivariant vector field having 50 limit cycles with a configuration of compound eyes are given.
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WU, YUHAI, and MAOAN HAN. "ON THE NUMBER AND DISTRIBUTIONS OF LIMIT CYCLES OF A PLANAR QUARTIC VECTOR FIELD." International Journal of Bifurcation and Chaos 23, no. 04 (April 2013): 1350069. http://dx.doi.org/10.1142/s0218127413500697.

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In this paper, the number and distributions of limit cycles of a planar quartic vector field are considered. Through perturbation technique and qualitative analysis of differential equation, it is shown that the quartic vector field has 21 limit cycles as parameters satisfy proper conditions. The distributions of limit cycles in the above perturbed planar quartic vector field are also presented.
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Gutierrez, C., and F. Sánchez-Bringas. "Planar vector field versions of Carathéodory's and Loewner's conjectures." Publicacions Matemàtiques 41 (January 1, 1997): 169–79. http://dx.doi.org/10.5565/publmat_41197_10.

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Bakhtin, Yuri, and Liying Li. "Weakly mixing smooth planar vector field without asymptotic directions." Proceedings of the American Mathematical Society 148, no. 11 (August 11, 2020): 4733–44. http://dx.doi.org/10.1090/proc/15147.

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Roitenberg, V. Sh. "Local Bifurcations of Reversible Piecewise Smooth Planar Dynamical Systems." Mathematics and Mathematical Modeling, no. 1 (June 9, 2020): 1–15. http://dx.doi.org/10.24108/mathm.0120.0000213.

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There are quite a few works, which consider local bifurcations of piecewise-smooth vector fields on the plane. A number of papers also studied the local bifurcations of smooth vector fields on the plane that are reversible with respect to involution. In the paper, we introduce reversible dynamical systems defined by piecewise-smooth vector fields on the coordinate plane (x, y) for which the discontinuity line y = 0 coincides with the set of fixed points of the system involution. We consider the generic one-parameter perturbations of such a vector field. The bifurcations of the singular point O lying on this line are described in two cases. In the first case, the point O is a rough saddle of the smooth vector fields that coincide with a piecewise smooth vector field in the half-planes y > 0 and y < 0. The parameter can be chosen so that for parameter values less than or equal to zero, the dynamical system has a unique singular point with four hyperbolic sectors in a vicinity of the point O. For positive values of the parameter in the vicinity of the point O, there are three singular points, a quasi-centre and two saddles, the separatrixes of which form a simple closed contour that bounds the cell from closed trajectories. In the second case, O is a rough node of the corresponding vector fields. The parameter can be chosen so that for values of the parameter less than or equal to zero, the dynamical system has a unique singular point in a vicinity of the point O, and all other trajectories are closed. For positive values of the parameter in the vicinity of the point O, there are three singular points, two nodes and a quasi-saddle, whose two separatrixes go to the nodes.
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de Carvalho, Tiago, and Durval José Tonon. "Normal Forms for Codimension One Planar Piecewise Smooth Vector Fields." International Journal of Bifurcation and Chaos 24, no. 07 (July 2014): 1450090. http://dx.doi.org/10.1142/s0218127414500904.

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In this paper, we are dealing with piecewise smooth vector fields in a 2D-manifold. In such a scenario, the main goal of this paper is to exhibit the homeomorphism that gives the topological equivalence between a codimension one piecewise smooth vector field and the respective C0-normal form.
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Dissertations / Theses on the topic "Planar vector field"

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Moretti, Junior Adimar [UNESP]. "Estudo de ciclos limites em sistemas diferenciais lineares por partes." Universidade Estadual Paulista (UNESP), 2012. http://hdl.handle.net/11449/92943.

