The diffusionless Lorenz equations; Shil’nikov bifurcations and reduction to an explicit map

A simplified, one-parameter version of the Lorenz model is obtained in the limit of high Rayleigh- and Prandtl-numbers, physically corresponding to diffusionless convection. It is argued that the bifurcation structure of this simplified Lorenz model essentially involves only Shil’nikov bifurcations. An exact solution to this simplified dynamical system is given which serves as the limit for strong forcing and appears to be a new integrable case of the Lorenz equations. For small values of the bifurcation parameter, an approximate, analytical and multipeaked map is obtained which gives successive periods of the pulse-like motion. This map leads to self-similar behaviour in parameter-space.