Artigos de revistas: "Planar vector field"

1

Roussarie, Robert. "A topological study of planar vector field singularities". Discrete & Continuous Dynamical Systems - A 40, n.º 9 (2020): 5217–45. http://dx.doi.org/10.3934/dcds.2020226.

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Algaba, Antonio, Cristóbal García e Jaume Giné. "Orbital Reversibility of Planar Vector Fields". Mathematics 9, n.º 1 (23 de dezembro de 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 e Peter Thompson. "Commuting planar polynomial vector fields for conservative Newton systems". Communications in Contemporary Mathematics 22, n.º 04 (3 de abril de 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 e SHUMIN LI. "BIFURCATIONS OF LIMIT CYCLES IN A Z2-EQUIVARIANT PLANAR POLYNOMIAL VECTOR FIELD OF DEGREE 7". International Journal of Bifurcation and Chaos 16, n.º 04 (abril de 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, e MAOAN HAN. "ON THE NUMBER AND DISTRIBUTIONS OF LIMIT CYCLES OF A PLANAR QUARTIC VECTOR FIELD". International Journal of Bifurcation and Chaos 23, n.º 04 (abril de 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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7

Gutierrez, C., e F. Sánchez-Bringas. "Planar vector field versions of Carathéodory's and Loewner's conjectures". Publicacions Matemàtiques 41 (1 de janeiro de 1997): 169–79. http://dx.doi.org/10.5565/publmat_41197_10.

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Bakhtin, Yuri, e Liying Li. "Weakly mixing smooth planar vector field without asymptotic directions". Proceedings of the American Mathematical Society 148, n.º 11 (11 de agosto de 2020): 4733–44. http://dx.doi.org/10.1090/proc/15147.

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9

Roitenberg, V. Sh. "Local Bifurcations of Reversible Piecewise Smooth Planar Dynamical Systems". Mathematics and Mathematical Modeling, n.º 1 (9 de junho de 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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10

de Carvalho, Tiago, e Durval José Tonon. "Normal Forms for Codimension One Planar Piecewise Smooth Vector Fields". International Journal of Bifurcation and Chaos 24, n.º 07 (julho de 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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Algaba, A., C. García e M. Reyes. "Invariant curves and analytic integrability of a planar vector field". Journal of Differential Equations 266, n.º 2-3 (janeiro de 2019): 1357–76. http://dx.doi.org/10.1016/j.jde.2018.07.074.

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Wu, Yu-hai, Li-xin Tian e Mao-an Han. "On the limit cycles of a quintic planar vector field". Science in China Series A: Mathematics 50, n.º 7 (julho de 2007): 925–40. http://dx.doi.org/10.1007/s11425-007-0045-0.

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Xiao, Dongmei. "Bifurcations on a Five-Parameter Family of Planar Vector Field". Journal of Dynamics and Differential Equations 20, n.º 4 (25 de abril de 2008): 961–80. http://dx.doi.org/10.1007/s10884-008-9109-2.

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Shi, Jianping, e Wentao Jiang. "Bifurcation of Limit Cycles in a 12-Degree Hamiltonian System Under Thirteenth-Order Perturbation". International Journal of Bifurcation and Chaos 28, n.º 01 (janeiro de 2018): 1850011. http://dx.doi.org/10.1142/s0218127418500116.

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This paper considers the weakened Hilbert’s 16th problem for symmetric planar perturbed polynomial Hamiltonian system. A [Formula: see text]-equivariant planar vector field of degree 12 is deduced to find as many as possible limit cycles and their configuration patterns. By using bifurcation theory of planar dynamical system and the method of detection function, we have obtained that, under the thirteenth-order perturbation, the above [Formula: see text]-equivariant planar perturbed Hamiltonian vector field of 12-degree has at least 117 limit cycles. Moreover, this paper also shows the configuration of compound eyes of the corresponding perturbed systems.

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Duffy, Charles J., e Robert H. Wurtz. "Planar Directional Contributions to Optic Flow Responses in MST Neurons". Journal of Neurophysiology 77, n.º 2 (1 de fevereiro de 1997): 782–96. http://dx.doi.org/10.1152/jn.1997.77.2.782.

