2026/03/09 by D. Sornette, V. R. Saiprasad, V. Troude · 1 voice
Physics and Astronomy · #nlin.CD #physics.app-ph
arxiv published 2026/03/09 · arxiv updated 2026/06/13
Deterministic chaos is usually associated with local spectral expansion: Jacobian eigenvalues are expected to exceed unity somewhere on the attractor. We show that this view is incomplete in dimensions d>1. For non-normal Jacobians, pointwise spectral stability can suggest everywhere local contraction, while non-orthogonal eigenvectors still allow transient singular-vector amplification. We construct four low-dimensional deterministic maps realizing this mechanism: partition-reinjected, phase-prescribed, feedback-driven, and affine-reinjected non-normal routes to chaos. In all cases, the instantaneous Jacobian remains spectrally stable on the attractor, with eigenvalues fixed inside the unit disk, while increasing non-normality drives the maximal Lyapunov exponent through zero. The positive exponent therefore describes sustained asymptotic chaos, not transient chaos. Across the four classes, the common signature is spectral radius ρtrajmax<1, singular value σtrajmax>1 maximum Lyapunov exponent λ1>0, and an increase of attractor dimension. These examples identify non-normality and recurrent reinjection of transiently amplified directions as a deterministic route to chaos distinct from eigenvalue instability.