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### Open Access Journals

DCDS

In this work we study the periodic solutions, their stability and
bifurcation for the class of Duffing differential equation $x''+ \epsilon C x'+ \epsilon^2 A(t) x +b(t) x^3 = \epsilon^3 \Lambda h(t)$, where $C>0$,
$\epsilon>0$ and $\Lambda$ are real parameter, $A(t)$, $b(t)$ and $h(t)$
are continuous $T$--periodic functions and $\epsilon$ is sufficiently
small. Our results are proved using the averaging method of first
order.

keywords:
bifurcation
,
stability.
,
Duffing differential
equation
,
averaging method
,
Periodic solution

DCDS

It is well known that invariant algebraic curves of polynomial
differential systems play an important role in questions regarding
integrability of these systems. But do they also have a role in
relation to limit cycles? In this article we show that not only they
do have a role in the production of limit cycles in polynomial
perturbations of such systems but that algebraic invariant curves
can even generate

*algebraic limit cycles*in such perturbations. We prove that when we perturb any quadratic system with an invariant ellipse surrounding a center (quadratic systems with center always have invariant algebraic curves and some of them have invariant ellipses) within the class of quadratic differential systems, there is at least one 1-parameter family of such systems having a limit cycle bifurcating from the ellipse. Therefore the cyclicity of the period annulus of such systems is at least one.
DCDS-B

We provide sufficient conditions for the existence of limit cycles
for the Floquet differential equations $\dot {\bf x}(t) = A{\bf
x}(t)+ε(B(t){\bf x}(t)+b(t))$, where ${\bf x}(t)$ and $b(t)$ are
column vectors of length $n$, $A$ and $B(t)$ are $n\times n$
matrices, the components of $b(t)$ and $B(t)$ are $T$--periodic
functions, the differential equation $\dot {\bf x}(t)= A{\bf x}(t)$
has a plane filled with $T$--periodic orbits, and $ε$ is a small
parameter. The proof of this result is based on averaging theory but
only uses linear algebra.

DCDS

We study the Hopf bifurcation from the singular point with
eigenvalues $a$ε$ \ \pm\ bi$ and $c $ε located at the origen
of an analytic differential system of the form $ \dot x= f(
x)$, where $x \in \R^3$. Under convenient assumptions we prove
that the Hopf bifurcation can produce $1$, $2$ or $3$ limit cycles.
We also characterize the stability of these limit cycles. The main
tool for proving these results is the averaging theory of first and
second order.

CPAA

We present two new families of polynomial differential systems of arbitrary degree with centers, a two--parameter family and a four--parameter family.

DCDS

The cyclicity of period annuli of some classes of reversible and
non-Hamiltonian quadratic systems under quadratic perturbations
are studied. The argument principle method and the centroid curve
method are combined to prove that the related Abelian integral has
at most two zeros.

DCDS-B

In this paper we apply the averaging theory to a class of
three-dimensional autonomous quadratic polynomial differential
systems of Lorenz-type, to show the existence of limit cycles
bifurcating from a degenerate zero-Hopf equilibrium.

DCDS

In this paper vector fields around the origin in dimension three
which are approximations of discontinuous ones are studied. In a
former work of Sotomayor and Teixeira [6] it is shown, via
regularization, that Filippov's conditions are the natural ones to
extend the orbit solutions through the discontinuity set for vector
fields in dimension two. In this paper we show that this is also the
case for discontinuous vector fields in dimension three. Moreover, we
analyse the qualitative dynamics of the local flow in a neighborhood
of the codimension zero regular and singular points of the
discontinuity surface.

DCDS

By rescaling the variables, the parameters and the periodic function
of the Vallis differential system we provide sufficient conditions
for the existence of periodic solutions and we also characterize
their kind of stability. The results are obtained using averaging
theory.

DCDS

In general the center--focus problem cannot be solved, but in the case that the singularity has purely imaginary eigenvalues there are algorithms to solving it.
The present paper implements one of these algorithms for the polynomial differential systems of the form
\[
\dot x= -y + x f(x) g(y),\quad \dot y= x+y f(x) g(y),
\]
where $f(x)$ and $g(y)$ are arbitrary polynomials.
These differential systems have constant angular speed and are also called rigid systems.
More precisely, in this paper we give the center conditions for these systems, i.e. the necessary and sufficient conditions in order that they have an uniform isochronous center.
In particular, the existence of a focus with the highest order is also studied.

keywords:
limit cycles
,
the center problem
,
Isochronous centers
,
Lyapunov quantities.
,
focal basis

## Year of publication

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