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

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We perform a phase-space analysis of the cosmological 3-fluid problem consisting of a barotropic fluid with an equation-of-state parameter γ-1, a pressureless dark matter fluid, plus a scalar field ϕ (representing dark energy) coupled to exponential potential V=V₀(-κλϕ). Besides the potential-kinetic-scaling solutions, which are not the unique late-time attractors whenever they exist for λ²≥ 3, we derive new attractors where both dark energy and dark matter coexist and the final density is shared in a way independent of the value of γ>1. The case of a pressureless barotropic fluid (γ=1) has a one-parameter family of attractors where all components coexist. New one-parameter families of matter-dark matter saddle points and kinetic-matter repellers exist. We investigate the stability of the ten critical points by linearization and/or Lyapunov's Theorems and a variant of the theorems formulated in this paper. A solution with two transient periods of acceleration and two transient periods of deceleration is derived.