Smoothing tautologies, hidden dynamics, and sigmoid asymptotics for piecewise smooth systems

TL;DR

This paper explores how nonlinear switching terms in piecewise smooth systems, modeled via sigmoid functions, reveal hidden dynamics and attractors that are overlooked by traditional instantaneous switch models.

Contribution

It extends Filippov's sliding mode method to include nonlinear effects through sigmoid-based switch modeling, uncovering hidden behaviors in real systems.

Findings

Nonlinear effects can significantly alter circuit behavior

Hidden attractors can exist within switching surfaces

Sigmoid series effectively model complex switch dynamics

Abstract

Switches in real systems take many forms, such as impacts, electronic relays, mitosis, and the implementation of decisions or control strategies. To understand what is lost, and what can be retained, when we model a switch as an instantaneous event, requires a consideration of so-called hidden terms. These are asymptotically vanishing outside the switch, but can be encoded in the form of nonlinear switching terms. A general expression for the switch can be developed in the form of a series of sigmoid functions. We review the key steps in extending the Filippov's method of sliding modes to such systems. We show how even slight nonlinear effects can hugely alter the behaviour of an electronic control circuit, and lead to `hidden' attractors inside the switching surface.