Recent progress of symbolic dynamics of one-and especially two-dimensional maps has enabled us to construct symbolic dynamics for systems of ordinary differential equations (ODEs). Numerical study under the guidance of symbolic dynamics is capable of yielding global results on chaotic and periodic regimes in systems of dissipative ODEs, which cannot be obtained either by purely analytical means or by numerical work alone. By constructing symbolic dynamics of one-and two-dimensional maps from the Poincare sections all unstable periodic orbits up to a given length at a fixed parameter set may be located and all stable periodic orbits up to a given length may be found in a wide parameter range. This knowledge, in turn, tells much about the nature of the chaotic limits. Applied to the Lorenz equations, this approach has made it possible to assign absolute periods and symbolic names to stable and unstable periodic orbits in this autonomous system. Symmetry breakings and restorations as well as coexistence of different regimes are also analyzed by using symbolic dynamics.
