Reversibility and branching of periodic orbits
Ana Cristina Mereu Marco Antonio Teixeira
Discrete & Continuous Dynamical Systems - A 2013, 33(3): 1177-1199 doi: 10.3934/dcds.2013.33.1177
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.
keywords: normal form Lyapunov center theorem. resonance reversible systems Periodic orbits
Regularization of discontinuous vector fields in dimension three
Jaume Llibre Marco Antonio Teixeira
Discrete & Continuous Dynamical Systems - A 1997, 3(2): 235-241 doi: 10.3934/dcds.1997.3.235
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.
keywords: Regularization and discontinuous vector fields.
On the similarity of Hamiltonian and reversible vector fields in 4D
Ricardo Miranda Martins Marco Antonio Teixeira
Communications on Pure & Applied Analysis 2011, 10(4): 1257-1266 doi: 10.3934/cpaa.2011.10.1257
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.
keywords: reversible vector field Hamiltonian vector field. Normal form
Divergent diagrams of folds and simultaneous conjugacy of involutions
Solange Mancini Miriam Manoel Marco Antonio Teixeira
Discrete & Continuous Dynamical Systems - A 2005, 12(4): 657-674 doi: 10.3934/dcds.2005.12.657
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.
keywords: singularities involution discontinuous vector fields reversible diffeomorphisms. Divergent diagram of folds normal form
On the birth of minimal sets for perturbed reversible vector fields
Jaume Llibre Ricardo Miranda Martins Marco Antonio Teixeira
Discrete & Continuous Dynamical Systems - A 2011, 31(3): 763-777 doi: 10.3934/dcds.2011.31.763
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

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