Non-bifurcation of critical periods from semi-hyperbolic polycycles of quadratic centres

2021 ◽  
pp. 1-11
Author(s):  
D. Marín ◽  
M. Saavedra ◽  
J. Villadelprat
Keyword(s):  
Open Subset ◽  
Critical Periods ◽  
Prove Theorem ◽  
Saddle Node ◽  
Bifurcation Of Critical Periods ◽  
Image Position ◽  
Compact Subsets

In this paper we consider the unfolding of saddle-node \[ X= \frac{1}{xU_a(x,y)}\Big(x(x^{\mu}-\varepsilon)\partial_x-V_a(x)y\partial_y\Big), \] parametrized by $(\varepsilon,\,a)$ with $\varepsilon \approx 0$ and $a$ in an open subset $A$ of $ {\mathbb {R}}^{\alpha },$ and we study the Dulac time $\mathcal {T}(s;\varepsilon,\,a)$ of one of its hyperbolic sectors. We prove (theorem 1.1) that the derivative $\partial _s\mathcal {T}(s;\varepsilon,\,a)$ tends to $-\infty$ as $(s,\,\varepsilon )\to (0^{+},\,0)$ uniformly on compact subsets of $A.$ This result is addressed to study the bifurcation of critical periods in the Loud's family of quadratic centres. In this regard we show (theorem 1.2) that no bifurcation occurs from certain semi-hyperbolic polycycles.

scholarly journals On uniformly distributed orbits of certain horocycle flows

1982 ◽  
Vol 2 (2) ◽  
pp. 139-158 ◽  
Author(s):  
S. G. Dani
Keyword(s):  
Probability Measure ◽  
Periodic Point ◽  
Open Subset ◽  
Invariant Probability Measure ◽  
Zero Measure ◽  
Lebesque Measure ◽  
Image Position

AbstractLet(where t ε ℝ) and let μ be the G-invariant probability measure on G/Γ. We show that if x is a non-periodic point of the flow given by the (ut)-action on G/Γ then the (ut)-orbit of x is uniformly distributed with respect to μ; that is, if Ω is an open subset whose boundary has zero measure, and l is the Lebesque measure on ℝ then, as T→∞, converges to μ(Ω).

scholarly journals TOPOLOGICAL CELL DECOMPOSITION AND DIMENSION THEORY IN P-MINIMAL FIELDS

2017 ◽  
Vol 82 (1) ◽  
pp. 347-358 ◽  
Author(s):  
PABLO CUBIDES KOVACSICS ◽  
LUCK DARNIÈRE ◽  
EVA LEENKNEGT
Keyword(s):  
Algebraic Geometry ◽  
Open Subset ◽  
Cell Decomposition ◽  
Real Algebraic Geometry ◽  
Dimension Theory ◽  
Definable Set ◽  
Image Position ◽  
Definable Function

AbstractThis paper addresses some questions about dimension theory for P-minimal structures. We show that, for any definable set A, the dimension of $\bar A\backslash A$ is strictly smaller than the dimension of A itself, and that A has a decomposition into definable, pure-dimensional components. This is then used to show that the intersection of finitely many definable dense subsets of A is still dense in A. As an application, we obtain that any definable function $f:D \subseteq {K^m} \to {K^n}$ is continuous on a dense, relatively open subset of its domain D, thereby answering a question that was originally posed by Haskell and Macpherson.In order to obtain these results, we show that P-minimal structures admit a type of cell decomposition, using a topological notion of cells inspired by real algebraic geometry.

scholarly journals Bifurcation of critical periods of polynomial systems

2015 ◽  
Vol 259 (8) ◽  
pp. 3825-3853 ◽  
Author(s):  
Brigita Ferčec ◽  
Viktor Levandovskyy ◽  
Valery G. Romanovski ◽  
Douglas S. Shafer
Keyword(s):  
Polynomial Systems ◽  
Critical Periods ◽  
Bifurcation Of Critical Periods

An extension of the Freiheitssatz

1981 ◽  
Vol 89 (1) ◽  
pp. 35-41 ◽  
Author(s):  
B. Baumslag ◽  
S. J. Pride
Keyword(s):  
Free Product ◽  
Natural Homomorphism ◽  
Prove Theorem ◽  
Image Position

Let I be a set and let H(i) (i ∈ I) be non-trivial groups. If J is a subset of I, we denote the free product of the H(j) (j∈J) by H(J). We denote H(I) simply by H. Let R be a cyclically reduced element of Hof length at least two, and letLet μ: H → G be the natural homomorphism. If J is a subset of I such that R ∉ H(J), we call H(J) a Magnus subgroup, or occasionally the J-Magnus subgroup (of H with respect to R). We will say that the Freiheitssatz holds if μ| M is an injection for each Magnus subgroup M. Magnus (4) showed that the Freiheitssatz holds if the H(i) are free, and this was extended by Pride (5) to the case where the H(i) are locally fully residually free. In this paper we prove Theorem 1. The Freiheitssatz holds if the H(i) are locally residually free.

