Instantaneous tangent-space geometry near Lyapunov-defined chaos onset: a comparative numerical study

Chaos onset is typically identified when the largest Lyapunov exponent becomes positive, but this asymptotic criterion may not capture structural changes in the finite-time, grid-resolved pre-onset regime considered here. We examine whether instantaneous tangent-space geometry exhibits measurable changes prior to the Lyapunov-based onset estimate. For five canonical systems (Lorenz–63, logistic map, forced Du!ng oscillator, H´enon map, and Ro¨ssler system), we compute a local expansion–contraction ratio r(t) = A(t)/(A(t)+B(t)) from instantaneous Jacobian eigenvalues, where A(t) sums positive real parts and B(t) sums absolute values of negative real parts. In four systems, the trajectory-averaged statistic ¯r(p) deviates from its preonset baseline at parameter distances ”plead = 0.024–0.41 before the estimated zero-crossing of the largest Lyapunov exponent. Detailed analysis indicates system-dependent geometric mechanisms: in Lorenz–63, the fraction of time with expanding directions increases from 29.8% to 31.7% prior to onset; in the H´enon map, the magnitude of expanding eigenvalues increases while the fraction remains constant at 50%. The Ro¨ssler system exhibits no pre-onset deviation under the same protocol, with the fraction jumping from 0% directly to 31% at onset, consistent with a more abrupt transition in the sampled regime. These observations align with the definition of Lyapunov exponents as Birkho# time-averages: the Oseledets theorem defines ωmax as a long-time limit, whereas ¯r(p) summarizes instantaneous geometric structure under finite sampling [1, 2, 3]. We additionally present a reproducible crosssystem comparison protocol using onset-aligned coordinates ”p = p→p→ and quantitative collapse scores. Overall, the local tangent-space diagnostic provides complementary structural information near the Lyapunov-defined onset in some systems, with detectability depending on the transition mechanism and sampled bifurcation structure.