This study presents a continuation method to calculate flow bifurcation with/without heat transfer in a two-sided lid-driven cavity with an aspect ratio of 1.96. The top and bottom lids of the cavity move in opposite directions and are allowed to be of different temperatures, thereby establishing a temperature gradient in the cavity flow and generating thermal transport. A comprehensive bifurcation diagram of the cavity flow is derived via the continuation method and linear stability analysis is used to identify the nature of the various flow solutions. For the isothermal flow case, the Reynolds number is used as the continuation parameter and three symmetric flows and two asymmetric flows are identified. For the non-isothermal flow case, the Grashof number is used as a continuation parameter. The flow evolution is studied for different temperature gradients, and bifurcation diagrams are constructed as a function of the continuation parameter. A thumb-shaped boundary line is established which identifies a restricted region defined in terms of the Grashof and Reynolds numbers within which a stable flow state exists. (c) 2006 Elsevier Ltd. All rights reserved.