Dynamical Systems Analysis of an Einstein-Cartan Ekpyrotic Nonsingular Bounce Cosmology

I construct an Einstein-Cartan ekpyrotic model (ECEM): a homogeneous, nearly Friedmann-Lemaître-Robertson-Walker (FLRW) background in Einstein-Cartan (EC) gravity whose spin-torsion sector, modeled phenomenologically as a Weyssenhoff fluid with stiff scaling $\propto a^{-6}$, is coupled to a scalar field with a steep exponential potential that interpolates between a negative ekpyrotic branch and a positive plateau. Extending the Copeland-Liddle-Wands (CLW) scalar-fluid dynamical system to a six-dimensional phase space including shear, curvature, and spin-torsion, I recast the equations in a compact deceleration-parameter form, compute the full Jacobian, and evaluate maximal Lyapunov exponents. Numerical solutions show that the ekpyrotic branch ($w_φ\gg1$) exponentially damps homogeneous shear, while the softened branch ($w_φ<1$) allows $ρ_s$ to overtake the scalar during contraction and trigger a torsion-supported bounce at high but finite densities where the EC spin-torsion term becomes dynamically dominant. Scans in a two-parameter softening plane $(φ_{\rm b},Δ)$ identify a finite region of nonsingular trajectories and quantify the required tuning; in the parameter ranges explored the maximal Lyapunov exponent on the constrained phase space is negative, giving no indication of chaotic behavior in this homogeneous truncation even when the usual curvature mode that destabilizes contracting General Relativity (GR) backgrounds is included. The construction is purely phenomenological and confined to homogeneous backgrounds: it does not address entropy accumulation, the cosmological arrow of time, or a complete cyclic cosmology.