THE EFFECT OF ASPECT RATIO ON THE BIFURCATION PROPERTIES OF A DOUBLE PARALLEL-CONNECTION LORENZ SYSTEM

A double parallel-connection (DPC) Lorenz system is developed by performing spectrum truncation of the Galerkin series expansion of the two-dimensional Rayleigh-Benard convection equation. Analyses of the equilibrium states indicate that a convective roll stems from a flow with a given wavenumber first losing its stability for a particular aspect ratio β after a stable laminar flow gets unstable; when β has the value βc able to deprive synchronously two flows with different wavenumbers of stability, occurrences of convective rolls with different wavenumbers depend entirely on the initial conditions, in good agreement with the relevant experimental results. The calculations of the unstablized rolls show that, with a smaller β (as compared with βc), the DPC Lorenz system has the same bifurcation properties as the ordinary Lorenz system; for a moderate β, the system has very complicated periodic, quasi-periodic and phase-locking motions; for a larger β, it results in intermittent chaos and causes mean flows with different numbers of vortices to occur alternately with time. All these indicate that β has substantial effect on the two Lorenz systems coupled through parallel connection in their interaction and the results.