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Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)
Neste trabalho temos como objetivo estudar o número e a distribuição de ciclos limites em sistemas diferenciais lineares por partes. Em particular estudamos o número de ciclos limites do sistema diferencial linear por partes planar ˙x = −y − ε φ ( x) , ˙y = x, onde ε 6= 0 é um parâmetro pequeno e φ é uma função periódica linear por partes ímpar de período 4 . Provamos que dado um inteiro arbitário positivo n, o sistema acima possui exatamente n ciclos limites na faixa |x| ≤ 2 (n + 1 ). Consequentemente, existem sistemas diferenciais lineares por partes contendo uma infinidade de ciclos limites no plano real. Inicialmente obtemos uma quota inferior par a o número destes ciclos limites na faixa | x| ≤ 2 (n + 1 ) via Teoria do Averaging . Em seguida , utilizando a Teoria de Campos de Vetores Rodados, verificamos que o sistema acima tem exatamente n ciclos limites na faixa | x| ≤ 2 (n + 1 )
The main goal of this work aim to study the number and distribution of limit cycles in piecewise linear differential systems. In particular we consider the planar piecewise linear differential system ˙x = −y − ε φ ( x) , ˙y = x, where ε 6= 0 is a small parameter and φ is an odd piecewise linear periodic function of period 4 . We prove that given an arbitrary positive integer n, the system above has exactly n limit cycles in the strip | x| ≤ 2 (n + 1 ) . Consequently, there are piecewise differential systems containing an infinite number of limit cycles in the real plane. First we get a lower bound on the number of limit cycles in the strip |x| ≤ 2 (n + 1 ) via Averaging Theory. In the following , using the Theory of Rotated Vector Fields, we see that above system has exactly n limit cycles in the strip | x| ≤ 2 (n + 1 )
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Moraes, Jaime Rezende de [UNESP]. "Ciclos limites de sistemas lineares por partes." Universidade Estadual Paulista (UNESP), 2011. http://hdl.handle.net/11449/92942.

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Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)
Consideramos dois casos principais de bifurcação de órbitas periódicas não hiperbólicas que dão origem a ciclos limite. Nosso estudo é feito para sistemas lineares por partes com três zonas em sua fórmula mais geral, que inclui situações sem simetria. Obtemos estimativas tanto para a amplitude como para o período do ciclo limite e apresentamos uma aplicação de interesse em engenharia: sistemas de controle.
We consider two main cases of bifurcation of non hyperbolic periodic orbits that give rise to limit cycles. Our study is done concerning piecewise linear systems with three zones in the more general formula that includes situations without symmetry. We obtain estimates for both the amplitude and the period of limit cycles and we present a applications of interest in engineering: control systems.
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Moraes, Jaime Rezende de. "Ciclos limites de sistemas lineares por partes /." São José do Rio Preto : [s.n.], 2011. http://hdl.handle.net/11449/92942.

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Orientador: Paulo Ricardo da Silva
Banca: Weber Flavio Pereira
Banca: Marcelo Messias
Resumo: Consideramos dois casos principais de bifurcação de órbitas periódicas não hiperbólicas que dão origem a ciclos limite. Nosso estudo é feito para sistemas lineares por partes com três zonas em sua fórmula mais geral, que inclui situações sem simetria. Obtemos estimativas tanto para a amplitude como para o período do ciclo limite e apresentamos uma aplicação de interesse em engenharia: sistemas de controle.
Abstract: We consider two main cases of bifurcation of non hyperbolic periodic orbits that give rise to limit cycles. Our study is done concerning piecewise linear systems with three zones in the more general formula that includes situations without symmetry. We obtain estimates for both the amplitude and the period of limit cycles and we present a applications of interest in engineering: control systems.
Mestre
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Maza, Sabido Susana. "Discrete and continuous symetries in planar vector fields." Doctoral thesis, Universitat de Lleida, 2008. http://hdl.handle.net/10803/81314.

