Lorenz System State Stability Identification using Neural Networks

The paper defines left/right regimes by the per-trajectory mean of x and labels stability via a 5-past/5-future window (a point is stable iff all 10 neighbors are in the same regime) . It then normalizes each Lorenz trajectory feature-wise by subtracting its own mean and dividing by a positive scale (called “standard deviation” in the text, but Eq. (5c) is actually the second central moment), and asserts—empirically via Fig. 13—that normalization does not change the relative positions of stable/unstable points, only the axis ranges . The candidate solution gives a concise proof: per-trajectory standardization is an affine, strictly increasing map that preserves the sign of x−μ and hence preserves regime membership and the window-based stability rule. It also notes the degenerate s=0 case and that normalizing y,z,derivatives is irrelevant since the rule depends only on x. Therefore, the claim is correct; the paper presents it empirically/heuristically, while the model supplies a clean formal proof.