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• This paper proposes a structure of a planar three-limb flexible mechanism. • The determination of equilibrium and stability by contour maps is introduced. • The variation of the multi-stability topology of the mechanism is analyzed. In general, the classical multi-stable mechanisms are made of single-loop bi-stable mechanism elements, but this paper proposes a novel structure of a planar three-limb flexible mechanism whose multi-stability is dependent on the structure parameters. The structure of this flexible mechanism is ingeniously designed to have a large selection range of the original limb length, resulting in the variation of the multi-stability distribution and the possible switching paths. To analyze the variation of this flexible mechanism, the paper introduces a method avoiding computational singularity to determine the stability by contour map and a method to draw the multi-stability topology to seek for the possible switching paths. Using these methods, the paper presents the variation of the multi-stability topology of an axisymmetric and two mirror-symmetric planar three-limb flexible mechanisms as the original length of the limbs increases. The results shows the mulita-stability of this flexible mechanism can be determined by the original length of the limbs and limb arrangement. The whole statements provide a theoretical basis for designing and applying such type of flexible multi-stable mechanism. [ABSTRACT FROM AUTHOR]