Laura Gardini

61029 Urbino, Italy
University of Urbino


Avrutin V., Schanz M., Gardini L.
Recently it has been demonstrated that the domain of robust chaos close to the periodic domain, which is organized by the period-adding structure, contains an infinite number of interior crisis bifurcation curves. These curves form the so-called bandcount adding scenario, which determines the occurrence of multi-band chaotic attractors. The analytical calculation of the interior crisis bifurcations represents usually a quite sophisticated and cumbersome task. In this work we demonstrate that, using the map replacement approach, the bifurcation curves can be calculated much easier. Moreover, using this approach recursively, we confirm the hypothesis regarding the self-similarity of the bandcount adding structure.
Keywords: piecewise-linear maps, crisis bifurcations, chaotic attractors, bandcount adding and doubling, self-similarityand renormalization
Citation: Avrutin V., Schanz M., Gardini L.,  Self-similarity of the bandcount adding structures: calculation by map replacement, Regular and Chaotic Dynamics, 2010, vol. 15, no. 6, pp. 685-703
Gardini L., Tramontana F.
Snap-back repellers in non-smooth functions
2010, vol. 15, nos. 2-3, pp.  237-245
In this work we consider the homoclinic bifurcations of expanding periodic points. After Marotto, when homoclinic orbits to expanding periodic points exist, the points are called snap-back-repellers. Several proofs of the existence of chaotic sets associated with such homoclinic orbits have been given in the last three decades. Here we propose a more general formulation of Marotto’s theorem, relaxing the assumption of smoothness, considering a generic piecewise smooth function, continuous or discontinuous. An example with a two-dimensional smooth map is given and one with a two-dimensional piecewise linear discontinuous map.
Keywords: snap back repellers, homoclinic orbits in noninvertible maps, orbits homoclinic to expanding points
Citation: Gardini L., Tramontana F.,  Snap-back repellers in non-smooth functions, Regular and Chaotic Dynamics, 2010, vol. 15, nos. 2-3, pp. 237-245

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