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Duffy, Charles J. and Robert H. Wurtz. Planar directional contributions to optic flow responses in MST neurons. J. Neurophysiol. 77: 782–796, 1997. Many neurons in the dorsal region of the medial superior temporal area (MSTd) of monkey cerebral cortex respond to optic flow stimuli in which the center of motion is shifted off the center of the visual field. Each shifted-center-of-motion stimulus presents both different directions of planar motion throughout the visual field and a unique pattern of global motion across the visual field. We investigated the contribution of planar motion to the responses of these neurons in two experiments. In the first, we compared the responses of 243 neurons to planar motion and to shifted-center-of-motion stimuli created by vector summation of planar motion and radial or circular motion. We found that many neurons preferred the same directions of motion in the combined stimuli as in the planar stimuli, but other neurons did not. When we divided our sample into one group with stronger directionality to both planar and vector combination stimuli and one group with weaker directionality, we found that the neurons with the stronger directionality were those that showed the greatest similarity in the preferred direction of motion for both the planar and combined stimuli. In a second set of experiments, we overlapped planar motion and radial or circular motion to create transparent stimuli with the same motion components as the vector combination stimuli, but without the shifted centers of motion. We found that the neurons that responded most strongly to the planar motion when it was combined with radial or circular motion also responded best when the planar motion was overlapped by a transparent motion stimulus. We conclude that the responses of those neurons with stronger directional responses to both the motion of planar and vector combination stimuli are most readily understood as responding to the total planar motion in the stimulus, a planar motion mechanism. Other neurons that had weaker directional responses showed no such similarity in the preferred directions of planar motion in the vector combination and the transparent overlap stimuli and fit best with a mechanism dependent on the global motion pattern. We also found that neurons having significant responses to both radial and circular motion also responded to the spiral stimuli that result from a vector combination of radial and circular motion. The preferred planar-spiral vector combination stimulus was frequently the one containing that neurons' preferred direction of planar motion, which makes them similar to other MSTd neurons.

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16

CHAN, H. S. Y., K. W. CHUNG e JIBIN LI. "BIFURCATIONS OF LIMIT CYCLES IN A Z3-EQUIVARIANT VECTOR FIELD OF DEGREE 5". International Journal of Bifurcation and Chaos 11, n.º 08 (agosto de 2001): 2287–98. http://dx.doi.org/10.1142/s0218127401003267.

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A concrete numerical example of Z3-equivariant planar perturbed Hamiltonian vector field of fifth degree having 23 limit cycles and a configuration of compound eyes are given, by using the bifurcation theory of planar dynamical systems and the method of detection functions. It gives rise to the conclusion: the Hilbert number H(5) ≥ 23 for the second part of Hilbert's 16th problem.

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WANG, SHARON, PEI YU e JIBIN LI. "BIFURCATION OF LIMIT CYCLES IN Z10-EQUIVARIANT VECTOR FIELDS OF DEGREE 9". International Journal of Bifurcation and Chaos 16, n.º 08 (agosto de 2006): 2309–24. http://dx.doi.org/10.1142/s0218127406016070.

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In this paper, we consider the weakened Hilbert's 16th problem for symmetric planar perturbed polynomial Hamiltonian systems. In particular, a perturbed Hamiltonian polynomial vector field of degree 9 is studied, and an example of Z10-equivariant planar perturbed Hamiltonian systems is constructed. With maximal number of closed orbits, it gives rise to different configurations of limit cycles. By applying the bifurcation theory of planar dynamic systems and the method of detection functions, with the aid of numerical simulations, we show that a polynomial vector field of degree 9 with Z10 symmetry can have at least 80 limit cycles, i.e. H(9) ≥ 92 - 1.

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18

Ma, Yunlei, e Yuhai Wu. "On the Limit Cycles of a Perturbed Z3-Equivariant Planar Quintic Vector Field". International Journal of Bifurcation and Chaos 25, n.º 05 (maio de 2015): 1550073. http://dx.doi.org/10.1142/s021812741550073x.

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In this paper, the number and distributions of limit cycles in a Z3-equivariant quintic planar polynomial system are studied. 24 limit cycles with two different configurations are shown in this quintic planar vector field by combining the methods of double homoclinic loops bifurcation, heteroclinic loop bifurcation and Poincaré–Bendixson Theorem. The results obtained are useful to the study of weakened 16th Hilbert problem.

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Algaba, A., C. García e M. Reyes. "Characterization of a monodromic singular point of a planar vector field". Nonlinear Analysis: Theory, Methods & Applications 74, n.º 16 (novembro de 2011): 5402–14. http://dx.doi.org/10.1016/j.na.2011.05.023.

