Abstract
Circle maps frequently arise in mathematical models of physical or biological systems. Motivated by Cherry flows and ‘threshold’ systems such as integrate and fire neuronal models, models of cardiac arrhythmias, and models of sleep/wake regulation, we consider how structural transitions in circle maps occur. In particular, we describe how maps evolve near the creation of a discontinuity. We show that the natural way to create discontinuities in the maps associated with both threshold systems and Cherry flows results in a singularity in the derivative of the map as the discontinuity is approached from either one or both sides. For the threshold systems, the associated maps have square root singularities and we analyse the generic properties of such maps with gaps, showing how border collisions and saddle-node bifurcations are interspersed. This highlights how the Arnold tongue picture for tongues bordered by saddle-node bifurcations is amended once gaps are present. We also show that a loss of injectivity naturally results in the creation of multiple gaps giving rise to a novel codimension two bifurcation.
Highlights
Circle maps arise in many mathematical models of reallife systems
We further show that in threshold systems there is a natural route to creating maps with multiple gaps that result in a novel codimension two bifurcation similar to the ‘big bang’ bifurcation seen in other contexts [26]
We have considered transitions from continuity to discontinuity, monotonicity to non-monotonicity and the creation of multiple gaps, and have described how these transitions can alter the bifurcations and dynamics of circle maps
Summary
Circle maps arise in many mathematical models of reallife systems. Our motivation for this paper comes from two classes of models. As parameters are varied, a pair of stationary points is created by saddle-node bifurcations the resulting flow is known as a Cherry flow [11] These can generate return maps which have either discontinuities or regions where the map is not defined [12]. Less is known about the details of the dynamics of non-injective discontinuous maps in general ( the techniques of kneading theory do apply), partly because it is less clear what results would be useful without further context Both threshold systems and the transition from Poincaré flows to Cherry flows provide natural settings to consider the transition from continuous circle maps to piecewise continuous circle maps with discontinuities.
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