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Aquesta tesi es situa en el marc de la teoriaqualitativadelssistemesd’equacionsdiferencials en el pla. Cada capítol conté un aspectediferent, però en totsells es tractenproblemes, la soluciódelsqualsestà basada en el rol que hi juguen les simetriesdiscretes i continues (reversibilitat o simetries de Lie) de campsvectorialsplans. A la introducció, es dóna un resumdelsresultatsmésconeguts i s’hiintrodueix la notació que es fa servir al llarg de la tesi. En el segon i tercer capítol, s’aborda el problema de trobarl’expressió explícita del canvilinealitzant o orbitalmentlinealitzantd#un camp vectorial suau a partir del coneixementd’uncommutador, en el cas de la linealització, o una simetria de Lie, en el cas de la linealització orbital. Cada capítol finalitzaambexemplesil.lustratius del procedimentconstructiudelscanvis. Al Capítol 5 s’apliquenelsresultatsdelscapítolsanteriors, combinatsamblinealitzacionsDarbouxianes. Concretament, es considera un sistema quadràtictipusLotka-Volterra i es caracteritzen les selles linealitzables i orbitalmentlinealitzablesmitjançant la troballadelscanvislinealitzants o orbitalmentlinealitzants. En el sisè capítol, s’utilitzal’existènciad’unàlgebra de simetriespuntuals de Lie per donar informació sobre l’existència i localitzaciód’òrbitesperiòdiques. En particular, quanl’àlgebra de simetriespuntuals de Lie d’unaequació diferencial escalar de segónordreautònoma i suau té dimensiómajor o igual a dos, definim les anomenadesfuncionsfonamentals que enspermeten estudiar les òrbitesperiòdiques al pla de fases. En el cas particular d’equacionspolinomials de Liénard, mostrem la no existència de cicles límit en tot el pla de fases. Finalment, al Capítol 7 es relacionen elssistemes reversibles amb el problema del centre aixícomamb el problema de la integrabilitat analítica. Consideremsistemesd’equacionsdiferencialsanalíticsamb centres degenerats i mostrem que poden transformar-se, després d’un reescalat del temps, en un sistema lineal i reversible. El coneixement de integralsprimeresens proporciona un procediment per caracteritzar, en alguns casos, la condició de reversibilitat del centre degenerat. D’altra banda, relacioneml’existència de integralsprimeresanalítiquesamb la reversibilitat orbital analítica en el cas de singularitatsdèbils no degenerades.
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Maza, Sabido Susanna. "Discrete and continuous symetries in planar vector fields." Doctoral thesis, Universitat de Lleida, 2008. http://hdl.handle.net/10803/81314.

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Aquesta tesi es situa en el marc de la teoriaqualitativadelssistemesd’equacionsdiferencials en el pla. Cada capítol conté un aspectediferent, però en totsells es tractenproblemes, la soluciódelsqualsestà basada en el rol que hi juguen les simetriesdiscretes i continues (reversibilitat o simetries de Lie) de campsvectorialsplans. A la introducció, es dóna un resumdelsresultatsmésconeguts i s’hiintrodueix la notació que es fa servir al llarg de la tesi. En el segon i tercer capítol, s’aborda el problema de trobarl’expressió explícita del canvilinealitzant o orbitalmentlinealitzantd#un camp vectorial suau a partir del coneixementd’uncommutador, en el cas de la linealització, o una simetria de Lie, en el cas de la linealització orbital. Cada capítol finalitzaambexemplesil.lustratius del procedimentconstructiudelscanvis. Al Capítol 5 s’apliquenelsresultatsdelscapítolsanteriors, combinatsamblinealitzacionsDarbouxianes. Concretament, es considera un sistema quadràtictipusLotka-Volterra i es caracteritzen les selles linealitzables i orbitalmentlinealitzablesmitjançant la troballadelscanvislinealitzants o orbitalmentlinealitzants. En el sisè capítol, s’utilitzal’existènciad’unàlgebra de simetriespuntuals de Lie per donar informació sobre l’existència i localitzaciód’òrbitesperiòdiques. En particular, quanl’àlgebra de simetriespuntuals de Lie d’unaequació diferencial escalar de segónordreautònoma i suau té dimensiómajor o igual a dos, definim les anomenadesfuncionsfonamentals que enspermeten estudiar les òrbitesperiòdiques al pla de fases. En el cas particular d’equacionspolinomials de Liénard, mostrem la no existència de cicles límit en tot el pla de fases. Finalment, al Capítol 7 es relacionen elssistemes reversibles amb el problema del centre aixícomamb el problema de la integrabilitat analítica. Consideremsistemesd’equacionsdiferencialsanalíticsamb centres degenerats i mostrem que poden transformar-se, després d’un reescalat del temps, en un sistema lineal i reversible. El coneixement de integralsprimeresens proporciona un procediment per caracteritzar, en alguns casos, la condició de reversibilitat del centre degenerat. D’altra banda, relacioneml’existència de integralsprimeresanalítiquesamb la reversibilitat orbital analítica en el cas de singularitatsdèbils no degenerades.
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Cardoso, Filho João Lopes. "A qualitative study of planar piecewise smooth vector fields." Universidade Federal de Goiás, 2018. http://repositorio.bc.ufg.br/tede/handle/tede/8577.