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20

Bateman, Michael. "Maximal averages along a planar vector field depending on one variable". Transactions of the American Mathematical Society 365, n.º 8 (12 de março de 2013): 4063–79. http://dx.doi.org/10.1090/s0002-9947-2013-05673-5.

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21

Aimi, A., G. Buffoni e M. Groppi. "Decomposition of a planar vector field into irrotational and rotational components". Applied Mathematics and Computation 244 (outubro de 2014): 63–90. http://dx.doi.org/10.1016/j.amc.2014.06.080.

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Duffy, C. J., e R. H. Wurtz. "Sensitivity of MST neurons to optic flow stimuli. II. Mechanisms of response selectivity revealed by small-field stimuli". Journal of Neurophysiology 65, n.º 6 (1 de junho de 1991): 1346–59. http://dx.doi.org/10.1152/jn.1991.65.6.1346.

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1. In these experiments we examined the receptive field mechanisms that support the optic flow field selective responses of neurons in the dorsomedial region of the medial superior temporal area (MSTd). Our experiments tested the predictions of two hypotheses of optic flow field selectivity. The direction mosaic hypothesis states that these receptive fields contain a set of planar direction-selective subfields that match the local directions of motion within optic flow fields. The vector field hypothesis states that these receptive fields are uniquely sensitive to distributed properties of planar, circular, or radial optic flow fields. 2. Experiments using large-field stimuli revealed that some neurons showed changes in optic flow field selectivity depending on the position of the stimulus in the receptive field; these are position-dependent responses. However, other neurons maintained the same optic flow field selectivities in spite of changes in stimulus position; these are position-invariant responses. We have used the position dependence or invariance of optic flow field selectivity as a way of testing the direction mosaic and vector field hypotheses. Position dependence is more consistent with the direction mosaic hypothesis, whereas position invariance is more consistent with the vector field hypothesis. 3. To test for position effects, we examined the optic flow field selectivity of small subfields within the large receptive fields of 160 MSTd neurons. First, we centered small-field optic flow stimuli of various sizes over the same position in the receptive field. Most MSTd neurons showed decreasing response amplitude with decreasing stimulus size but maintained optic flow field selectivity. 4. We then placed small-field stimuli at various positions within the large receptive field of these MSTd neurons. Position-invariant response selectivity was most prominent in single-component neurons, suggesting that they were more consistent with the vector field hypothesis. Position-dependent response selectivity was most prominent in triple-component neurons, suggesting that they were more consistent with the direction mosaic hypothesis. However, the variations in planar direction preference throughout the receptive field of these triple-component neurons were not consistent with a direction mosaic explanation of the large-field circular or radial selectivity observed. 5. Small-field position studies also demonstrated the existence of zones within the receptive field in which either direction-selective inhibitory or direction-selective excitatory responses predominated. The degree of overlap between these zones increased from nonselective to triple- to double- and finally to single-component neurons.(ABSTRACT TRUNCATED AT 400 WORDS)

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ALGABA, A., C. GARCÍA e M. REYES. "A note on analytic integrability of planar vector fields". European Journal of Applied Mathematics 23, n.º 5 (23 de abril de 2012): 555–62. http://dx.doi.org/10.1017/s0956792512000113.

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We give a new characterisation of integrability of a planar vector field at the origin. This allows us to prove that the analytic systemswhereh,K, Ψ and ξ are analytic functions defined in the neighbourhood ofOwithK(O) ≠ 0 or Ψ(O) ≠ 0 andn≥ 1, have a local analytic first integral at the origin. We show new families of analytically integrable systems that are held in the above class. In particular, this class includes all the nilpotent and generalised nilpotent integrable centres that we know.

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Speissegger, Patrick. "Quasianalytic Ilyashenko Algebras". Canadian Journal of Mathematics 70, n.º 1 (1 de fevereiro de 2018): 218–40. http://dx.doi.org/10.4153/cjm-2016-048-x.

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AbstractWe construct a quasianalytic field of germs at +∞ of real functions with logarithmic generalized power series as asymptotic expansions, such that is closed under differentiation and log-composition; in particular, is a Hardy field. Moreover, the field o (−log) of germs at 0+ contains all transition maps of hyperbolic saddles of planar real analytic vector fields.

