In this dissertation, firstly, we show that a polynomial difeerential system of even
order does not have a global center. Next, we characterize all Liénard polynomial systems
having a global center at the origin. In particular, we provide an explicit expression of all
Liénard polynomial systems of degree three with a global center at the origin. Finally, we
classify all degree three and degree five Kukles systems with a global center at the origin.

In this work, we present a study of the abstract theory of multivalued monotone dynamical systems on a complete metric space and the application of this theory to a differential equation without uniqueness of solution.

The problem of secant defectivity consists in studying the dimension of the h-secant variety of an algebraic variety. We say that an algebraic variety is h-defective when its h-secant variety does not have the expected dimension. In this work we will study the secant defectivity of the Segre-Veronese varieties of the products of the projective line.

In this work, we performed a qualitative study of a model consisting of three ordinary differential equations that describe the interaction between leukemic stem cells and immune cells, where the immune functional response against leukemia exhibits an optimal activation window. We investigated the stability of the equilibrium points with respect to the system parameters and the existence of bifurcations. We rigorously demonstrate that the model exhibits at least two types of bifurcations. The first is the transcritical bifurcation around the tumor-free equilibrium point. The second is the Hopf bifurcation around a biologically plausible equilibrium point. We focused our attention on the latter, examining the emergence of limit cycles and analyzing their stability through the sign of the Lyapunov coefficient. We verified the theoretical results through numerical simulations using the Mathematica software.

In this paper we will give a detailed proof of the Variational Principle which states that
the topological entropy of a continuous application defined in a compact metric space
is equal to the supremum of the entropies of invariant measures. We will also establish
topological, metric and conditional entropies in a more explicit way. All of was done with
reference to [11]. We will also show an example of the Variational Principle which shows
that the entropy of the set of non-errant points is the same as the entropy of the set of
non-errant points is the same as the entropy of space, that is, the entropy of a set is loaded
on the set of wandering points.

This work proves the local and global existence of solutions for semi-diffusive partial inclusion systems with m-accretive operators seen in Compactness Methods for Nonlinear Evolutions [7] Diaz and Vrabie. Based on this article, we prove the local existence of solutions for semi-diffusive partial differential inclusion systems with monotonous maximal operator of the form $div(|\nabla u|^{p(.)-2}\nabla u)$ with external forces F and G multivalued maps, F is upper semicontinuous, the pair (F,G) is positively sublinear and G with separable variables.

This work studies the existence of steady-state solutions for the reaction-diffusion problem
below
⎧⎪⎨
⎪⎩
𝑢_𝑡 = 𝜖²𝑢_𝑥𝑥 − 𝑢(𝑢² − 𝛼²(𝑥)), (𝑥, 𝑡) ∈ (0, 1) × (0,∞)
𝑢_𝑥(0, 𝑡) = 𝑢_𝑥(1, 𝑡) = 0, 𝑡 ∈ (0,∞),
where 𝛼 is a function of class 𝐶_2. We use the sub and super-solutions technique to guarantee
the existence of stable steady-state solutions that develop transition layers near to
the points where 𝛼(𝑥) has a local minimum.

This work has its initial motivation in the article \textbf{``On the motion under focal attraction in a rotating medium''}, by J. Sotomayor \cite{soto}, which models the following problem of planar differential equations present, for example, in biology: Inside a shallow circular container, we put a liquid and several species of flatworms that swim with different speeds in the direction of a luminous point fixed on the edge of this container. The objective is to subject the liquid to a constant rotation in order to isolate each of the different species present in the experiment. After this initial study, two modifications were made in the model of differential equations, adding a radial drift to the system. Through these modifications, we studied the bifurcation diagram of each of these systems, which involved bifurcations of Bogdanov-Takens, Hopf and Saddle-node types.

The problem of secant defectivity consists in studying the dimension of the h-secant
variety of an algebraic variety. We say that an algebraic variety is h-defective when its
h-secant variety does not have the expected dimension. In this work we will study the
secant defectivity of the Segre-Veronese varieties using the osculating spaces to determine
the defect of the h-secant varieties.

The approach analyzed the density of continuous functions in the space of integrable Riemann function and the space of square integrable Riemann function. The analysis shows that the space of continuous compact support functions is dense in the space of the integrable Lebesgue functions, considering the fixed exponent greater than or equal to zero and finite, the domain of functions a Hausdorff space locally compact and the measure defined as in the representation theorem of Riesz . We explore the density of space of continuous compact support functions in the generalized Lebesgue spaces, with variable exponent being a measurable and essentially limited function. Considering Sobolev spaces with variable exponent, we discuss conditions about the exponent that guarantee the density of continuous functions space. One result merges a monotonicity condition and a continuous log-Hölder condition. Another result discusses the density using two corollaries, exponent depends only on the nth coordinate of each point and another where the exponent depends only on the distance from the point to the origin.

In this text we study the Einstein equations in the spatially flat spacetimes (Bianchi-I)which are endowed with an extra locally rotational symmetry (LRS). We develop a newlocal coordinate representation where we use the components of the energy-momentumtensor directly in the metric, the energy densityρbeing the "time" coordinate. As anapplication, some general classes of exact solutions are obtained which are of physical andmathematical interest. In particular, the general barotropic perfect fluid solution is given, p = p(ρ).

We study the relationship that exists between complex dynamics and
symbolic dynamics. Specifically, we describe a geometric method, with which, based on the complex polynomial dynamics of degree d, we can find a minimal set of shift automorphisms that generates the group Aut_d.

Given an asymptotically autonomous system, we present a theorem that shows the convergence of the pullback attractor to the global attractor if and only if the pullback attractor is forward compact. Other results with different sufficient conditions also highlight this convergence. Results with necessary conditions are also presented. Moreover, we define the limit-set and the lower limit-set of pullback attractor and we present results that show the relationship between these and the global attractor. Finally, we apply the results obtained to a quasi-linear parabolic equation with variable exponent in which the principal operator depends on time.

The Strong Real Jacobian Conjecture, related to the Jacobian Conjecture, states that a
planar polynomial map with non vanishing Jacobian is injective. In a celebrated construction, Pinchuk provided a counterexample to this conjecture which has guided much
research in this area. The polynomial function F = (p, q), obtained by Pinchuk, consists
of a pair of polynomial functions, p of degree 10 and q of degree 40. Campbell showed
that the degree of q can be reduced to 25. In this work, we present the global phase portraits of the Hamiltonian polynomial vector fields Hp and Hq associated to the Pinchuk
polynomial map.

The main objective of this work is to present different versions of the Poincaré-Bendixson Theorem. We will present the well-known classical version for smooth vector fields in the plane, a version for continuous vector fields in the plane, a version for piecewise smooth vector fields in the plane, and a version for continuous vector fields on the Klein bottle.

In this paper we present a method for constructing developable surfaces containing a given curve as a curvature line and then we study the total torsion of closed curvature lines of surfaces in R3 and construct a smooth local surface containing a given closed curve with total torsion equal to an integer multiple of 2 as the hyperbolic main cycle. We conclude the work with a study on the behavior of curvature lines near an umbilical point of a parameterized surface.

In this work, we study the definite positivity of functions, whose domains are Cartecian spacetime