Simpson DJW2024-10-232024-10-232024-06Simpson DJW. (2024). The necessity of the sausage-string structure for mode-locking regions of piecewise-linear maps. Physica D: Nonlinear Phenomena. 462.0167-2789https://mro.massey.ac.nz/handle/10179/71829Piecewise-smooth maps are used as discrete-time models of dynamical systems whose evolution is governed by different equations under different conditions (e.g. switched control systems). By assigning a symbol to each region of phase space where the map is smooth, any period-p solution of the map can be associated to an itinerary of p symbols. As parameters of the map are varied, changes to this itinerary occur at border-collision bifurcations (BCBs) where one point of the periodic solution collides with a region boundary. It is well known that BCBs conform broadly to two cases: persistence, where the symbolic itinerary of a periodic solution changes by one symbol, and a nonsmooth-fold, where two solutions differing by one symbol collide and annihilate. This paper derives new properties of periodic solutions of piecewise-linear continuous maps on Rn to show that under mild conditions BCBs of mode-locked solutions on invariant circles must be nonsmooth-folds. This explains why Arnold tongues of piecewise-linear maps exhibit a sausage-string structure whereby changes to symbolic itineraries occur at codimension-two pinch points instead of codimension-one persistence-type BCBs. But the main result is based on the combinatorical properties of the itineraries, so the impossibility of persistence-type BCBs also holds when the periodic solution is unstable or there is no invariant circle.(c) 2024 The Author/sCC BY 4.0https://creativecommons.org/licenses/by/4.0/Piecewise-smoothBorder-collision bifurcationInvariant circleArnold tongueJournal article10.1016/j.physd.2024.1341421872-8022journal-article134142S0167278924000939