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### Open Access Journals

DCDS

This paper studies questions regarding the local and global asymptotic stability of analytic
autonomous ordinary differential equations in $\mathbb{R}^n$. It is well-known that such stability
can be characterized in terms of Liapunov functions. The authors prove similar results for the
more geometrically motivated Dulac functions. In particular it holds that any analytic autonomous
ordinary differential equation having a critical point which is a global attractor admits a Dulac
function. These results can be used to give criteria of global attraction in two-dimensional
systems.

DCDS

We give an effective method for controlling the maximum number of
limit cycles of some planar polynomial systems. It is based on a
suitable choice of a Dulac function and the application of the
well-known Bendixson-Dulac Criterion for multiple connected regions.
The key point is a new approach to control the sign of the functions
involved in the criterion.
The method is applied to several examples.

CPAA

For any positive decreasing to zero sequence $a_n$ such that $\sum a_n$ diverges we consider the related series $\sum k_na_n$ and $\sum j_na_n.$ Here, $k_n$ and $j_n$ are real sequences such that $k_n∈\{0,1\}$ and $j_n∈\{-1,1\}.$ We study their convergence and characterize it in terms of the density of 1's in the sequences $k_n$ and $j_n.$ We extend our results to series $\sum m_na_n,$ with $m_n∈\{-1,0,1\}$ and apply them to study some associated random series.

DCDS

We deal with $m$-periodic, $n$-th order difference equations and
study whether they can be globally linearized. We give an
affirmative answer when $m=n+1$ and for most of the known examples
appearing in the literature. Our main tool is a refinement of the
Montgomery-Bochner Theorem.

DCDS

Explicit upper and lower bounds for the traveling wave solutions of
Fisher-Kolmogorov type equations

It is well-known that the existence of traveling
wave solutions for reaction-diffusion partial differential equations can be proved
by showing the existence of certain heteroclinic orbits for related autonomous
planar differential equations. We introduce a method for finding explicit upper
and lower bounds of these heteroclinic orbits. In particular, for the classical
Fisher-Kolmogorov equation we give rational upper and lower bounds which allow to
locate these solutions analytically and with very high accuracy. These results
allow one to construct analytical approximate expressions for the traveling wave
solutions with a rigorous control of the errors for arbitrary values of the
independent variables. These explicit expressions are very simple and tractable
for practical purposes. They are constructed with exponential and rational
functions.

CPAA

We study the center problem for planar systems with a linear center at the origin that in complex
coordinates have a nonlinearity formed by the sum of two monomials. Our first result lists several
centers inside this family. To the best of our knowledge this list includes a new class of Darboux
centers that are also persistent centers. The rest of the paper is dedicated to try to prove that the
given list is exhaustive. We get several partial results that seem to indicate that this is the case.
In particular, we solve the question for several general families with arbitrary high degree and for
all cases of degree less or equal than 19. As a byproduct of our study we also obtain the highest
known order for weak-foci of planar polynomial systems of some given degrees.

DCDS

This paper studies the differential equation $\dot z=f(z)$, where
$f$ is an analytic function in $\mathbb C$ except, possibly, at
isolated singularities. We give a unify treatment of well known
results and provide new insight into the local normal forms and
global properties of the solutions for this family of differential
equations.

DCDS

We obtain a criterion for determining the stability of singular limit cycles of Abel equations
$x'=A(t)x^3+B(t)x^2$. This stability controls the possible saddle-node bifurcations of limit
cycles. Therefore, studying the Hopf-like bifurcations at $x=0$, together with the bifurcations at
infinity of a suitable compactification of the equations, we obtain upper bounds of their number
of limit cycles. As an illustration of this approach, we prove that the family $x'=a t(t-t_A)x^3+b
(t-t_B)x^2$, with $a ,b>0$, has at most two positive limit cycles for any $t_B,t_A$.

DCDS

We study the number of limit cycles and the
bifurcation diagram in the Poincaré sphere of a one-parameter family of planar differential
equations of degree five $\dot {\bf x}=X_b({\bf x})$ which has been already considered in previous
papers. We prove that there is a value $b^*>0$ such that the limit cycle exists only when
$b\in(0,b^*)$ and that it is unique and hyperbolic by using a rational Dulac function. Moreover we
provide an interval of length $27/1000$ where $b^*$ lies. As far as we know the tools used to
determine this interval are new and are based on the construction of algebraic curves without
contact for the flow of the differential equation. These curves are obtained using analytic
information about the separatrices of the infinite critical points of the vector field. To prove
that the Bendixson--Dulac Theorem works we develop a method for studying whether one-parameter
families of polynomials in two variables do not vanish based on the computation of the so called
double discriminant.

DCDS-B

We consider an autonomous differential system in $\mathbb{R}^n$
with a periodic orbit and we give a new method for computing the
characteristic multipliers associated to it. Our method works when
the periodic orbit is given by the transversal intersection of
$n-1$ codimension one hypersurfaces and is an alternative to the
use of the first order variational equations. We apply it to study
the stability of the periodic orbits in several examples,
including a periodic solution found by Steklov studying the rigid
body dynamics.

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