Non-Normal Route to Chaos

TL;DR

This paper demonstrates that chaos can arise in deterministic systems without spectral criticality, through non-normal amplification mechanisms, challenging traditional views on the spectral conditions for chaos.

Contribution

It introduces a new route to chaos via non-normality in dynamical systems, independent of spectral criticality, supported by a constructed 3D example.

Findings

Chaos can develop with all eigenvalues inside the stability region.

Non-normal amplification can induce chaos without spectral criticality.

The mechanism applies broadly to deterministic dynamical systems.

Abstract

Deterministic chaos is commonly associated with spectral criticality: exponential sensitivity is expected when Jacobian eigenvalues exceed unity in parts of the attractor, producing the local expansion that offsets contraction elsewhere. We show that this paradigm is incomplete in dimensions d>1. We construct a bounded 3D dynamical system whose Jacobian is pointwise spectrally contracting, namely all instantaneous eigenvalues remain strictly inside the stability region, yet the system develops a positive maximal Lyapunov exponent and undergoes a transition to chaos as a non-normality index increases at fixed spectral radius. The mechanism relies on the repeated regeneration of transient non-normal amplification through endogenous switching that reinjects trajectories into amplifying non-orthogonal directions. Although demonstrated here for a discrete-time map, the mechanism is geometric and applies more broadly to deterministic dynamical systems. These results show that chaos can emerge without spectral criticality and identify non-normality as an independent route to deterministic chaos.