Systematic designing of bi-rhythmic and tri-rhythmic models in families of Van der Pol and Rayleigh oscillators

The paper derives first‑order averaged amplitude equations for four LLS oscillator families and states parity‑dependent upper bounds on the number of limit cycles; its worked examples (including the five‑cycle S–U–S–U–S tri‑rhythmic cases) match the model’s equations, parameter choices, root counts, and stability patterns exactly. For example, the extended Van der Pol amplitude equation ṙ = (μ r/1024)[−21δ r^10 + 28γ r^8 − 40β r^6 + 64α r^4 − 128 r^2 + 512] in the paper coincides with the model’s B1, and the reported radii/stabilities at α=0.144, β=0.005, γ=5.862×10^−5, δ=2.13×10^−7 agree to the stated digits and S–U–S–U–S pattern . Likewise, the alternative Van der Pol (ẋ^3 damping) amplitude equation ṙ = (μ r^3/2048)[−9δ r^10 + 14γ r^8 − 24β r^6 + 48α r^4 − 128 r^2 + 768] and its five-cycle example at α=0.139317, β=0.00454603, γ=2.402×10^−5, δ=3.058×10^−8 coincide with the model’s B2/C2 values and stability sequence . The Rayleigh cases (standard and ẋ^3) also match exactly, including amplitude equations and the tri‑rhythmic parameter sets with the same radii and alternation of stability . Conceptually, both rely on first‑order Krylov–Bogolyubov averaging and on small μ; the model adds a clear explanation that G(x) does not contribute to ṙ at O(μ) and a monomial‑parity argument for the degree bound, which strengthens the paper’s heuristic counting rules stated in its general scheme .