We investigate homeomorphisms of a compact interval, applied randomly. We consider this system as a skew product with the two-sided Bernoulli shift in the base. If on the open interval there is a metric in which almost all maps are contractions, then (with mild additional assumptions) there exists a global pullback attractor, which is a graph of a function from the base to the fiber. It is also a forward attractor. However, the value of this function depends only on the past, so when we take the one-sided shift in the base, it disappears. We illustrate those phenomena on an example, where there are two piecewise linear homeomorphisms, one moving points to the right and the other one to the left.

We investigate the local integrability in $\mathbb{C}^3$ of some three-dimensional Lotka-Volterra equations at the origin with $(p:q:r)$-resonance,

dot x &= P = x(p +ax+by+cz),

dot y &= Q = y(q + dx +ey + fz),

dot z &= R = z(r + gx + hy +kz).

Recent work on this problem has centered on the case where the resonance is of "rank\guio{$2$}". That is, there are two independent linear dependencies of $p$, $q$ and $r$ over~$\mathbb{Q}$. Here, we consider some situations where there is only one such dependency. In particular, we give necessary and sufficient conditions for integrability for the case of $(i,-i,\lambda)$-resonance with $\lambda\notin i\mathbb{R}$ (after a scaling, this is just the case $p+q=0$ with $q/r \notin \mathbb{R}$), and also the case of $(i-1,-i-1,2)$-resonance (a subcase of $p+q+r=0$) under the additional assumption that $a=k=0$.

Our necessary and sufficient conditions for integrability are given via the search for two independent first integrals of the form $x^\alpha y^\beta z^\gamma (1+O(x,y,z))$. However, a new feature in the case of rank-$1$ resonance is that there is a distinguished choice of analytic first integral, and hence it makes sense to seek conditions for just one (analytic) first integral to exist. We give necessary and sufficient conditions for just one first integral for the two families of systems mentioned above, except that for the second family some of the cases of sufficiency have been left as conjectural.

It follows from Birkhoff's Ergodic Theorem that the irregular set of points for which the Birkhoff averages of a given continuous function diverge has zero measure with respect to any finite invariant measure. In strong contrast, for systems with the weak specification property, we show here that if the irregular set is nonempty, then it is residual. This includes topologically transitive topological Markov chains, sofic shifts and more generally shifts with the specification property. We consider also the more general case of ratios of Birkhoff averages of continuous functions and the case when the set of accumulation points of the ratios of Birkhoff averages is a prescribed closed interval. Finally, we give an application of our work to the pointwise dimension of a Gibbs measure on a repeller of a conformal map.

Let $F=(F_1,F_2,F_3)\colon\mathbb{R}^3\to\mathbb{R}^3$ be a $C^\infty$ local diffeomorphism. We prove that each of the following conditions are sufficient to the global injectivity of $F$:

A) The foliations $\mathcal{F}_{F_i}$ made up by the connected components of the level surfaces $F_i=\text{constant}$, consist of leaves without half-Reeb components induced by $F_j$, $j\in\{1,2,3\}\setminus\{i\}$, for $i\in\{1,2,3\}$.

B) For each $i\neq j\in\{1,2,3\}$, $F_i\rvert_L\colon L\to\mathbb{R}$ satisfy the Palais--Smale condition, for all $L\in\mathcal{F}_{F_j}$.

We also prove that B) implies A) and give examples to show that the converse is not true. Further, we give examples showing that none of these conditions is necessary to the global injectivity of F.

In this paper we study the topology of the hyperbolic component of the parameter plane for the Newton's method applied to $n$-degree Bring--Jerrard polynomials given by $P_n(z) = z^n-cz +1$, $c\in\mathbb{C}$. For $n=5$, using the Tschirnhaus--Bring--Jerrard nonlinear transformations, this family controls, at least theoretically, the roots of all quintic polynomials. We also study a bifurcation cascade of the bifurcation locus by considering $c\in \mathbb{R}$.

