Non-Normal Route to Chaos

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 $\ddot{I}{}_{\mathrm{traj}}^{\max }<1$, singular value $\ddot{I}{ƒ}_{\mathrm{traj}}^{\max }>1$ maximum Lyapunov exponent $\hat{I}{»}_1>0$, 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.

17 pages, 11 figures