We study the family of piecewise linear differential systems in the plane with two pieces separated by a cubic curve. Our main result is that 7 is a lower bound for the Hilbert number of this family. In order to get our main result, we develop the Melnikov functions for a class of nonsmooth differential systems, which generalizes, up to order 2, some previous results in the literature. Whereas the first order Melnikov function for the nonsmooth case remains the same as for the smooth one (i.e. the first order averaged function) the second order Melnikov function for the nonsmooth case is different from the smooth one (i.e. the second order averaged function). We show that, in this case, a new term depending on the jump of discontinuity and on the geometry of the switching manifold is added to the second order averaged function.2010 Mathematics Subject Classification. Primary 34C07, 37G15, 34A36.
We present a generalization of the most usual symmetries in differential equations known as the time-reversibility and the equivariance ones. We check that the typical properties are also valid for the new definition that unifies both. With it, we are able to present new families of planar polynomial vector fields having equilibrium points of center type. Moreover, we provide the highest lower bound for the local cyclicity of an equilibrium point of polynomial vector fields of degree 6, M (6) ≥ 48.
In this paper, we study arithmetical and topological properties for a class of Rauzy fractals R a given by the polynomial x 3 − ax 2 + x − 1 where a ≥ 2 is an integer. In particular, we prove the number of neighbors of R a in the periodic tiling is equal to 8. We also give explicitly an automaton that generates the boundary of R a . As a consequence, we prove that R 2 is homeomorphic to a topological disk.
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