Existence of delay-induced switched states near the balance point

Let $f\in C^{1}(\mathbb{R})$ satisfy conditions (H1)–(H3), and let $\beta$ be the balance point. Consider initial histories satisfying $x(\theta)\in[\chi^{-},\chi^{+}]$ for $-1\leq\theta\leq 0$, with at least one sign change. Switched-state conjecture. For all $\varepsilon>0$ and $\eta>0$ sufficiently small, all initial conditions with $y(0)$ sufficiently close to $\beta$ and such an initial history lead to a delay-induced switched state. The conjecture proposes that the stabilization mechanism established in the paper produces delay-induced switched states near the balance point; the source gives rigorous supporting analysis and numerical evidence but no proof of this full assertion.