Impact of on-site potentials on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>q</mml:mi></mml:math> breathers in nonlinear chains

On-site potentials are ubiquitous in physical systems and strongly influence their heat transport and energy localization. These potentials will inevitably affect the dynamical properties of q breathers (QBs), defined as periodic orbits exponentially localized in normal mode space. By integrating on-site terms into the Fermi-Pasta-Ulam-Tsingou-β system, this work utilizes numerical simulations and Floquet analysis to systematically explore the influence of on-site potentials on QB stability. For most QBs, except those at the phonon band edges, the instability is primarily governed by parametric resonance and effectively described by coupled Mathieu equations. This approach provides a theoretical expression for the instability thresholds, which aligns well with numerical results. The instability threshold is highly sensitive to the seed mode, in stark contrast to systems without on-site potentials. In addition, a systematic stability analysis across three-dimensional parameter space shows that quartic intersite nonlinearity monotonically enhances QB instability, while quartic on-site nonlinearity yields nonmonotonic effects, in particular, an initial destabilization followed by restabilization at high amplitudes. The quadratic on-site potential uniformly promotes stability. Furthermore, these instability phase diagrams highlight both the individual contributions and combined effects of each possible component in regulating the QB dynamics. These findings offer valuable insights into QB stability and the manipulation of localized excitations in diverse physical systems with on-site potentials.