## Journals

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### Open Access Journals

DCDS

We study the dynamics near an equilibrium point of a $2$-parameter family of a reversible system in $\mathbb{R}^6$. In particular, we exhibit conditions for the existence of periodic orbits near the equilibrium of systems having the form $x^{(vi)}+ \lambda_1 x^{(iv)} + \lambda_2 x'' +x = f(x,x',x'',x''',x^{(iv)},x^{(v)})$. The techniques used are Belitskii normal form combined with Lyapunov-Schmidt reduction.

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.

CPAA

We study the existence of formal conjugacies between reversible vector fields and Hamiltonian vector fields in 4D around a generic singularity. We construct conjugacies for a generic class of reversible vector fields. We also show that reversible vector fields are formally orbitally equivalent to polynomial decoupled Hamiltonian vector fields. The main tool we employ is the normal form
theory.

DCDS

The results in this paper fit into a program to study the existence
of periodic orbits, invariant cylinders and tori filled with periodic
orbits in perturbed reversible systems. Here we focus on bifurcations
of one-parameter families of periodic orbits for reversible vector
fields in $\mathbb{R}^4$. The main used tools are normal forms theory,
Lyapunov-Schmidt method and averaging theory.

keywords:
Invariant torus
,
Limit cycle
,
Averaging method
,
Isochronous center
,
Reversible system.
,
Periodic orbit

DCDS

In this work we show that the smooth
classification of divergent diagrams of folds $(f_1, \ldots, f_s)
: (\mathbb R^n,0) \to (\mathbb R^n \times \cdots \times \mathbb
R^n,0)$ can be reduced to the classification of the $s$-tuples
$(\varphi_1, \ldots, \varphi_s)$ of associated involutions. We
apply the result to obtain normal forms when $s \leq n$ and
$\{\varphi_1, \ldots, \varphi_s\}$ is a transversal set of linear
involutions. A complete description is given when $s=2$ and $n\geq 2$. We also present a brief discussion on applications of our
results to the study of discontinuous vector fields and discrete
reversible dynamical systems.

## Year of publication

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