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Made available in DSpace on 2018-06-15T10:25:16Z (GMT). No. of bitstreams: 2 Tese - João Lopes Cardoso Filho - 2018.pdf: 1729607 bytes, checksum: 8279e98ec23b68bab062f8c812957bf4 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2018-05-18
Fundação de Amparo à Pesquisa do Estado de Goiás - FAPEG
In this work we exhibit canonical forms for 2D codimension one piecewise smooth vector Fields (PSVF). All possible orientations and codimension one scenarios were covered. Also the intrinsic objects that characterize each one of the canonical forms were presented. Also we present topological distinct canonical forms for a larger class for symmetric PSVF where the set of fixed points is contained in the variety os discontinuity. Finally we analyze the simultaneous occurrence of sliding and crossing limit cycle in the case where the piecewise linear vector fields presents a continuum of periodic orbits.
Neste trabalho exibiremos inicialmente as formas canônicas para campos vetoriais suaves por partes (PSVF) no plano. Todas os possíveis cenários de codimensão um são abordados. Também apresentamos formas canônicas topologicamente distintas para uma classe de PSVF com simetria onde o conjunto de pontos fixos está contido na variedade de descontinuidade. Finalmente, analisaremos a ocorrência simultânea de ciclos limite costurantes e deslizantes no caso linear por partes que apresentam um contínuo de órbitas periódicas.
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Brigitzer, Björn Tim. "Semi-local linearization for flat saddles of planar vector fields." Thesis, Uppsala University, Department of Mathematics, 2006. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-121363.

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Guo, Shaoming [Verfasser]. "Hilbert transforms and maximal operators along planar vector fields / Shaoming Guo." Bonn : Universitäts- und Landesbibliothek Bonn, 2015. http://d-nb.info/1077290233/34.

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Santallusia, Esvert Xavier. "Contribution to the center and integrability problems in planar vector fields." Doctoral thesis, Universitat de Lleida, 2017. http://hdl.handle.net/10803/402941.

