Nontwist Hamiltonian systems have shearless invariant curves that act like barriers in phase space [1, 2]. Recently, secondary shearless curves have also been identified in the phase space of twist maps, in the neighbourhood of peculiar bifurcations of elliptic fixed points [3]. We use Slater’s theorem to develop a qualitative and quantitative numerical approach to determine the breakup of invariant curves in the phase space of area-preserving maps [4]. We also determine the breakup critical parameters, of the shearless curves, with a procedure based on the determinism analysis performed on the recurrence plot of orbits near the critical transition [5]. Finally, we present evidences of transport barriers in plasmas confined in the tokamak TCABR [6] and in the Texas Helimak [7]. References 1- P. J. Morrison, Physics of Plasmas 7, 2279 [2000]. 2- D. Del-Castillo-Negrete, Physics of Plasmas 7, 1702 [2000]. 3- C. V. Abud, I. L. Caldas. Chaos 22, 033142 (2012). 4- C. V. Abud, I. L. Caldas. Physica D 308, 34 (2015) 5- M. S. Santos, M. Mugnaine, J. D. Szezech Jr, A. M. Batista, I. L. Caldas, M. S. Baptista, R. L. Viana. Chaos 28, 085717 (2018). 6-A. F. Marcus, I. L. Caldas, Z. O. Guimarães-Filho, P. J. Morrison, W. Horton, I. C. Nascimento, Yu. K. Kuznetsov. Phys. Plasmas 15, 112304 (2008). 7- D. L. Toufen, Z. O. Guimarães-Filho, I. L. Caldas, F. A. Marcus, K. W. Gentle. Phys. Plasmas (2012).