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WU, YUHAI, YONGXI GAO e MAOAN HAN. "ON THE NUMBER AND DISTRIBUTIONS OF LIMIT CYCLES IN A QUINTIC PLANAR VECTOR FIELD". International Journal of Bifurcation and Chaos 18, n.º 07 (julho de 2008): 1939–55. http://dx.doi.org/10.1142/s0218127408021464.

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This paper is concerned with the number and distributions of limit cycles in a Z2-equivariant quintic planar vector field. By applying qualitative analysis method of differential equation, we find that 28 limit cycles with four different configurations appear in this special planar polynomial system. It is concluded that H(5) ≥ 28 = 52+ 3, where H(5) is the Hilbert number for quintic polynomial systems. The results obtained are useful to the study of the second part of 16th Hilbert problem.

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Sadovskaia, N., e R. Ramírez. "Inverse approach to study the planar polynomial vector field with algebraic solutions". Journal of Physics A: Mathematical and General 37, n.º 12 (9 de março de 2004): 3847–68. http://dx.doi.org/10.1088/0305-4470/37/12/009.

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YU, PEI, e MAOAN HAN. "CRITICAL PERIODS OF PLANAR REVERTIBLE VECTOR FIELD WITH THIRD-DEGREE POLYNOMIAL FUNCTIONS". International Journal of Bifurcation and Chaos 19, n.º 01 (janeiro de 2009): 419–33. http://dx.doi.org/10.1142/s0218127409022981.

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In this paper, we consider local critical periods of planar vector field. Particular attention is given to revertible systems with polynomial functions up to third degree. It is assumed that the origin of the system is a center. Symbolic and numerical computations are employed to show that the general cubic revertible systems can have six local critical periods, which is the maximal number of local critical periods that cubic revertible systems may have. This new result corrects that in the literature: general cubic revertible systems can at most have four local critical periods.

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Petrera, Matteo, Jennifer Smirin e Yuri B. Suris. "Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 475, n.º 2223 (março de 2019): 20180761. http://dx.doi.org/10.1098/rspa.2018.0761.

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Kahan discretization is applicable to any quadratic vector field and produces a birational map which approximates the shift along the phase flow. For a planar quadratic canonical Hamiltonian vector field, this map is known to be integrable and to preserve a pencil of cubic curves. Generically, the nine base points of this pencil include three points at infinity (corresponding to the asymptotic directions of cubic curves) and six finite points lying on a conic. We show that the Kahan discretization map can be represented in six different ways as a composition of two Manin involutions, corresponding to an infinite base point and to a finite base point. As a consequence, the finite base points can be ordered so that the resulting hexagon has three pairs of parallel sides which pass through the three base points at infinity. Moreover, this geometric condition on the base points turns out to be characteristic: if it is satisfied, then the cubic curves of the corresponding pencil are invariant under the Kahan discretization of a planar quadratic canonical Hamiltonian vector field.

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CARRILLO, FRANCISCO A., FERNANDO VERDUZCO e JOAQUÍN DELGADO. "ANALYSIS OF THE TAKENS–BOGDANOV BIFURCATION ON m-PARAMETERIZED VECTOR FIELDS". International Journal of Bifurcation and Chaos 20, n.º 04 (abril de 2010): 995–1005. http://dx.doi.org/10.1142/s0218127410026277.

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Given an m-parameterized family of n-dimensional vector fields, such that: (i) for some value of the parameters, the family has an equilibrium point, (ii) its linearization has a double zero eigenvalue and no other eigenvalue on the imaginary axis, sufficient conditions on the vector field are given such that the dynamics on the two-dimensional center manifold is locally topologically equivalent to the versal deformation of the planar Takens–Bogdanov bifurcation.

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Baider, Alberto, e Richard Churchill. "Uniqueness and non-uniqueness of normal forms for vector fields". Proceedings of the Royal Society of Edinburgh: Section A Mathematics 108, n.º 1-2 (1988): 27–33. http://dx.doi.org/10.1017/s0308210500026482.

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SynopsisThe use of normal forms in the study of equilibria of vector fields and Hamiltonian systems is a well-established practice and is described in standard references (e.g. [1], [7] or [10]). Also well known is the fact that such normal forms are not unique, and the relationship between distinct normal forms of the same vector field has also been investigated, in particular by M. Kummer [8] and A. Brjuno [2,3] (also see [12]). In this paper we use this relationship to extract invariants of the vector field directly from an arbitrary normal form. The treatment is sufficiently general to handle the vector field and Hamiltonian cases simultaneously, and applications in these contexts are presented.The formulation of our main result (Theorem 1.1) is reminiscent of, and was heavily influenced by, work of Shi Songling on planar vector fields [11]. Additional inspiration was provided by M. Kummer's contributions to the 1:1 resonance problem in [9]. The authors are grateful to Richard Cushman for comments on an earlier version of this paper.

