STRICTLY NEGATIVE POINCARÉ DISPLACEMENT AND POWER-LAW DECAY IN A FAMILY OF GENERALIZED ABEL EQUATIONS

Аннотация

We study a one-parameter family of generalized Abel equations with trigonometric
coefficients and quadratic and quartic nonlinearities. For 0 < μ ≤ 2, an exact integral identity establishes
strictly negative Poincaré displacement throughout its domain, ruling out positive periodic solutions. Every
positive solution defined over the first complete period extends globally forward and converges to zero. An
expansion of the return map near the origin then yields algebraic decay with exponent 1/4 and an explicit
parameter-dependent leading coefficient. The proof combines a global displacement-sign estimate with a
local return-map expansion to determine the decay rate.