Dynamics of the Henon Map in the Digital Domain

Nonlinear maps are usually implemented using finite-precision floating-point formats both in practical applications and in theoretical investigations. In the digital domain, the size of the state space is finite and every trajectory after a finite number of iterations reaches a cycle. It is therefore important to study the influence of rounding errors and the finiteness of the state space on properties of nonlinear maps such as the number of cycles, their periods, sizes of basins of attraction of cycles, and average convergence times. In this work, a thorough analysis of the dynamics of finite-precision implementations of the Henon map is carried out. Six computational formulas and three popular finite-precision floating-point formats are considered. An exhaustive search is performed to find all cycles existing for single-precision floating-point implementations. Interval methods are used to reduce the number of initial conditions that must be considered. An efficient graph-based algorithm is designed to find basins of attraction. For the double-precision and extended-precision implementations, statistical methods are utilized to find cycles and to estimate sizes of their basins of attraction. Properties of observed cycles and corresponding dynamical phenomena are thoroughly discussed.