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Aquesta tesi consta d'un primer capítol introductori, set capítols amb diferents resultats i una bibliografia. El primer capítol conté la definició i els resultats previs necessaris per abordar la resta de la memòria. El capítol 2 i 3 estan molt relacionats. En el primer es descriu un mètode alternatiu per al còmput de les constants de Poincaré--Liapunov. A diferència de mètodes anteriors, el mètode presentat no requereix el càlcul d'integrals i d'ona de forma explícita les constants de Poincaré--Liapunov. En el tercer capítol es descriu com s'ha implementat aquest nou mètode i els resultats que d'ona per a sistemes quadràtics i sistemes amb termes no lineals cúbics homogenis. El quart capítol es centra en equacions d'Abel i la seva integrabilitat. Es descriu la forma d'una integral primera que sigui algebraica en funcíó de les variables dependents i es donen múltiples exemples d'equacions d'Abel integrables en aquest sentit. En el cinquè capítol també s'aborda el problema de la integrabilitat però per a equacions diferencials en el pla definides per funcions analítiques. Es fa un reescalat de les variables dependents i de la variable independent amb un paràmetre "epsilon" que està elevat a potències senceres (blow-up paramètric) de forma que el sistema resultant sigui analític en "epsilon". Es d'ona un mètode que aprofita que una integral primera, si existeix, ha de ser analítica en el paràmetre a fi de trobar condicions per a l'existència d'aquesta integral primera. D'aquesta manera es defineix el que s'anomenen variables essencials del sistema. Els darrers tres capítols versen sobre les equacions d'Abel i el problema del centre. En general es consideren equacions d'Abel trigonomètriques. En el sisè capítol es donen algunes condicions necessàries i suficients per a que una equació d'Abel trigonomètrica definida per polinomis trigonomètrics de grau fins a 3 tingui un centre. Tots els exemples donats en aquest capítol tenen un centre universal. En el capítol setè es d'ona un exemple d'una equació d'Abel trigonomètrica definida per polinomis trigonomètrics de grau 3 que té un centre que no és universal. D'aquesta manera es resol un problema obert: determinar el grau més petit pel qual un equació d'Abel trigonomètrica amb centre no és de composició. El darrer capítol tracta equacions d'Abel trigonomètriques i polinomials i d'ona un compendi dels darrers resultats coneguts i conjectures sobre el problema del centre en aquestes equacions. També es donen exemples nous d'equacions d'Abel amb centre.
Esta tesis consta de un primer capítulo introductorio, siete capítulos con diferentes resultados y una bibliografía. El primer capítulo contiene la definición y los resultados previos necesarios para abordar el resto de la memoria. Los capítulos 2 y 3 están muy relacionados. En el primero se describe un método alternativo para el cómputo de las constantes de Poincaré--Liapunov. A diferencia de métodos anteriores, el método presentado no requiere el cálculo de integrales y da de forma explícita las constantes de Poincaré--Liapunov. En el tercer capítulo se describe cómo se ha implementado este nuevo método y los resultados que da para sistemas cuadráticos y sistemas con términos no lineales cíbicos homogéneos. El cuarto capítulo se centra en ecuaciones de Abel y su integrabilidad. Se describe la forma de una integral primera que sea algebraica en función de las variables dependientes y se dan múltiples ejemplos de ecuaciones de Abel integrables en este sentido. En el quinto capítulo también se aborda el problema de la integrabilidad pero para ecuaciones diferenciales en el plano definidas por funciones analíticas. Se hace un reescalado de las variables dependientes y de la variable independiente con un parámetro "epsilon" que está elevado a poténcias enteras (blow-up paramétrico) de forma que el sistema resultante sea analítico en "epsilon". Se da un método que aprovecha que una integral primera, si existe, debe ser analítica en el parámetro con el fin de encontrar condiciones para la existéncia de esta integral primera. De esta manera se define lo que se llaman variables esenciales del sistema. Los últimos tres capítulos versan sobre las ecuaciones de Abel y el problema del centro. En general se consideran ecuaciones de Abel trigonométricas. En el sexto capítulo se dan algunas condiciones necesarias y suficientes para que una ecuación de Abel definida por polinomios trigonométricos de grado hasta 3 tenga un centro. Todos los ejemplos dados en este capítulo tienen un centro universal. En la capítulo séptimo se da un ejemplo de una ecuación de Abel definida por polinomios trigonométricos de grado 3 que tiene un centro que no es universal. De esta manera se resuelve un problema abierto: determinar el grado mas pequeño por el que una ecuación de Abel trigonométrica con centro no es de composición. El último capítulo trata ecuaciones de Abel trigonométricas y polinomiales y da un compendio de los últimos resultados conocidos y conjeturas sobre el problema del centro en estas ecuaciones. También se dan ejemplos nuevos de ecuaciones de Abel con centro.
This thesis consists of a first introductory chapter, seven chapters with different results and a bibliography. The first chapter contains the definition and the previous results necessary to address the rest of the memory. Chapters 2 and 3 are closely related. In the first one, an alternative method is described for the computation of the Poincaré--Liapunov constants. Unlike previous methods, the presented method does not require the computation of primitives and gives an explicit expression of the Poincaré--Liapunov constants. The third chapter describes how this new method has been implemented and the results that it gives for quadratic systems and systems with homogeneous, cubic, non-linear terms. The fourth chapter focuses on Abel equations and their integrability. We describe the form of a first integral that is algebraic in function of the dependent variables and give more examples of equations of Abel integrable from this point of view. The fifth chapter also discusses the integrability problem but for differential equations in the plane defined by analytical functions. A rescaling of the dependent and the independent variables with a parameter "epsilon" which is elevated to integer powers (parametrical blow up) so that the resulting system is analytical in "epsilon". A method is given that takes advantage that a first integral, if it exists, it must be analytical in the parameter in order to find conditions for the existence of this first integral. In this way we define what are called essential variables of the system. The last three chapters deal with Abel equations and the center problem. In general, we consider Abel trigonometric equations. In the sixth chapter some necessary and sufficient conditions for an Abel equation defined by trigonometric polynomials of degree up to 3 have a center are given. All the examples given in this chapter have a universal center. In the seventh chapter it is given an example of an Abel equation defined by trigonometric polynomials of degree 3 with a center which is not universal. In this way an open problem is solved: to determine the lowest degree such that a trigonometric Abel equation has a center which is not a composition center. The last chapter deals with trigonometric and polynomial Abel equations and gives a survey of the last known results and conjectures about the center problem for these equations. Besides some new examples of Abel differential equations with a center are given.
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Cardin, Pedro Toniol [UNESP]. "Ciclos limites e a equação de van der Pol." Universidade Estadual Paulista (UNESP), 2008. http://hdl.handle.net/11449/94213.

