The attractors in two-dimensional (2D) phase space of the strongly dissipative Hénon map are reduced to those of an effectively 1D map. From the grammar for this effectively 1D map we can generate all of the unstable periodic orbits, which are exactly consistent with those from the 2D map obtained directly both with a Newton procedure and the techniques of Biham and Wenzel [Phys. Rev. Lett. 63, 819 (1989)]. This idea provides a method to give a very precise and not too cumbersome estimate of the characteristic quantities of strange attractors of strongly dissipative systems. It is also helpful to understand many observations on high-dimensional systems both experimentally and numerically.

• Received 6 January 1994

DOI:https://doi.org/10.1103/PhysRevE.49.5025