To analyze attractor behavior, we cast the evolution equations into autonomous dynamical form using dimensionless variables:
x=2H2,y=2H2,z=s2,=eff3H2.x = \frac{\sigma^2}{H^2}, \quad y = \frac{\omega^2}{H^2}, \quad z = \frac{\alpha s^2}{\rho}, \quad \Omega = \frac{\rho_{\text{eff}}}{3H^2}.
The fixed points (x,y,z)(x^*, y^*, z^*) satisfy:
dxdN=f1(x,y,z),dydN=f2(x,y,z),dzdN=f3(x,y,z),\frac{dx}{dN} = f_1(x,y,z), \quad \frac{dy}{dN} = f_2(x,y,z), \quad \frac{dz}{dN} = f_3(x,y,z),
where N=lnaN = \ln a is the e-fold time variable.
Linear stability analysis shows:
Isotropic FLRW is an attractor when s0s \to 0 and 0\omega \to 0,
Anisotropic torsion-driven expansion is a transient phase,
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Vorticity-dominated fixed points are unstable or saddle-like, confirming decay of rotation at late times,
Curvature-stabilized solutions (Bianchi IX with small positive kk) can remain close to flatness.
This implies that EC-Bianchi IX cosmologies are dynamically self-regulating: torsion induces early anisotropies but eventually yields to isotropic expansion, consistent with observed large-scale structure.
