Two mechanisms of the transition to chaos in finite-dimensional models of convection

Abstract

The onset of chaos in systems of ordinary differential equations possessing a stationary solution with a one-dimensional unstable manifold is studied both numerically and qualitatively with the help of an auxiliary piecewise monotomic discontinuous recursion relation. A connection is established between the route to chaos and the ratio of two leading eigenvalues of the vector field linearized near the fixed point. This connection is confirmed by numerical data obtained from the investigation of differential equations originating from a hydrodynamical problem. Two routes are considered — a well-known mechanism suggested by Lorenz and another one which is due to the accumulation of bifurcations corresponding to the emergence of homoclinic orbits of a saddle-point. The asymptotical properties of the latter route prove to be entirely determined by the above ratio.