<p style='text-indent:20px;'>In this paper we investigate the simultaneous existence of isochronous centers for a family of quartic polynomial differential systems under four different types of symmetry. Firstly, we find the normal forms for the system under each type of symmetry. Next, the conditions for the existence of isochronous bi-centers are presented. Finally, we study the global phase portraits of the systems possessing isochronous bi-centers. The study shows the existence of seven non topological equivalent global phase portraits, where three of them are exclusive for quartic systems under such conditions.</p>

No abstract

In the paper Pappus's theorem and the modular group, R. Schwartz constructed a 2-dimensional family of faithful representations ρΘ of the modular group PSL(2, Z) into the group G of projective symmetries of the projective plane via Pappus Theorem. The image of the unique index 2 subgroup PSL(2, Z)o of PSL(2, Z) under each representation ρΘ is in the subgroup PGL(3, R) of G and preserves a topological circle in the flag variety, but ρΘ is not Anosov. In her PhD Thesis, V. P. Valério elucidated the Anosov-like feature of Schwartz representations: For every ρΘ, there exists a 1-dimensional family of Anosov representations ρ ε Θ of PSL(2, Z)o into PGL(3, R) whose limit is the restriction of ρΘ to PSL(2, Z)o. In this paper, we improve her work: For each ρΘ, we build a 2-dimensional family of Anosov representations of PSL(2, Z)o into PGL(3, R) containing ρ ε Θ and a 1-dimensional subfamily of which can extend to representations of PSL(2, Z) into G . Schwartz representations are therefore, in a sense, the limits of Anosov representations of PSL(2, Z) into G .