Abstract. We consider the asymptotic behavior of the normalized Ricci flow on generalized Wallach spaces that could be considered as special planar dynamical systems. All non symmetric generalized Wallach spaces can be naturally parametrized by three positive numbers a1, a2, a3. Our interest is to determine the type of singularity of all singular points of the normalized Ricci flow on all such spaces. Our main result gives a qualitative answer for almost all points (a1, a2, a3) in the cube (0, 1/2] × (0, 1/2] × (0, 1/2].

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This paper is devoted to the study of the evolution of positively curved metrics on the Wallach spaces SU(3)/T max , Sp(3)/Sp(1) × Sp(1) × Sp(1), and F 4 /Spin(8). We prove that for all Wallach spaces, the normalized Ricci flow evolves all generic invariant Riemannian metrics with positive sectional curvature into metrics with mixed sectional curvature. Moreover, we prove that for the spaces Sp(3)/Sp(1) × Sp(1) × Sp(1) and F 4 /Spin(8), the normalized Ricci flow evolves all generic invariant Riemannian metrics with positive Ricci curvature into metrics with mixed Ricci curvature. We also get similar results for some more general homogeneous spaces.

We study the behavior of a three-dimensional dynamical system with respect to some set S given in 3-dimensional euclidian space. Geometrically such a system arises from the normalized Ricci flow on some class of generalized Wallach spaces that can be described by a real parameter a ∈ (0, 1/2), as for S it represents the set of invariant Riemannian metrics of positive sectional curvature on the Wallach spaces. Establishing that S is bounded by three conic surfaces and regarding the normalized Ricci flow as an abstract dynamical system we find out the character of interrelations between that system and S for all a ∈ (0, 1/2). These results can cover some well-known results, in particular, they can imply that the normalized Ricci flow evolves all generic invariant Riemannian metrics with positive sectional curvature into metrics with mixed sectional curvature on the Wallach spaces corresponding to the cases a ∈ {1/9, 1/8, 1/6} of generalized Wallach spaces.

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