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We perform a phase-space analysis of the cosmological 3-fluid problemconsisting of a barotropic fluid with an equation-of-state parameter$\gamma-1$, a pressureless dark matter fluid, plus a scalar field $\phi$(representing dark energy) coupled to exponential potential$V=V_0\exp{(-\kappa\lambda\phi)}$. Besides the potential-kinetic-scalingsolutions, which are not the unique late-time attractors whenever they existfor $\lambda^2\geq 3\ga$, we derive new attractors where both dark energy anddark matter coexist and the final density is shared in a way independent of thevalue of $\gamma >1$. The case of a pressureless barotropic fluid ($\gamma =1$)has a one-parameter family of attractors where all components coexist. Newone-parameter families of matter-dark matter saddle points and kinetic-matterrepellers exist. We investigate the stability of the ten critical points bylinearization and/or Lyapunov's Theorems and a variant of the theoremsformulated in this paper. A solution with two transient periods of accelerationand two transient periods of deceleration is derived.

Phase-space analysis of the cosmological 3-fluid problem: Families of attractors and repellers