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Nesta dissertação estudamos critérios para determinar a existência, a não existência e a unicidade de ciclos limites de campos de vetores planares. Mais especificamente, estudamos equações de Lienard Äx + f(x; _ x) _ x + g(x) = 0; onde f e g satisfazem determinadas hip¶oteses. Em particular estudamos a equa»c~ao de van der Pol Äx + (x2 ¡ 1) _ x + x = 0; a qual é conhecida da teoria dos circuitos elétricos. Provamos a existência e a unicidade de ciclos limites para estas equações. Por fim estudamos a equação de van der Pol com o parâmetro 1 e o fenômeno canard que ocorre ao considerarmos um parâmetro adicional ®: As técnicas utilizadas s~ao as usuais de Análise Assintótica.
In this work we study the existence, the non existence and the uniqueness of limit cycles of planar vector felds. More specifically, we study Lienard equations Äx+f(x; _ x) _ x+g(x) = 0; where f and g satisfy some hypothesis. In particular we study the van der Pol equation Äx + (x2 ¡ 1) _ x + x = 0; which is knew of the circuit theory. We prove the existence and the uniqueness of limit cycles for these equations. In the last part we study the van der Pol equation with the parameter 1 and the canard phenomenon which appears when we consider an additional parameter ®: The techniques employed are the usual in the Asymptotic Analysis.
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Books on the topic "Planar vector field"

1

Françoise, Jean-Pierre, and Robert Roussarie, eds. Bifurcations of Planar Vector Fields. Berlin, Heidelberg: Springer Berlin Heidelberg, 1990. http://dx.doi.org/10.1007/bfb0085387.

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Dumortier, Freddy, Robert Roussarie, Jorge Sotomayor, and Henryk Żaładek. Bifurcations of Planar Vector Fields. Berlin, Heidelberg: Springer Berlin Heidelberg, 1991. http://dx.doi.org/10.1007/bfb0098353.

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Chow, Shui-Nee. Normal forms and bifurcation of planar vector fields. Cambridge: Cambridge University Press, 1994.

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Chow, Shui-Nee. Normal forms and bifurcation of planar vector fields. Cambridge: Cambridge University Press, 2008.

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Roussarie, Robert H. Bifurcation of planar vector fields and Hilbert's sixteenth problem. Basel: Birkhäuser, 1998.

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Bifurcations of planar vector fields and Hilbert's sixteenth problem. Basel: Birkhäuser, 1998.

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Roussarie, Robert. Bifurcation of Planar Vector Fields and Hilbert’s Sixteenth Problem. Basel: Birkhäuser Basel, 1998. http://dx.doi.org/10.1007/978-3-0348-8798-4.

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Roussarie, Robert. Bifurcations of Planar Vector Fields and Hilbert's Sixteenth Problem. Basel: Springer Basel, 1998. http://dx.doi.org/10.1007/978-3-0348-0718-0.

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Desingularization of Nilpotent Singularities in Families of Planar Vector Fields. American Mathematical Society, 2002.

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Bifurcations Of Planar Vector Fields And Hilberts Sixteenth Problem. Springer Basel, 2013.

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Book chapters on the topic "Planar vector field"

1

Ilyashenko, Yulij, and Sergei Yakovenko. "Singular points of planar analytic vector fields." In Graduate Studies in Mathematics, 143–254. Providence, Rhode Island: American Mathematical Society, 2007. http://dx.doi.org/10.1090/gsm/086/02.

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Artés, Joan C., Jaume Llibre, Dana Schlomiuk, and Nicolae Vulpe. "Invariant theory of planar polynomial vector fields." In Geometric Configurations of Singularities of Planar Polynomial Differential Systems, 99–132. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-50570-7_5.

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Jarque, Xavier, and Jaume Llibre. "Global Structural Stability of Planar Hamiltonian Vector Fields." In Hamiltonian Dynamical Systems, 171–80. New York, NY: Springer New York, 1995. http://dx.doi.org/10.1007/978-1-4613-8448-9_12.

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Roussarie, Robert. "Families of Two-dimensional Vector Fields." In Bifurcation of Planar Vector Fields and Hilbert’s Sixteenth Problem, 1–15. Basel: Birkhäuser Basel, 1998. http://dx.doi.org/10.1007/978-3-0348-8798-4_1.

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Roussarie, Robert. "Families of Two-dimensional Vector Fields." In Bifurcations of Planar Vector Fields and Hilbert's Sixteenth Problem, 1–15. Basel: Springer Basel, 1998. http://dx.doi.org/10.1007/978-3-0348-0718-0_1.

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Lam, Wai Yeung, and Ulrich Pinkall. "Holomorphic Vector Fields and Quadratic Differentials on Planar Triangular Meshes." In Advances in Discrete Differential Geometry, 241–65. Berlin, Heidelberg: Springer Berlin Heidelberg, 2016. http://dx.doi.org/10.1007/978-3-662-50447-5_7.

