Dynamical systems analysis of an Einstein–Cartan ekpyrotic nonsingular bounce cosmology

Abstract 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 $$\rho _s\propto a^{-6}$$ ρ s ∝ 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 ( $$w_\phi \gg 1$$ w ϕ ≫ 1 ) exponentially damps homogeneous shear, while the softened branch ( $$w_\phi