Bifurcations of Critical Periods for a Class of Quintic Liénard Equation

2020 ◽  
Vol 30 (14) ◽  
pp. 2050201
Author(s):  
Zhiheng Yu ◽  
Lingling Liu
Keyword(s):  
Phase Portraits ◽  
Liénard Equation ◽  
Local Bifurcation ◽  
Exact Order ◽  
Critical Periods ◽  
Isochronous Centers ◽  
Bifurcation Of Critical Periods ◽  
Lienard Equation

In this paper, we investigate a quintic Liénard equation which has a center at the origin. We give the conditions for the parameters for the isochronous centers and weak centers of exact order. Then, we present the global phase portraits for the system having isochronous centers. Moreover, we prove that at most four critical periods can bifurcate and show with appropriate perturbations that local bifurcation of critical periods occur from the centers.

scholarly journals A THEORY OF PAIRS FOR NON-VALUATIONAL STRUCTURES

2019 ◽  
Vol 84 (02) ◽  
pp. 664-683
Author(s):  
ELITZUR BAR-YEHUDA ◽  
ASSAF HASSON ◽  
YA’ACOV PETERZIL
Keyword(s):  
Open Subset ◽  
Minimal Structure ◽  
Image Position

AbstractGiven a weakly o-minimal structure${\cal M}$and its o-minimal completion$\bar{{\cal M}}$, we first associate to$\bar{{\cal M}}$a canonical language and then prove thatTh$\left( {\cal M} \right)$determines$Th\left( {\bar{{\cal M}}} \right)$. We then investigate the theory of the pair$\left( {\bar{{\cal M}},{\cal M}} \right)$in the spirit of the theory of dense pairs of o-minimal structures, and prove, among other results, that it is near model complete, and every definable open subset of${\bar{M}^n}$is already definable in$\bar{{\cal M}}$.We give an example of a weakly o-minimal structure interpreting$\bar{{\cal M}}$and show that it is not elementarily equivalent to any reduct of an o-minimal trace.

Unique continuation properties of the nonlinear Schrödinger equation

1997 ◽  
Vol 127 (1) ◽  
pp. 191-205 ◽  
Author(s):  
Bing-Yu Zhang
Keyword(s):  
Open Subset ◽  
Function Theory ◽  
Unique Continuation ◽  
Inverse Scattering Transform ◽  
Nls Equation ◽  
Nonlinear Schrödinger ◽  
Scattering Transform ◽  
Hardy Function ◽  
Continuation Problem ◽  
Image Position

Consider the unique continuation problem for the nonlinear Schrödinger (NLS) equationBy using the inverse scattering transform and some results from the Hardy function theory, we prove that if u ∈ C(R; H1(R)) is a solution of the NLS equation, then it cannot have compact support at two different moments unless it vanishes identically. In addition, it is shown under certain conditions that if u is a solution of the NLS equation, then u vanishes identically if it vanishes on two horizontal half lines in the x–t space. This implies that the solution u must vanish everywhere if it vanishes in an open subset in the x–t space.

The Dirichlet Problem for the Subelliptic Laplacian on the Heisenberg Group II

1985 ◽  
Vol 37 (4) ◽  
pp. 760-766 ◽  
Author(s):  
Bernard Gaveau ◽  
Jacques Vauthier
Keyword(s):  
Dirichlet Problem ◽  
Continuous Function ◽  
Heisenberg Group ◽  
Open Subset ◽  
Martingale Problem ◽  
Three Dimensions ◽  
True Solution ◽  
Group Ii ◽  
Image Position

Let H3 be the Heisenberg group in three dimensions, Δ the fundamental subelliptic laplacian on H3 (see Section 1 for notations and definitions) and U be an open subset of H3 If φ is a continuous function on the boundary ∂U of U, the Dirichlet problem is thus,(1)In [3], p. 104, it was asserted by the first author that, when dU is regular (see Section 1 for this definition), the problem (1) has a solution continuous on D and a probabilistic formula was given. In [3], we prove that our probabilistic formula gives a solution of the so called “martingale problem” associated to Δ on U (see [5] for this notion). But it appears that the connection between the solution in the martingale problem sense and the true solution is not at all clear in the subelliptic case; in particular it is not obvious at all that the probabilistic formula is a C2 function.

Gδ Sets IN σ-IDEALS GENERATED BY COMPACT SETS

2019 ◽  
Vol 84 (02) ◽  
pp. 781-797
Author(s):  
MAYA SARAN
Keyword(s):  
Compact Subset ◽  
Polish Space ◽  
Compact Sets ◽  
Natural Class ◽  
Image Position ◽  
Compact Subsets

AbstractGiven a compact Polish space E and the hyperspace of its compact subsets ${\cal K}\left( E \right)$, we consider Gδ σ-ideals of compact subsets of E. Solecki has shown that any σ-ideal in a broad natural class of Gδ ideals can be represented via a compact subset of ${\cal K}\left( E \right)$; in this article we examine the behaviour of Gδ subsets of E with respect to the representing set. Given an ideal I in this class, we construct a representing set that recognises a compact subset of E as being “small” precisely when it is in I, and recognises a Gδ subset of E as being “small” precisely when it is covered by countably many compact sets from I.

Distance Functions and Orlicz-Sobolev Spaces

1986 ◽  
Vol 38 (5) ◽  
pp. 1181-1198 ◽  
Author(s):  
D. E. Edmunds ◽  
R. M. Edmunds
Keyword(s):  
Sobolev Space ◽  
Distance Function ◽  
Sobolev Spaces ◽  
Open Subset ◽  
Distance Functions ◽  
The Mean ◽  
Image Position

Let ∧ be a bounded, non-empty, open subset of Rn and given any x in Rn, letlet k ∊ N and suppose that p ∞ (1, ∞). It is known (c.f. e.g. [4]) that if u belongs to the Sobolev space WKp(∧) and u/dk ∊ Lp(∧), then . Further results in this direction are given in [5] and [9]. Moreover, if m is the mean distance function in the sense of [2], then it turns out thatUnder appropriate smoothness conditions on the boundary of ∧, m and d are equivalent, and thus may in this case be characterized as the subspace of W1,2(∧) consisting of all functions u ∊ W1,2(∧) such that u/d ∊ L2(∧). Further results in this direction are given in [5] and [9]. Moreover, if m is the mean distance function in the sense of [2], then it turns out that

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