Alignment indices: a new, simple method for determining the ordered or chaotic nature of orbits

We introduce a new, simple and efficient method for determining the ordered or chaotic nature of orbits in two-dimensional (2D), four-dimensional (4D) and six-dimensional (6D) symplectic maps: the computation of the alignment indices. For a given orbit we follow the evolution in time of two different initial deviation vectors computing the norms of the difference d(-) (parallel alignment index) and the addition d(+) (antiparallel alignment index) of the two vectors. The time evolution of the smaller alignment index reflects the chaotic or ordered nature of the orbit. In 2D maps the smaller alignment index tends to zero for both ordered and chaotic orbits but with completely different time rates, which allows us to distinguish between the two cases. In 4D and 6D maps the smaller alignment index tends to zero in the case of chaotic orbits, while it tends to a positive non-zero value in the case of ordered orbits. The efficiency of the new method is also shown in a case of weak chaos and a comparison with other known methods that separate chaotic from regular orbits is presented.