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Gasull, A., and J. Sotomayor. "On the basin of attraction of dissipative planar vector fields." In Lecture Notes in Mathematics, 187–95. Berlin, Heidelberg: Springer Berlin Heidelberg, 1990. http://dx.doi.org/10.1007/bfb0085393.

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Dumortier, Freddy, Robert Roussarie, Jorge Sotomayor, and Henryk Żaładek. "Abelian integrals in unfoldings of codimension 3 singular planar vector fields." In Lecture Notes in Mathematics, 165–224. Berlin, Heidelberg: Springer Berlin Heidelberg, 1991. http://dx.doi.org/10.1007/bfb0098361.

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Roussarie, Robert. "Limit Periodic Sets." In Bifurcation of Planar Vector Fields and Hilbert’s Sixteenth Problem, 17–31. Basel: Birkhäuser Basel, 1998. http://dx.doi.org/10.1007/978-3-0348-8798-4_2.

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Roussarie, Robert. "The 0-Parameter Case." In Bifurcation of Planar Vector Fields and Hilbert’s Sixteenth Problem, 33–49. Basel: Birkhäuser Basel, 1998. http://dx.doi.org/10.1007/978-3-0348-8798-4_3.

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Conference papers on the topic "Planar vector field"

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ROUSSARIE, ROBERT. "BIFURCATIONS OF PLANAR VECTOR FIELD UNFOLDINGS." In Proceedings of the International Conference on Differential Equations. WORLD SCIENTIFIC, 2005. http://dx.doi.org/10.1142/9789812702067_0010.

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Pereira, Fernando Lobo, Teresa Grilo, and SIlvio Gama. "Optimal Control Framework for AUV’s Motion Planning in Planar Vortices Vector Field." In 2018 IEEE/OES Autonomous Underwater Vehicle Workshop (AUV). IEEE, 2018. http://dx.doi.org/10.1109/auv.2018.8729782.

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Ruan, Hengxin, Libo Wang, and Lianlin Li. "Theoretical study of vector-vortex field generated by quasi-periodic planar structure." In 2014 XXXIth URSI General Assembly and Scientific Symposium (URSI GASS). IEEE, 2014. http://dx.doi.org/10.1109/ursigass.2014.6929100.

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Liang, Yueqian, Yingmin Jia, Zhuo Wang, and Fumitoshi Matsuno. "Combined vector field approach for planar curved path following with fixed-wing UAVs." In 2015 American Control Conference (ACC). IEEE, 2015. http://dx.doi.org/10.1109/acc.2015.7172278.

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Harutyunyan, S., D. J. Hasanyan, and R. B. Davis. "Magnetoelastic Interactions at the Planar Interface of Two Ferromagnetic Solids: A Theoretical Study." In ASME 2010 International Mechanical Engineering Congress and Exposition. ASMEDC, 2010. http://dx.doi.org/10.1115/imece2010-39444.

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A model for the phenomena in which magnetic waves excite elastic waves and vice versa, is formulated and used to explore magnetic and anti-plane elastic interactions in piecewise homogeneous ferromagnetic media. The magnetization vectors in each medium are perpendicular to the wave propagation direction. The space is in an external homogeneous magnetic field, the direction of which coincides with direction of the initial magnetization vector. A joint contact on the interface of two ferromagnetic semi-spaces is considered. The existence of new type of waves is shown. These waves are localized on the interface of two ferromagnetic media and accompany to the reflection and transmission waves. These types of waves named as an accompanying surface magnetoelastic (ASM) waves. An analytical expression for the coefficients of reflection, transmission, and accompanying surface magnetoelastic waves is derived. Coefficients of mentioned waves are strongly dependent on applied magnetic field, material properties of each medium as well as the frequency and the angle of an incident elastic wave. Conclusions and directions for the design of magnetoelastic devises are outlined.
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Molaee-Ardekani, Behnam, Mohammad-Bagher Shamsollahi, Lotfi Senhadji, Bijan Vosoughi-Vahdat, and Eric Wodey. "Designing a planar vector field to investigate the role of a slow variable in an enhanced mean-field model during general anesthesia." In Conference Proceedings. Annual International Conference of the IEEE Engineering in Medicine and Biology Society. IEEE, 2006. http://dx.doi.org/10.1109/iembs.2006.260801.

