Computation of Lyapunov-type numbers for invariant curves of planar maps (Q2780570)

A study of the persistence of invariant manifolds in dynamical systems under small perturbations is carried out. The paper starts with a clear presentation of the subject for dynamical systems defined by a diffeomorphism \( f: {\mathbb R}^2 \to {\mathbb R}^2 \) which has a smooth simply closed curve \( \Gamma \) invariant under \(f\). NEWLINENEWLINENEWLINEIn this context the authors motivate the introduction of the Lyapunov-type numbers which are scalar quantities that measure the attractivity of the dynamics towards \( \Gamma. \) The behaviour of the Lyapunov-type numbers for several types of points \( p \in \Gamma \) are studied. NEWLINENEWLINENEWLINENext the authors focus their attention in the delayed logistic map defined by \( f_{\lambda}(x,y)= ( y, \lambda y ( 1 - x)) \) where \( \lambda > 0 \) is a parameter. This map has, apart of the origin, the fixed point \( P_2= (1- 1/ \lambda, 1- 1/ \lambda)\) which is asymptotically stable for \( \lambda \in (0,2)\) and unstable for \( \lambda > 2 .\) Then they study the Lyapunov numbers of invariant curves \( \Gamma_{\lambda} \) around \( \lambda = 2\) together with an approximate computation of these curves. NEWLINENEWLINENEWLINEThe paper ends with a review of the Hadamard graph transformation that has been used by the authors in their approach to construct explicitly the invariant curves and also with some comments on the difficulties that appear in practical computations.