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Streich, Rita, Jan van der Kruk e Alan G. Green. "Vector-migration of standard copolarized 3D GPR data". GEOPHYSICS 72, n.º 5 (setembro de 2007): J65—J75. http://dx.doi.org/10.1190/1.2766466.

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The derivation of ground-penetrating radar (GPR) images in which the amplitudes of reflections and diffractions are consistent with subsurface electromagnetic property contrasts requires migration algorithms that correctly account for the antenna radiation patterns and the vectorial character of electromagnetic wavefields. Most existing vector-migration techniques are based on far-field approximations of Green’s functions, which are inappropriate for the majority of GPR applications. We have recently developed a method for rapidly computing practically exact-field Green’s functions in a multicomponent vector migration scheme. A significant disadvantage of this scheme is the extra effort needed to record, process, and migrate at least two components of GPR data. By making straightforward modifications to the multi-component algorithm, we derive an equivalent exact-field sin-gle-component vector-migration scheme that can be applied to most standard GPR data acquired using a single copolarized antenna pair. Our new formulation is valid for polarization-independent features (e.g., most point scatterers, spheres, and planar objects), but not for polarization-dependent ones (e.g., most underground public utilities). A variety of tests demonstrates the stability of the exact-field single-component operators. Applications of the new scheme to synthetic and field-recorded GPR data containing dipping planar reflections produce images that are virtually identical to the corresponding multicomponent vector-migrated images and are almost invariant to the relative orientations of antennas and reflectors. The new single-component vector-migration scheme is appropriate for migrating the majority of GPR data acquired by researchers and practitioners.

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JOHNSON, TOMAS, e WARWICK TUCKER. "AN IMPROVED LOWER BOUND ON THE NUMBER OF LIMIT CYCLES BIFURCATING FROM A QUINTIC HAMILTONIAN PLANAR VECTOR FIELD UNDER QUINTIC PERTURBATION". International Journal of Bifurcation and Chaos 20, n.º 01 (janeiro de 2010): 63–70. http://dx.doi.org/10.1142/s0218127410025405.

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The limit cycle bifurcation of a Z2 equivariant quintic planar Hamiltonian vector field under Z2 equivariant quintic perturbation is studied. We prove that the given system can have at least 27 limit cycles. This is an improved lower bound on the possible number of limit cycles that can bifurcate from a quintic planar Hamiltonian system under quintic perturbation.

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Hirsch, Morris W. "Systems of differential equations that are competitive or cooperative. VI: A localCrClosing Lemma for 3-dimensional systems". Ergodic Theory and Dynamical Systems 11, n.º 3 (setembro de 1991): 443–54. http://dx.doi.org/10.1017/s014338570000626x.

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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.

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Guíñez, Víctor, Eduardo Sáez e Iván Szántó. "Limit Cycles Close to Infinity of Certain Non-Linear Differential Equations". Canadian Mathematical Bulletin 33, n.º 1 (1 de março de 1990): 55–59. http://dx.doi.org/10.4153/cmb-1990-009-8.

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Zhang, Xiao, Yun Gang Li, Hu Cheng e Heng Kun Liu. "Analysis of the Planar Magnetic Field of Linear Permanent Magnet Halbach Array". Applied Mechanics and Materials 66-68 (julho de 2011): 1336–41. http://dx.doi.org/10.4028/www.scientific.net/amm.66-68.1336.

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In this paper, the analytical solution to the planar magnetic field of a linear permanent magnet Halbach array is researched on. First, the magnetic field of a current surface is derived from the magnetic field produced by the molecular current of the permanent magnet. The magnetic fields of pieces of permanent magnet with vertical and diagonal magnetization direction are both solved by combining the magnetic field of current surfaces with different directions. Then the planar magnetic field of the entire linear permanent magnet Halbach array is obtained by calculating the vector addition of the magnetic field of all the permanent magnet cubes using Denavit-Hartenberg (D-H) transform technique, which yields the closed-form analytical expressions. The analytical algorithm proposed in this paper can be utilized to design and optimize the Halbach array, which in result can greatly simplify the calculation and expedite the progress. The effectiveness of the proposed method is evaluated by the finite element analysis software Maxwell.

