Symbolic coding for noninvertible systems: uniform approximation and numerical computation (Q2834292)

The rough contents of this large and dense article are as follows: 1. Introduction, 2. Numerical computation of the maximal invariant set, 3. Uniform dichotomies of variational equations, 4. Multi humped orbits on finite and infinite intervals, 5. The homoclinic theorem for finite and infinite orbits in noninvertible systems and 6. Computation and visualization of the maximal invariant set and its coding. Two appendices, the first devoted to the theory of exponential dichotomies in a noninvertible setup, and references (42 entries) are also available. The third and the fourth sections are devoted to some theoretical aspects, namely the roughness theorem for exponential dichotomies in the noninvertible case, the pseudo-orbits, the exponential dichotomy of the variational equations, the solvability of the discrete boundary value problems on both finite and infinite time intervals, etc. Then, as the main result, the authors introduce and thoroughly justify a numerical algorithm in order to compute the maximal invariant set close to a given point of a homoclinic orbit of a smooth map depending on a one-dimensional parameter. They validate the performances of this algorithm by carrying out various dynamical features of two- and three-imensional systems.NEWLINENEWLINEAn impressive picture of homoclinic orbits along with approximations of stable and unstable sets of a fixed point for a wild chaos model is displayed.