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Abstract. In [1] the Sharkovskiȋ Theorem was extended to periodic orbits of strips of quasiperiodic skew products in the cylinder.In this paper we deal with the following natural question that arises in this setting: Does Sharkovskiȋ Theorem holds when restricted to curves instead of general strips?We answer this question in the negative by constructing a counterexample: We construct a map having a periodic orbit of period 2 of curves (which is, in fact, the upper and lower circles of the cylinder) and without any invariant curve.In particular this shows that there exist quasiperiodic skew products in the cylinder without invariant curves.

This contribution proposes a study of the mechanical behavior from damage to fracture of normalized and annealed samples of AISI 4340 steel. Besides, a numerical study in a security device of offshore industry is conducted. Experimental tests were initially performed on smooth and notched cylindrical specimens subjected to monotonic tension as well as on rectangular specimens subjected to pure shear and a combination of shear and tension loads. Such experimental tests were selected to observe different combinations of the stress triaxiality and the normalized third invariant concerning the stress state. The two types of heat treatments were considered to achieve different levels of ductility and verify their influence on the fracture mode of the alloy. The approach based on the second invariant of the deviatoric stress tensor ( J 2), with nonlinear isotropic hardening, is assumed to describe the mechanical behavior of the material in axisymmetric and three-dimensional numerical simulations. Finite elements simulations were carried out to analyze the performance of a simplified version of a mechanism of a blow-out preventer-BOP valve. The experimental and numerical curves of force versus displacement were compared, highlighting the effect of the stress triaxiality, the third invariant and calibration condition on the behavior of the material. The evolution of the equivalent plastic strain was plotted, and the fracture onset compared with its maximum value. The results show that the AISI 4340 alloy is highly dependent on the stress triaxiality and third invariant. Moreover, it can be concluded that different calibration conditions for the isotropic hardening curve can result in different levels of forces for the correct performance of the BOP in a critical situation.

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We extend the results and techniques from [7] to study the combinatorial dynamics (forcing) and entropy of quasiperiodically forced skew-products on the cylinder. For these maps we prove that a cyclic permutation τ forces a cyclic permutation ν as interval patterns if and only if τ forces ν as cylinder patterns. This result gives as a corollary the Sharkovskiȋ Theorem for quasiperiodically forced skew-products on the cylinder proved in [7]. Next, the notion of s-horseshoe is defined for quasiperiodically forced skew-products on the cylinder and it is proved, as in the interval case, that if a quasiperiodically forced skew-product on the cylinder has an s-horseshoe then its topological entropy is larger than or equals to log(s). Finally, if a quasiperiodically forced skew-product on the cylinder has a periodic orbit with pattern τ , then h(F ) ≥ h(f τ ), where f τ denotes the connect-the-dots interval map over a periodic orbit with pattern τ . This implies that if the period of τ is 2 n q with n ≥ 0 and q ≥ 1 odd, then h(F ) ≥ log(λq ) 2 n , where λ 1 = 1 and, for each q ≥ 3, λ q is the largest root of the polynomial x q − 2x q−2 − 1. Moreover, for every m = 2 n q with n ≥ 0 and q ≥ 1 odd, there exists a quasiperiodically forced skew-product on the cylinder F m with a periodic orbit of period m such that h(F m ) = log(λq ) 2 n . This extends the analogous result for interval maps to quasiperiodically forced skew-products on the cylinder.