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

Martha Alvarez Ramírez

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 Poincaré-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.

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

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 Poincaré-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 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.

Paper Structure

This paper contains 5 sections, 1 theorem, 9 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Apparent chaos and regularization in the Cummings-Dixon-Kaus model: From heuristics to rigorous analysis
  3. A two-dimensional nonsmooth model exhibiting chaos
  4. Bifurcation Analysis
  5. Conclusions

Key Result

Proposition 1

Let $X$ denote the vector field of the original (singular) planar CDK system eq_chaos2d and $\widetilde{X}$ the vector field of the regularized polynomial system eq:CDK_poly_reduced. For any annulus the flows of $X$ and $\widetilde{X}$ are topologically equivalent: there exists a homeomorphism $h:A\to A$ mapping the orbits of $X$ onto those of $\widetilde{X}$ and preserving their $\alpha$- and $\

Figures (7)

  • Figure 1: Phase portraits of the three-dimensional CDK system for representative parameter ratios. The projections onto different coordinate planes illustrate the irregular dynamics and sensitivity to initial conditions near the singularity, while the time series highlights the erratic temporal behavior.
  • Figure 2: Phase portrait and some trajectories of the two-dimensional system \ref{['eq_chaos2d']} in the $(x,z)$-plane for parameter values $a = 0.6$ and $b = 0.7$.
  • Figure 3: Phase portrait of the regularized two-dimensional system \ref{['eq:CDK_poly_reduced']} in the $(x,z)$-plane for parameter values $a = 0.6$ and $b = 0.7$. The dynamics reveal elliptic sectors with infinitely many homoclinic loops, showing how regularization removes spurious chaotic behavior while preserving recurrent complexity.
  • Figure 4: Phase portrait and representative trajectories of the two-dimensional system \ref{['eq_chaos2d']} projected onto the $(x,y)$-plane for $a = 10$ and $b = 10$. The plots display a butterfly-shaped structure characterized by positive Lyapunov exponents, fractal geometry, and aperiodic switching, thereby supporting its interpretation as a strange attractor in the sense of dynamical systems theory.
  • Figure 5: Time series of $x(t)$ and $y(t)$ for system \ref{['eq_chaos2d']}, illustrating sustained aperiodic oscillations.
  • ...and 2 more figures

Theorems & Definitions (2)

  • Proposition 1
  • proof : Sketch of proof