In this article we deal with singularly perturbed Filippov systems $Z_{\varepsilon}$:

(1) \dot{x} = \begin{cases} F(x,y,\varepsilon) &\text{if } h(x,y,\varepsilon) \leq 0,

G(x,y,\varepsilon) &\text{if } h(x,y,\varepsilon) \geq 0,

\end{cases} \quad \varepsilon \dot{y} = H(x,y,\varepsilon)

where $\varepsilon \in \mathbb{R}$ is a small parameter, $x \in \mathbb{R}^n$, $n\geq2,$ and $y \in \mathbb{R}$ denote the slow and fast variables, respectively, and $F$, $G$, $h$, and $H$ are smooth maps. We study the effect of singular perturbations at typical singularities of $Z_0$. Special attention will be dedicated to those points satisfying $q \in \{h(x,y,0) = 0\} \cap \{H(x,y,0) = 0\}$ where $F$ or $G$ is tangent to $\{h(x,y,0) = 0\}$. The persistence and the stability properties of those objects are investigated.

Let $X$ be a germ of holomorphic vector field at $0\in\mathbb{C}^n$ and let $E$ be a linear subspace of $\mathbb C^n$ which is invariant for the linear part of $X$ at $0$. We give a sufficient condition that imply the existence of a non-singular invariant manifold tangent to $E$ at $0$. It generalizes to higher dimensions the conditions in the classical Briot--Bouquet's Theorem: roughly speaking, we impose that the convex hull of the eigenvalues $\mu_i$ corresponding to $E$ does not contain $0$ and there are no resonances between the $\mu_i$ and the complementary eigenvalues. As an application, we propose an elementary proof of the analyticity of the local stable and unstable manifolds of a real analytic vector field at a singular point.

Epidemiological models with bilinear incidence rate usually have an asymptotically stable trivial equilibrium corresponding to the disease-free state, or an asymptotically stable nontrivial equilibrium (i.e. interior equilibrium) corresponding to the endemic state. In this paper, we consider an epidemiological model, which is a SIRS (susceptible-infected-removed-susceptible) model in uenced by random perturbations. We prove that the solutions of the system are positive for all positive initial conditions and that the solutions are global, that is, there is no finite explosion time. We present necessary and sufficient condition for the almost sure asymptotic stability of the steady state of the stochastic system.

We prove a global asymptotic stability result for maps coming from n-th order dfference equation and satisfying a Markus-Yamabe type condition. We also show that this result is sharp.

We consider an outer billiard around a Reulaux triangle. We prove the existence of infinitely many periodic points accumulating at infinity. To do so we con- struct a return map from a strip into itself and we study its properties. We also show some numerical simulations which, in particular, display heteroclinic intersections and Smale's horseshoes.

We prove that the limit cycle oscillations of the celebrated Rosenzweig-MacArthur differential system and other predator-prey models are non-algebraic.

This is a survey relating several subjects bridging algebraic differential equations and their integrability, pertubative approach based on algebraic moments, the Jacobian conjecture and optimal transport of measures defined on the complement of an arrangement of hyperplanes. This text was written in relation with Jaume Llibre's 60th birthday and it touches several fields in which Jaume gave important contributions. It also contains some results wich have not yet been published before.

The existence and stability of limit cycles in discontinuous piecewise linear systems obtained by the aggregation of two linear systems of focus type and having only one equilibrium point is considered. By using an adequate canonical form with five parameters, a thorough study of some Poincarç maps is performed. Different bifurcations which are responsible for the appearance of crossing limit cycles are detected and parameter regions with none, one, two and three crossing limit cycles are found.

In particular, a first analytical proof of the existence, for certain differential systems in the considered family, of at least three homotopic crossing limit cycles surrounding the equilibrium point, is included. This fact has recently been numerically discovered in a particular example by S.-M. Huan and X.-S. Yang.

In this paper we find necessary and sufficient conditions in order that a planar homogeneous polynomial differential system has a polynomial or rational first integral. We apply these conditions to linear and quadratic homogeneous polynomial differential systems.

Complex limit cycle located in a neighborhood of a hyperbolic polycycle can not vanish under a small deformation that preserves the characteristic values of the vertexes of the polycycle. The cycles either change holomorphically under the change of the parameter, or come to the boundary of the fixed neighborhood of the polycycle. The present paper makes these statements rigorous and proves them.

First, doing a combination of analytical and algebraic computations, we determine by first time an explicit normal form depending only on three parameters for all cubic homogeneous polynomial differential systems having a center.

After using the averaging method of first order we show that we can obtain at most one limit cycle bifurcating from the periodic orbits of the mentioned centers when they are perturbed inside the class of all cubic polynomial differential systems. Moreover, there are examples with one limit cycles.