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Bonckaert, Patrick. "Gevrey series in compensators linearizing a planar resonant vector field and its unfolding". Bulletin of the Belgian Mathematical Society - Simon Stevin 26, n.º 1 (março de 2019): 21–62. http://dx.doi.org/10.36045/bbms/1553047227.

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Wu, Yu Hai, Xue Di Wang e Li Xin Tian. "Bifurcations of limit cycles in a Z 4-equivariant quintic planar vector field". Acta Mathematica Sinica, English Series 26, n.º 4 (15 de fevereiro de 2010): 779–98. http://dx.doi.org/10.1007/s10114-010-6487-2.

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Wu, Yuhai, Yongxi Gao e Maoan Han. "Bifurcations of the limit cycles in a z3-equivariant quartic planar vector field". Chaos, Solitons & Fractals 38, n.º 4 (novembro de 2008): 1177–86. http://dx.doi.org/10.1016/j.chaos.2007.02.019.

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Şahan, Gökhan, e Vasfi Eldem. "Well posedness conditions for planar conewise linear systems". Transactions of the Institute of Measurement and Control 41, n.º 8 (11 de outubro de 2018): 2093–99. http://dx.doi.org/10.1177/0142331218780212.

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In this study, we give well-posedness conditions for planar conewise linear systems where the vector field is not necessarily continuous. It is further shown that, for a certain class of planar conewise linear systems, well posedness is independent of the conic partition of [Formula: see text]2. More specifically, the system is well posed for any conic partition of [Formula: see text]2.

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Tian, Yun, e Pei Yu. "Seven Limit Cycles Around a Focus Point in a Simple Three-Dimensional Quadratic Vector Field". International Journal of Bifurcation and Chaos 24, n.º 06 (junho de 2014): 1450083. http://dx.doi.org/10.1142/s0218127414500837.

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In this paper, we show that a simple three-dimensional quadratic vector field can have at least seven small-amplitude limit cycles, bifurcating from a Hopf critical point. This result is surprisingly higher than the Bautin's result for quadratic planar vector fields which can only have three small-amplitude limit cycles bifurcating from an elementary focus or an elementary center. The methods used in this paper include computing focus values, and solving multivariate polynomial systems using modular regular chains. In order to obtain higher-order focus values for nonplanar dynamical systems, computationally efficient approaches combined with center manifold computation must be adopted. A recently developed explicit, recursive formula and Maple program for computing the normal form and center manifold of general n-dimensional systems is applied to compute the focus values of the three-dimensional vector field.

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LIU, YIRONG, e JIBIN LI. "CENTER PROBLEM AND MULTIPLE HOPF BIFURCATION FOR THE Z5-EQUIVARIANT PLANAR POLYNOMIAL VECTOR FIELDS OF DEGREE 5". International Journal of Bifurcation and Chaos 19, n.º 06 (junho de 2009): 2115–21. http://dx.doi.org/10.1142/s0218127409023810.

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This paper proves that a Z5-equivariant planar polynomial vector field of degree 5 has at least five symmetric centers, if and only if it is a Hamltonian vector field. The characterization of a center problem is completely solved. The shortened expressions of the first four Lyapunov constants are given. Under small Z5-equivariant perturbations, the conclusion that the perturbed system has at least 25 limit cycles with the scheme 〈5 ∐ 5 ∐ 5 ∐ 5 ∐ 5〉 is rigorously proved.

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JOHNSON, TOMAS, e WARWICK TUCKER. "AN IMPROVED LOWER BOUND ON THE NUMBER OF LIMIT CYCLES BIFURCATING FROM A HAMILTONIAN PLANAR VECTOR FIELD OF DEGREE 7". International Journal of Bifurcation and Chaos 20, n.º 05 (maio de 2010): 1451–58. http://dx.doi.org/10.1142/s0218127410026599.

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The limit cycle bifurcations of a Z2 equivariant planar Hamiltonian vector field of degree 7 under Z2 equivariant degree 7 perturbation is studied. We prove that the given system can have at least 53 limit cycles. This is an improved lower bound for the weak formulation of Hilbert's 16th problem for degree 7, i.e. on the possible number of limit cycles that can bifurcate from a degree 7 planar Hamiltonian system under degree 7 perturbation.

