The Lorenz Attractor Exists – An Auto-Validated Proof

We present an algorithm for computing rigorous solutions to a large class of ordinary differential equations. The main algorithm is based on a partitioning process and the use of interval arithmetic with directed rounding. As an application, we prove that the Lorenz equations support a strange attractor, as conjectured by Edward Lorenz in 1963. This conjecture was recently listed by Steven Smale as one of several challenging problems for the 21st century. We also prove that the attractor is robust, i.e., it persists under small perturbations of the coefficients in the underlying differential equations. Furthermore, the flow of the equations admits a unique SRB measure, whose support coincides with the attractor. The proof is based on a combination of normal form theory and rigorous computations. 1 Background to the Problem The following non-linear system of differential equations, ẋ1 = −σx1 + σx2 ẋ2 = %x1 − x2 − x1x3 (1) ẋ3 = −βx3 + x1x2, was introduced in 1963 by Edward Lorenz, see [5]. As a crude model of atmospheric dynamics, these equations led Lorenz to the discovery of sensitive dependence of initial conditions an essential factor of unpredictability in many systems. Numerical simulations for an open neighbourhood of the classical parameter values σ = 10, β = 8/3 and % = 28 suggest that almost all points in phase space tend to a strange attractor the Lorenz attractor. For % > 1, there are three fixed points: the origin and the two “twin points” C± = (± √ β(%− 1),± √ β(%− 1), %− 1).