Non-Normal Route to Chaos

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.

Figures (3)

• Figure 1: Bifurcation portraits of the NNSRT map \ref{['eq:NNSRT']}-\ref{['eq:A_param']} shown through coordinate-wise maxima $x_{\max}$ (a), $y_{\max}$ (b), and $z_{\max}$ (c) versus the normalised non-normality index $K/K_c$ for $\alpha=0.7$, $\beta=0.2$, $\alpha_z=0.5$, and $\epsilon=10^{-3}$. The vertical dashed line marks the analytically predicted threshold $K/K_c=1$.
• Figure 2: Transition to chaos without spectral criticality for the dynamical system \ref{['eq:NNSRT']}-\ref{['eq:A_param']} with the same parameter values as in Fig. 1. Solid line: maximal Lyapunov exponent $\lambda_1$ versus $K/K_c$. Dashed line: $\ln(\rho)$, where $\rho$ is the spectral radius of the Jacobian $\mathbf{Df}(\mathbf{x}_n,z_n)$ along the trajectory, (pointwise spectral contraction). Dotted line: $\ln(\sigma)$, where $\sigma$ is the largest singular value of $\mathbf{Df}(\mathbf{x}_n,z_n)$, quantifying transient amplification enabled by non-normality; the shaded band indicates its variability along the orbit. The vertical line marks $K=K_c$ at which $\lambda_1$ crosses zero: chaos emerges while eigenvalues remain uniformly inside the unit disk.
• Figure 3: Fractal-dimension diagnostics of the attractor versus the normalised non-normal index $K/K_c$ for the dynamical system \ref{['eq:NNSRT']}-\ref{['eq:A_param']} with the same parameter values as in Fig. 1. The thick black curve shows the box-counting fractal dimension BC(xyz) of the attractor in the $(x,y,z)$ space, while the colored curves show the Kaplan-Yorke dimension (KY), the Box-Counting fractal dimension BC (xy) of the projection of the attractor in the $(x,y)$ plane, and the correlation dimension $D_2$. Insets show representative $(x,y)$ projections of the attractor below ($K/K_c=0.92$) and above ($K/K_c=1.64$) the transition.