The necessity of the sausage-string structure for mode-locking regions of piecewise-linear maps
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Date
2024-06
Open Access Location
Authors
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Publisher
Elsevier BV
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(c) 2024 The Author/s
CC BY 4.0
CC BY 4.0
Abstract
Piecewise-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.
Description
Keywords
Piecewise-smooth, Border-collision bifurcation, Invariant circle, Arnold tongue
Citation
Simpson DJW. (2024). The necessity of the sausage-string structure for mode-locking regions of piecewise-linear maps. Physica D: Nonlinear Phenomena. 462.