Linearizing around the fixed points gives the Jacobian matrix JJ. The eigenvalues i\lambda_i of JJ determine stability:
(i)<0\Re(\lambda_i) < 0: stable (attractor),
(i)>0\Re(\lambda_i) > 0: unstable,
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Mixed signs: saddle point.
Typical results:
The isotropic FLRW point (x=0,y=0,z=0x=0, y=0, z=0) is a future attractor,
Early-universe torsion-dominated point (z1z \gg 1) is unstable but necessary for seeding anisotropies,
Intermediate fixed points correspond to transition regimes with decaying shear and vorticity.
This confirms that the Einstein--Cartan--Bianchi IX cosmology naturally evolves from a highly anisotropic, rotating early phase to a stable isotropic late-time phase.
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