A system is spectrally stable when all eigenvalues of its Jacobian lie inside the unit disk (discrete time) or have negative real parts (continuous time). Every perturbation eventually shrinks. The long-run prediction is unambiguous: the system returns to its fixed point. Spectral analysis — the decomposition into eigenvalues and eigenvectors — is the standard tool for this assessment.
Troude and Sornette (2026, arXiv:2603.07206) construct a linearly stable system with no oscillatory components that nevertheless exhibits synchronized behavior. There are no Hopf bifurcations, no limit cycles, no eigenvalues with positive real parts. The system should relax to a fixed point and stay there. Instead, it produces intermittent episodes of phase-coherent oscillation that satisfy every standard measure of synchronization: Kuramoto order parameter, broken time-reversal symmetry, positive entropy production.
Lakshmivarahan, Lewis, and Soleimani (2026, arXiv:2603.08191) construct a bounded three-dimensional system with a spectrally contracting Jacobian that nevertheless develops positive Lyapunov exponents and transitions to chaos. All eigenvalues are inside the stability region at every point in state space. The system should contract uniformly. Instead, it produces deterministic chaos — sensitive dependence on initial conditions, exponential divergence of nearby trajectories.
The mechanism in both cases is non-normality: the Jacobian does not commute with its transpose. The eigenvectors are not orthogonal.
When eigenvectors are orthogonal, the eigenvalues tell the full story. A perturbation along one eigenvector evolves independently of the others, growing or shrinking according to its eigenvalue. No interaction between modes.
When eigenvectors are non-orthogonal, perturbations along one mode can transiently amplify motion along another. The amplification is algebraic, not exponential — it eventually decays. But if the system's dynamics reinject trajectories into the amplifying region before the transient dies (through noise in the synchronization case, through endogenous switching in the chaos case), the transient amplification becomes self-sustaining. The individual episodes are temporary. The process of generating them is not.
The result is that non-normality can push a spectrally stable system in either direction. Add noise, and the transient amplification creates coherent oscillation from a system with no oscillators — order from stability. Remove noise but add nonlinear reinjection, and the same transient amplification creates chaos from a system with no spectral instability — chaos from stability.
The eigenvalues are the same in both cases: stable. The eigenvector geometry is what differs, and it is what determines the dynamics. Spectral analysis, which discards eigenvector angles and retains only eigenvalue magnitudes, is blind to this axis of variation.
The structural lesson: the angle between basis vectors is as dynamically important as the rate of contraction along each one. A system whose modes contract at rate -1 along orthogonal directions is stable. The same contraction rates along non-orthogonal directions can produce synchronization, chaos, or both — depending on how the non-orthogonality interacts with the system's other features. The invisible axis — the angle between modes — is doing the work that spectral analysis attributes to the eigenvalues. When the two analyses disagree, the geometry wins.