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Molaee-Ardekani, Behnam, Mohammad-Bagher Shamsollahi, Lotfi Senhadji, Bijan Vosoughi-Vahdat, and Eric Wodey. "Designing a planar vector field to investigate the role of a slow variable in an enhanced mean-field model during general anesthesia." In Conference Proceedings. Annual International Conference of the IEEE Engineering in Medicine and Biology Society. IEEE, 2006. http://dx.doi.org/10.1109/iembs.2006.4398880.

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Rockwell, R. D., P. E. Allaire, and M. E. F. Kasarda. "Radial Planar Magnetic Bearing Analysis With Finite Elements Including Rotor Motion and Power Losses." In ASME 1997 International Gas Turbine and Aeroengine Congress and Exhibition. American Society of Mechanical Engineers, 1997. http://dx.doi.org/10.1115/97-gt-503.

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No literature is currently available which has evaluated finite element power loss models for magnetic bearings and compared the results to experimental results. In this paper a finite element model of the magnetic and electric fields in magnetic bearings, including the motion of the magnetic material in the rotor, is developed. It evaluates the two dimensional magnetic vector potential, magnetic flux density, electric field, eddy current, and power losses in an example magnetic bearing configuration. Results were obtained for both a solid rotor and a laminated rotor. For a solid rotor, both the magnetic flux density and eddy current plots at high rotational speeds are concentrated at the outer edge of the rotor. The ratio of calculated solid to laminated losses is found to be in the range of measured results by other authors. An effective axial conductivity was employed to model a laminated rotor and compared to experimental loss measurements. The correlation between measured and calculated results is quite good for a range of rotor speeds, magnetic flux density, and air gap thickness.
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Carvalho, Tiago De, Claudio A. Buzzi, and Rodrigo D. Euzébio. "On Poincaré-Bendixson Theorem in Planar Nonsmooth Vector Fields." In XXXV CNMAC - Congresso Nacional de Matemática Aplicada e Computacional. SBMAC, 2015. http://dx.doi.org/10.5540/03.2015.003.01.0023.

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Lyko, Christoph, Dirk Michaelis, Dieter Peitsch, and Mirko Dittmar. "Investigation of Bypass Transition in the Presence of a Separation Bubble With Tomographic PIV." In ASME Turbo Expo 2016: Turbomachinery Technical Conference and Exposition. American Society of Mechanical Engineers, 2016. http://dx.doi.org/10.1115/gt2016-57884.

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Low pressure turbines of small and medium sized engines may operate at very low Reynolds numbers. In consequence transition is delayed to an extend where laminar separation, detached transition and reattachment occur. The wakes from upstream blade rows lead to overall high turbulence levels which play a key role in the transition process. Freestream eddies buffeting the laminar boundary layer induce streamwise vortices known as Klebanoff Modes. To investigate this type of flow a flat plate was exposed to a pressure distribution. It is based on the PAK-B suction side and was created by a contoured wall facing the plate. The PAK-B is a Pratt & Whitney design and a Mach number scaled version of a highly aft loaded low pressure turbine airfoil. Due to the latter it suffers from a large separation bubble at low Reynolds numbers. The flow has been intensively investigated by hot-wire anemometry with a very high spatial resolution. This allows obtaining very precise information about the location of characteristic flow areas; for instance the separation and reattachment positions. Based on this information, Tomographic PIV was employed to expose detailed features in specific areas of the flow field. This technique provides the velocity vector information inside a flow volume. It complements hot-wire results, which give a time resolved information but only planar velocity magnitudes. Combining these techniques and comparing their results is therefore an excellent way to raise the physical understanding of the flow behaviour. This has been done using velocity profiles, skin friction coefficients and integral boundary layer parameters. As the 3D-PIV information allows calculation of derived quantities, like the vector field rotation, a picture of the coherent structures can be drawn.
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Reports on the topic "Planar vector field"

1

Guckenheimer, John. Investigation of Global Bifurcations in Planar Vector Fields. Fort Belvoir, VA: Defense Technical Information Center, March 1988. http://dx.doi.org/10.21236/ada204159.

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Baggenstoss, Paul M. A Baum-Welch Algorithm for Noisy Vector Fields for Classification and Synthesis of Textures Using Non-Symmetric Half-Plane. Fort Belvoir, VA: Defense Technical Information Center, August 2008. http://dx.doi.org/10.21236/ada494616.

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