We discuss the global dynamics of some logistic models governed by delay-differential equations. We focus on models of exploited populations, and study the changes in the dynamics as the harvesting effort is increased. We get new results and highlight the link among different logistic equations usually employed in population models.

We classify the links of basic periodic orbits of integrable vector fields on~$\mathbb{S}^3$ generalizing results on two degree of freedom Hamiltonian systems. We also study the case of completely integrable systems and define invariants for the two classes of vector fields.

Theorems on the existence of invariant KAM tori are established for perturbations of Hamiltonian systems which are circle bundle flows. By averaging the perturbation over the bundle flow one obtains a Hamiltonian system on the orbit (quotient) space by a classical theorem of Reeb. A non-degenerate critical point of the system on the orbit space gives rise to a family of periodic solutions of the perturbed system. Conditions on the critical points are given which insure KAM tori for the perturbed flow.

These general theorems are used to show that the near circular periodic solutions of the planar lunar problem are orbitally stable and are surrounded by KAM 2-tori.

For the spatial case it is shown that there are periodic solutions of two types, either near circular equatorial, that is, the infinitesimal particle moves close to the plane of the primaries following near circular trajectories and the other family where the infinitesimal particle moves along the axis perpendicular to the plane of the primaries following near rectilinear trajectories. We prove that the two solutions are elliptic and are surrounded by invariant 3-tori applying a recent theorem of Han, Li, and Yi.

In the spatial case a second averaging is performed, and the corresponding or bit space (called the twice-reduced space) is constructed. The flow of the averaged Hamiltonian on it is given and several families of invariant 3-tori are determined using Han, Li, and Yi Theorem.

We provide sufficient conditions for the existence of limit cycles of non-smooth perturbed planar centers when the discontinuity set is a union of regular curves. We introduce a mechanism which allows us to deal with such systems. The main tool used in this paper is the averaging method. Some applications are explained with careful details.

For stably dissipative Lotka-Volterra equations the dynamics on the attractor are Hamiltonian and we argue that complex dynamics can occur. We also present examples and properties of some infinite dimensional Volterra systems with applications related with stably dissipative Lotka-Volterra equations. We finish by mentioning recent contributions on the subject.

Starting from a class of isochronous Hamiltonian centers, we produce a new class of Hamiltonian isochronous centers by using non-canonical transformations.

We describe the origin and evolution of ideas on topological and polynomial invariants and their interaction, in problems of classification of polynomial vector fields. The concept of moduli space is discussed in the last section and we indicate its value in understanding the dynamics of families of such systems. Our interest here is in the concepts and the way they interact in the process of topologically classifying polynomial vector fields. We survey the literature giving an ample list of references and we illustrate the ideas on the testing ground of families of quadratic vector fields. In particular, the role of polynomial invariants is illustrated in the proof of our theorem in the section next to last. These concepts have proven their worth in a number of classification results, among them the most recent work on the geometric classification of the whole class of quadratic vector fields, according to their configurations of infinite singularities. An analog work including both finite and infinite singularities of the whole quadratic class, joint work with J. C. Artés, J. Llibre, and N. Vulpe, is in progress.

We find a uniform method to establish Poincare type linearization theorems for regular systems including classical autonomous, random and almost periodic ones via modified majorant norm methods.

We consider families of analytic functions with Taylor coefficients\guio{polynomials} in the parameter $\lambda$: $f_\lambda(z)=\sum_{k=0}^\infty a_k(\lambda) z^k$, $a_k \in {\mathbb C}[\lambda]$. Let $R(\lambda)$ be the radius of convergence of $f_\lambda$. The "Taylor domination'' property for this family is the inequality of the following form: for certain fixed~$N$ and $C$ and for each $k\geq N+1$ and $\lambda,

$|a_{k}(\lambda)|R^{k}(\lambda)\leq C \max_{i=0,\dotsc,N} |a_{i}(\lambda)|R^{i}(\lambda).$

Taylor domination property implies a uniform in $\lambda$ bound on the number of zeroes of~$f_\lambda$. In this paper we discuss some known and new results providing Taylor domination (usually, in a smaller disk) via the Bautin approach. In particular, we give new conditions on $f_\lambda$ which imply Taylor domination in the full disk of convergence. We discuss Taylor domination property also for the generating functions of the Poincar\'e type linear recurrence relations.