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arXiv · 2603.08191

Non-Normal Route to Chaos

D. Sornette·V. R. Saiprasad·V. Troude
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Abstract

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 $\rho_{\mathrm{traj}}^{\max}<1$, singular value $\sigma_{\mathrm{traj}}^{\max}>1$ maximum Lyapunov exponent $\lambda_1>0$, 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.

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D. Sornette, V. R. Saiprasad, V. Troude. 2026-03-09. Non-Normal Route to Chaos. https://arxiv.org/abs/2603.08191

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