Affine-Periodic Solutions by Asymptotic Method

Abstract

We consider the existence of affine-periodic solutions to the nonlinear ordinary differential equation: 0.1 $\label {ae} x^{\prime }=f(t,x) $ in ℝn, where f is continuous and ensures the existence and uniqueness of solutions with respect to initial conditions, and there exist T > 0 and Q ∈ GL(n) such that: 0.2 $\label {me} f(t+T,x)=Qf(t,Q^{-1}x) \quad \forall (t,x)\in \mathbb R\times \mathbb R^{n}. $ We utilize the asymptotic method to study the existence of Q-affine T-periodic solutions of Eqs. 0.1–0.2. Affine-periodic solutions exhibit a rich variety of complex spatiotemporal pattern, might be periodic, anti-periodic, quasi-periodic, and even unbounded spiral motions.