Dynamic invisibility of implicit and nested state-dependent delays in a class of externally forced delay equations

Delayed feedback is a fundamental mechanism in many systems and is often modeled by delay differential equations (DDEs). While delays are frequently assumed to be constant, in realistic feedback processes they may depend on the state and even be defined implicitly or in a nested fashion by the feedback relation itself. In this work, we address the question of how such implicit and nested state-dependent delays affect the observed behavior of a class of externally forced systems with negative switched feedback. Our motivating example and principal case study is the piecewise-smooth limiting form of a conceptual DDE model for the El Niño-Southern Oscillation (ENSO) with a negative feedback term subject to implicit state-dependent delay. We prove, for a wider prototypical class of DDEs with switched negative feedback and more general forcing, that (subject to an explicit condition on the parameters) any finite level of nested state-dependent delay and the limiting implicit delay are dynamically invisible: they all generate exactly the same trajectory, namely, that of the corresponding constant-delay DDE. We also discuss how the delayed argument functions organize the temporal order of reactions (TOR) in this regime, and show that a more stringent condition is required to ensure TOR for nested state-dependent delays.