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Ji, Qing, Jin Fang Cheng e Da Wei Xiao. "Research on near field acoustic source localization algorithm based on arbitrary planar vector array". Vibroengineering PROCEDIA 13 (26 de setembro de 2017): 227–32. http://dx.doi.org/10.21595/vp.2017.18983.

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Shi, Jian-ping, e Ji-bin Li. "Bifurcations of limit cycles in a Z6-equivariant planar vector field of degree 7". Applied Mathematics and Computation 244 (outubro de 2014): 191–200. http://dx.doi.org/10.1016/j.amc.2014.06.091.

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Zhou, Hongxian, e Yanmin Zhao. "Bifurcations of limit cycles in a -equivariant planar polynomial vector field of degree 7". Applied Mathematics and Computation 216, n.º 1 (março de 2010): 35–50. http://dx.doi.org/10.1016/j.amc.2009.12.055.

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Villarini, Massimo. "regularity properties of the period function near a center of a planar vector field". Nonlinear Analysis: Theory, Methods & Applications 19, n.º 8 (outubro de 1992): 787–803. http://dx.doi.org/10.1016/0362-546x(92)90222-z.

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Li, Jibin, e Mingqiang Zhang. "Bifurcations of Limit Cycles in a Z8-Equivariant Planar Vector Field of Degree 7". Journal of Dynamics and Differential Equations 16, n.º 4 (outubro de 2004): 1123–39. http://dx.doi.org/10.1007/s10884-004-7835-7.

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LIU, YIRONG, e JIBIN LI. "CENTER PROBLEM AND MULTIPLE HOPF BIFURCATION FOR THE Z6-EQUIVARIANT PLANAR POLYNOMIAL VECTOR, FIELDS OF DEGREE 5". International Journal of Bifurcation and Chaos 19, n.º 05 (maio de 2009): 1741–49. http://dx.doi.org/10.1142/s0218127409023950.

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This paper proves that a Z6-equivariant planar polynomial vector field of degree 5 has at least six symmetric centers, if and only if it is a Hamiltonian vector field. The characterization of a center problem is completely solved. The shortened expressions of the first four Lyapunov constants are given. Under small Z6-equivariant perturbations, the conclusion that the perturbed system has at least 24 limit cycles with the scheme 〈 4 ∐ 4 ∐ 4 ∐ 4 ∐ 4 ∐ 4〉 is rigorously proved. Two schemes of distributions of limit cycles are given.

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ZHAO, LIQIN, e XUEXING WANG. "THE SEPARATRIX VALUES OF A PLANAR HOMOCLINIC LOOP". International Journal of Bifurcation and Chaos 19, n.º 07 (julho de 2009): 2233–47. http://dx.doi.org/10.1142/s0218127409024037.

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It is well known that the stability of a homoclinic loop for planar vector fields is closely related to the cyclicity of this homoclinic loop. For a planar homoclinic loop consisting of a hyperbolic saddle, the loop values are crucial to the stability. The loop values are divided into two classes: saddle values and separatrix values. The saddle values are related to Dulac map near the saddle, and the separatrix values are related to the regular map near the homoclinic loop. The alternation of these quantities determines the stability of the homoclinic loop. So, it is important to investigate the separatrix values in both theory and for practical applications. For a given planar vector field, we can try to calculate the saddle values by means of dual Liapunov constants or by finding elementary invariants developed by Liu and Li [1990]. The first separatrix value was obtained by Dulac. The second separatrix value was given by Han and Zhu [2007] and by Hu and Feng [2001] independently. The third separatrix value was obtained by Luo and Li [2005] by means of Tkachev's method. In this paper, we shall establish the formulae for the third and fourth separatrix values. As applications, we will give an example with the homoclinic bifurcation of order 9 and prove that the cyclicity of homoclinic loop together with double homoclinic loops is 57.

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Wu, Yusen, Wentao Huang e Yongqiang Suo. "Weak Center and Bifurcation of Critical Periods in a Cubic Z2-Equivariant Hamiltonian Vector Field". International Journal of Bifurcation and Chaos 25, n.º 11 (outubro de 2015): 1550143. http://dx.doi.org/10.1142/s0218127415501436.

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This paper focuses on the problems of weak center and local bifurcation of critical periods for a class of cubic Z2-equivariant planar Hamiltonian vector fields. By computing the period constants carefully, one can see that there are three weak centers: (±1, 0) and the origin. The corresponding weak center conditions are also derived. Meanwhile, we address the problem of the coexistence of bifurcation of critical periods that occurred from (±1, 0) and the origin.

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