Nearly periodic slow motion in deterministic fully coupled averaging

Let $w_{-}(r)$ and $w_{+}(r)$ be strictly increasing and decreasing functions, respectively, satisfying

$$R_{12}(w_{-}(r))=R_{21}(w_{+}(r))=r.$$

Suppose that $w_{-}(\frac{}{\lambda})=w_{+}(\lambda)=w^*$ for some $\lambda>0$, with $w_{-}(r)<w^*<w_{+}(r)$ for $r<\lambda$. Assume that $\delta\to0$ and $\varepsilon\to0$ so that

$$\lim_{\varepsilon,\delta\to0}\varepsilon\ln(\delta\varepsilon)=-\rho>-\lambda.$$

For the slowest motion $\tilde W^{\varepsilon,\delta}_{w,x,y}$, and for every $w,x$, there exists $t_0>0$ such that $\tilde W^{\varepsilon,\delta}_{w,x,y}(t+t_0)$, $t\geq0$, converges weakly as a random process on $({\mathcal W},m_{\mathcal W})$ to a periodic function $\psi(t)$ satisfying $\psi(t+T)=\psi(t)$, where

$$T=T(\rho)=\int_{w_{-}(\rho)}^{w_{+}(\rho)}\frac{dw}{|\bar A_1(w)|}+\int_{w_{-}(\rho)}^{w_{+}(\rho)}\frac{dw}{|\bar A_2(w)|}.$$

This describes the proposed nearly periodic behavior of the slowest variable in the deterministic fully coupled averaging regime; the statement is presented as a conjectural prediction, and no resolution is supplied in the source.