### Oscillatory Solutions in the Planar Restricted Three-Body Problem.

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We classify the phase portraits of the cubic systems in the plane such that they do not have finite critical points, and the critical points on the equator of the Poincaré sphere are isolated and have linear part non-identically zero.

Let X be a homogeneous polynomial vector field of degree 2 on S having finitely many invariant circles. Then, we prove that each invariant circle is a great circle of S, and at most there are two invariant circles. We characterize the global phase portrait of these vector fields. Moreover, we show that if X has at least an invariant circle then it does not have limit cycles.

Symmetric piecewise linear bi-dimensional systems are very common in control engineering. They constitute a class of non-differentiable vector fields for which classical Hopf bifurcation theorems are not applicable. For such systems, sufficient and necessary conditions for bifurcation of a limit cycle from the periodic orbit at infinity are given.

We consider the following topological spaces: $\mathbf{O}=\{z\in \u2102:|z+i|=1\}$, ${\mathbf{O}}_{3}=\mathbf{O}\cup \{z\in \u2102:{z}^{4}\in [0,1],Im\phantom{\rule{0.166667em}{0ex}}z\ge 0\}$, ${\mathbf{O}}_{4}=\mathbf{O}\cup \{z\in \u2102:{z}^{4}\in [0,1]\}$, ${\infty}_{1}=\mathbf{O}:|z-i|=1\}\cup \{z\in \u2102:z\in [0,1]\}$, ${\infty}_{2}={\infty}_{1}\cup \{z\in \u2102:{z}^{2}\in [0,1]\}$, et $\mathbf{T}=\{z\in \u2102:z=\mathrm{cos}(3\theta ){e}^{i\theta},\phantom{\rule{3.33333pt}{0ex}}0\le \theta \le 2\pi \}$. Set $E\in \{{\mathbf{O}}_{3},{\mathbf{O}}_{4},{\infty}_{1},{\infty}_{2},\mathbf{T}\}$. An $E$ map $f$ is a continuous self-map of $E$ having the branching point fixed. We denote by $Per\left(f\right)$ the set of periods of all periodic points of $f$. The set $K\subset \mathbb{N}$ is the of $E$ if it satisfies the following two conditions: (1) If $f$ is an $E$ map and $K\subset Per\left(f\right)$, then $Per\left(f\right)=\mathbb{N}$. (2) If $S\subset \mathbb{N}$ is a set such that for every $E$ map $f$, $S\subset Per\left(f\right)$ implies $Per\left(f\right)=\mathbb{N}$, then $K\subset S$. In this paper we compute the full periodicity kernel of ${\mathbf{O}}_{3},{\mathbf{O}}_{4},{\infty}_{1},{\infty}_{2}$ and $\mathbf{T}$.

Let f be a continuous map on a compact connected Riemannian manifold M. There are several ways to measure the dynamical complexity of f and we discuss some of them. This survey contains some results and open questions about relationships between the topological entropy of f, the volume growth of f, the rate of growth of periodic points of f, some invariants related to exterior powers of the derivative of f, and several invariants measuring the topological complexity of f: the degree (for the case...

The aim of this paper is to describe the set of periods of a Morse-Smale diffeomorphism of the two-dimensional sphere according to its homotopy class. The main tool for proving this is the Lefschetz fixed point theory.

We study phase portraits of quadratic systems with a unique finite singularity. We prove that there are 111 different phase portraits without limit cycles and that 13 of them are realizable with exactly one limit cycle. In order to finish completely our study two problems remain open: the realization of one topologically possible phase portrait, and to determine the exact number of limit cycles for a subclass of these systems.

We study phase portraits of quadratic vector fields with a weak focus of third order at the origin. We show numerically the existence of at least 20 different global phase portraits for such vector fields coming from exactly 16 different local phase portraits available for these vector fields. Among these 20 phase portraits, 17 have no limit cycles and three have at least one limit cycle.

Let $X$ be a connected closed manifold and $f$ a self-map on $X$. We say that $f$ is almost quasi-unipotent if every eigenvalue $\lambda $ of the map ${f}_{*k}$ (the induced map on the $k$-th homology group of $X$) which is neither a root of unity, nor a zero, satisfies that the sum of the multiplicities of $\lambda $ as eigenvalue of all the maps ${f}_{*k}$ with $k$ odd is equal to the sum of the multiplicities of $\lambda $ as eigenvalue of all the maps ${f}_{*k}$ with $k$ even. We prove that if $f$ is ${C}^{1}$ having finitely many periodic points all of them hyperbolic,...

In this paper we provide the greatest lower bound about the number of (non-infinitesimal) limit cycles surrounding a unique singular point for a planar polynomial differential system of arbitrary degree.

We classify the braid types that can occur for finite unions of periodic orbits of diffeomorphisms of surfaces of genus one with zero topological entropy.

We prove that the lowest upper bound for the number of isolated zeros of the Abelian integrals associated to quadratic Hamiltonian vector fields having a center and an invariant straight line after quadratic perturbations is one.

Given the plane triangle with vertices (0,0), (0,4) and (4,0) and the transformation F: (x,y) ↦ (x(4-x-y),xy) introduced by A. N. Sharkovskiĭ, we prove the existence of the following objects: a unique invariant curve of spiral type, a periodic trajectory of period 4 (given explicitly) and a periodic trajectory of period 5 (described approximately). Also, we give a decomposition of the triangle which helps to understand the global dynamics of this discrete system which is linked with the behavior...

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