In this paper we extend the periodic structure obtained for σ-permutation maps in the n-dimensional cube I n to the same class of maps defined in the n-dimensional torus T n .

Dedicated to Professor Alexander N. Sharkovsky on the occasion of his 65th Birthday.In this paper, we construct a triangular map F on I 3 ðI ¼ ½0; 1Þ holding the following uniform property: All points ða; b; c Þ [ I 3 except those of the face I 2 0 ¼ {0} £ I 2 ; which are fixed by F, have as v-limit set the face I 2 0 : So, we are able to describe the family WðF Þ ¼ {v F ða; b; c Þ : ða; b; c Þ [ I 3 } for a continuous endomorphism defined in a compact metric space of dimensión higher than two, establishing that WðF Þ ¼ I 2 0 < {ð0; b; c Þ} ðb;c Þ[I 2 :

The Frenkel-Kontorova-Tomlinson (FKT) model represents mechanical systems in which the atomic smooth surfaces of two bodies slide against each other. The model is very sensitive to changes of the system parameters, and ranges from simple stable harmonic to chaotic solutions. The design of the model between two bodies for the dynamic problem, following the network method rules, is explained with precision and run on standard electrical circuit simulation software. It provides the phase diagrams of atom displacement for each atom and the total friction force by the summation of all the atom displacements. This article is focused on studying the effect of the selected time step on the result and in the lack of sensitivity of Lyapunov exponents to assess chaotic behaviour.

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This paper generalizes the nondegenerated conditions that imply the most common bifurcations in uniparametric families of maps defined on ‫.ޒ‬ It also presents a new very simple proof of the Hopf bifurcation theorem of maps on ‫ޒ‬ 2 Ž Ž . . see G. Iooss, ''Mathematical Studies, '' Vol. 36, 1979 , based on one of the results obtained in one dimension and generalizes one of the nondegenerated conditions Ž . the Hopf condition of the theorem. ᮊ