SYMBOLIC DYNAMICS OF THE LORENZ EQUATIONS

The Lorenz equations are investigated in a wide range of parameters by using the method of symbolic dynamics. First, the systematics of stab le periodic orbits in the Lorenz equations is compared with that of th e one-dimensional (1D) cubic map, which shares the same discrete symme try with the Lorenz model. It encompasses all the known periodic windo ws of the Lorenz equations with only one exception. Second, in order t o justify the above approach and to understand the exceptions, another 1D map with a discontinuity is extracted from an extension of the geo metric Lorenz attractor and its symbolic dynamics is constructed. All this has to be done in the light of symbolic dynamics of two-dimension al maps. Finally, symbolic dynamics for the actual Poincare return map of the Lorenz equations is constructed in a heuristic way. New period ic windows of the Lorenz equations and their parameters can be predict ed from this symbolic dynamics in combination with the 1D cubic map. T he extended geometric 2D Lorenz map and the 1D antisymmetric map with a discontinuity describe the topological aspects of the Lorenz equatio ns to high accuracy.