On distinguishing genuine from spurious chaos in planar singular and nonsmooth systems: A diagnostic approach

TL;DR

This paper develops a diagnostic protocol combining regularization and numerical analysis to reliably distinguish genuine chaos from artifacts in planar singular and nonsmooth systems.

Contribution

It introduces a standardized method for identifying true chaos in low-dimensional nonsmooth systems, addressing issues caused by singularities and discretization.

Findings

Regularization restores smoothness, validating Poincaré-Bendixson theorem applicability.

Recomputed bifurcation sequences show convergence to Feigenbaum's constant.

Confirmed robust chaos through Lyapunov exponents, spectra, and fractal dimensions.

Abstract

We present a rigorous reassessment of chaotic behavior in two-dimensional autonomous systems with singular or nonsmooth dynamics. For the Cummings-Dixon-Kaus (CDK) model, we show that blow-up regularization restores smoothness and renders the hypotheses of the Poincar\'e-Bendixson theorem applicable, thereby excluding chaotic attractors away from the singular set. We prove topological equivalence between the original and regularized flows on annular domains, ensuring that no spurious invariant sets are introduced by desingularization. In contrast, for a nonsmooth system with a $|x|$ term, we recompute the entire period-doubling cascade, obtain a seven-term sequence of bifurcation values converging to Feigenbaum's constant, and confirm robust chaos through positive Lyapunov exponents, broadband spectra, and fractal dimension estimates. As a main outcome, we propose a diagnostic protocol integrating regularization, numerical refinement, and invariant-set criteria. This protocol provides a reproducible standard for distinguishing genuine planar chaos from artifacts caused by singularities or discretization, and offers a benchmark for future studies of low-dimensional nonsmooth systems.