Symbolic Dynamics Analysis of the Lorenz Equations

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 to yield global results on chaotic and periodic regimes in systems of dissipative ODEs which cannot be obtained neither by purely analytical means nor by numerical work alone. By constructing symbolic dynamics of 1D and 2D 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 led to a nomenclature, i.e., absolute periods and symbolic names, of stable and unstable periodic orbits for an autonomous system. Symmetry breakings and restorations as well as coexistence of different regimes are also analyzed by using symbolic dynamics.