Periodic orbits and invariant surfaces of nonlinear mappings

The accurate computation of periodic orbits and the knowledge of their stability properties are very important for studying the behavior of many physically interesting dynamical systems. In this contribution, we describe first an efficient numerical method for computing periodic orbits of nonlinear mappings in two and four dimensions of any period and to any desired accuracy. This method always converges to a periodic orbit virtually independently of the initial guess, which is very useful when the mapping has many periodic orbits close to each other, as in the case of conservative maps. We illustrate this method on Henon's 2-D mapping for real and complex arguments as well as on a 4-D symplectic mapping, by computing some of its periodic orbits and determining their particular arrangement in the 4-D space, according to their stability characteristics. We then obtain periodic orbits whose sequences of (rational) winding numbers converge on the irrational winding numbers of an invariant surface and discuss the possible existence of an analogue of Greene's 2-D criterion in 4-D symplectic mappings.