DCDS
Reversibility and branching of periodic orbits
Ana Cristina Mereu Marco Antonio Teixeira
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
DCDS-B
Limit cycles in uniform isochronous centers of discontinuous differential systems with four zones
Jackson Itikawa Jaume Llibre Ana Cristina Mereu Regilene Oliveira

We apply the averaging theory of first order for discontinuous differential systems to study the bifurcation of limit cycles from the periodic orbits of the uniform isochronous center of the differential systems $\dot{x}=-y+x^2, \;\dot{y}=x+xy$, and $\dot{x}=-y+x^2y, \;\dot{y}=x+xy^2$, when they are perturbed inside the class of all discontinuous quadratic and cubic polynomials differential systems with four zones separately by the axes of coordinates, respectively.

Using averaging theory of first order the maximum number of limit cycles that we can obtain is twice the maximum number of limit cycles obtained in a previous work for discontinuous quadratic differential systems perturbing the same uniform isochronous quadratic center at origin perturbed with two zones separately by a straight line, and 5 more limit cycles than those achieved in a prior result for discontinuous cubic differential systems with the same uniform isochronous cubic center at the origin perturbed with two zones separately by a straight line. Comparing our results with those obtained perturbing the mentioned centers by the continuous quadratic and cubic differential systems we obtain 8 and 9 more limit cycles respectively.

keywords: Limit cycle averaging theory uniform isochronous center discontinuous polynomial system
DCDS
Isochronicity for trivial quintic and septic planar polynomial Hamiltonian systems
Francisco Braun Jaume Llibre Ana Cristina Mereu
In this paper we completely characterize trivial polynomial Hamiltonian isochronous centers of degrees $5$ and $7$. Precisely, we provide simple formulas, up to linear change of coordinates, for the Hamiltonians of the form $H = \left(f_1^2 + f_2^2 \right)/2$, where $f = (f_1, f_2): \mathbb{R}^2\to \mathbb{R}^2$ is a polynomial map with $\det D f = 1$, $f(0,0) = (0,0)$ and the degree of $f$ is $3$ or $4$.
keywords: polynomial Hamiltonian systems Isochronous centers Jacobian conjecture.

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