Existence of the Lorenz attractor

Consider the classical three-dimensional Lorenz system of ordinary differential equations $\dot{x} = \sigma(y - x)$, $\dot{y} = \rho x - y - xz$, $\dot{z} = xy - \beta z$ on $\mathbb{R}^3$ with the standard parameters $\sigma = 10$, $\beta = 8/3$, and $\rho = 28$. Does this flow possess a robust singular-hyperbolic strange attractor whose dynamics are topologically and statistically modeled by the geometric Lorenz attractor (supporting a unique physical Sinai–Ruelle–Bowen measure)?

Solved in the affirmative by Warwick Tucker (announced in Comptes Rendus de l'Académie des Sciences in 1999; full paper published in Foundations of Computational Mathematics in 2002). Because trajectories of the Lorenz system pass arbitrarily close to the saddle equilibrium at the origin $(0,0,0)$—where return times diverge—purely numerical integration cannot rule out stable periodic orbits. Tucker combined analytic normal form theory in a small cube around the origin with a rigorous interval-arithmetic ODE solver on a transverse Poincaré section $\Sigma \subset \{z = 27\}$ partitioned into small rectangles, proving the existence of a forward-invariant trapping region and a uniformly expanding invariant cone field.