Let T be a competitive map on a rectangular region R ⊂ R 2 , and assume T is C 1 in a neighborhood of a fixed point x ∈ R. The main results of this paper give conditions on T that guarantee the existence of an invariant curve emanating from x when both eigenvalues of the Jacobian of T at x are nonzero and at least one of them has absolute value less than one, and establish that C is an increasing curve that separates R into invariant regions. The results apply to many hyperbolic and nonhyperbolic cases, and can be effectively used to determine basins of attraction of fixed points of competitive maps, or equivalently, of equilibria of competitive systems of difference equations. Several applications to planar systems of difference equations with non-hyperbolic equilibria are given.

With the new FIG Code of Points for men (2006) based on the philosophy of open ended difficulty score, point advantages have been given, again, to those who are in search for and willing to perform new elements. Each element in the Code of Points can be developed by changing its start and its final position, the start and the final grip with the apparatus, the body position during the element, by adding a flight phase or a rotation around the frontal, the longitudinal or the sagital axis. The Tkachev is quite an old release element (approximately 40 years old) on high bar. In line with the knowledge available to us today, we have been looking into the possibility of performing the Tkachev salto. Following series of biomechanical analysis with consideration of the gymnast's safety, we calculated that the Tkachev salto could be performed by those gymnasts who can perform the straight Tkachev with a high amplitude. Gymnast who will be able to perform the Tkachev salto at a major competition will enter the gymnastics history and have huge chances of wining the most prestigious competitions.