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A three-equation model describing the evolution of the turbulence level, averaged shear flow and sheared zonal flow is analyzed using topological properties of the asymptotic solutions. An exploration in parameter space is done, identifying the attractor sets, which are fixed points and limit cycles. Then a more detailed analysis of all Morse sets is conducted using topological-combinatorial computations. This model allows the description of different types of transitions to improved plasma confinement regimes.