Asymptotic pseudo-periodicity of solutions for nonlinear fractional evolution equations

Abstract

This work focuses on the dynamic behaviors, particularly the asymptotic periodicity, of solutions to nonlinear fractional-order dynamical systems modeled by Caputo fractional evolution equations in Banach spaces. We propose a new class of weighted pseudo $S$-asymptotically $(ξ,τ,a)$-periodic functions, which unifies well-established function types such as standard periodic functions and $S$-asymptotically periodic functions as its special instances. We then systematically derive the fundamental properties of this new function family, and rigorously prove its completeness, corresponding convolution theorems, and nonlinear composition rules. Building on these newly established results, we demonstrate that under reasonable hypotheses, the mild solutions to a broad class of nonlinear fractional-order dynamical systems --- both delay-free and with finite delay --- in Banach spaces admit the (weighted pseudo) $S$-asymptotically $(ξ,τ,a)$-periodic property, which further implies that no bifurcation or chaotic behavior emerges in these fractional-order systems. The theoretical conclusions are finally validated by illustrative examples describing memory-dependent complex dynamic processes, supplemented with supporting numerical simulations. Additionally, the presented weighted pseudo $S$-asymptotically $(ξ,τ,a)$-periodic framework can be extended to functions with first-kind discrete discontinuities, so that systems with discontinuous nonlinear terms can be handled via the same analytical paradigm.