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We explore the complicated dynamics arising in a neighbourhood of a homoclinic point associated with a homoclinic bifurcation of a two-parameter family of three-dimensional dissipative diffeomorphisms. We address the case in which the unstable manifold of the periodic saddle involved in the homoclinic bifurcation has dimension two. Besides proving the existence of strange attractors with two positive Lyapounov exponents for the associated limit return map, we also select a curve in the space of parameters in order to numerically detect the presence of possible new families of one-dimensional and two-dimensional strange attractors. The end of this curve of parameters corresponds to a return map which is conjugate to a 'bidimensional tent map'.

The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.Typesetting: Camera-ready TEX output by the author SPIN: 10520379 46/3142-543210 -Printed on acid-free paper PrefaceFor dissipative dynamics, chaos is defined as the existence of strange attractors.Chaotic behaviour was often numerically observed, but the first mathematical proof of the existence, with positive probability (persistence), of a strange attractor was

For a two parameter family of two-dimensional piecewise linear maps and for every natural number n, we prove not only the existence of intervals of parameters for which the respective maps are n times renormalizable but also we show the existence of intervals of parameters where the coexistence of at least 2 n strange attractors takes place. This family of maps contains the two-dimensional extension of the classical one-dimensional family of tent maps. 2010

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