On the alleged chaos in periodically forced traveling-wave reductions of fractional WBBM models: a quantitative re-examination

Abstract

A large literature applies a standard pipeline to fractional nonlinear evolution equations -- traveling-wave reduction to a planar Hamiltonian system, addition of periodic forcing, and visual inspection of phase portraits -- to claim bifurcations, quasi-periodicity, and chaos, often without any quantitative diagnostic. This is particularly problematic because the reduced systems are undamped, near-integrable oscillators for which KAM theory confines chaos, if present at all, to thin stochastic layers, so the burden of proof for a chaos claim is high. We analyze these pitfalls and quantitatively re-examine a representative example, Ullah, Ali and Roshid's study of the second fractional Wazwaz-Benjamin-Bona-Mahony (WBBM) model [PLoS ONE 19(7): e0307565 (2024)], using analytical arguments and four independent diagnostics (Benettin largest Lyapunov exponents in two configurations, a separation-growth test over 3x10^6 time units, stroboscopic Poincare sections, spectral analysis). We find that (i) its linear stability analysis concludes "unstable propagation" from a dispersion relation that is real for every real wave number: all modes are neutrally stable, and the reported singularity is a pole, not a temporal instability; (ii) its unforced "quasi-periodic" system is necessarily periodic, being a planar autonomous Hamiltonian system; (iii) all four chaos assertions fail every diagnostic -- exponents bounded by 5x10^{-7}, linear-in-time separation growth, smooth closed invariant Poincare curves -- while the quasi-periodic assertion is confirmed; (iv) its equilibrium classification and phase portraits are correct. We also document a false positive of the Gottwald-Melbourne 0-1 test on a regular orbit and show that stroboscopic parameter sweeps of conservative systems yield "chaotic-looking" diagrams for purely regular tori, closing with a checklist of standards for chaos claims.