Periods 1 + 2 imply chaos in steep or nonsmooth maps

Abstract

Maps with discontinuities can be shown to have many of the same properties of continuous maps if we include hidden orbits - solutions that include points lying on a discontinuity. We show here how the well known property that ‘period 3 implies chaos’ also applies to maps with discontinuities, but with a twist, namely that if a map has a hidden fixed point and a hidden period 2 orbit, then it has hidden orbits of all periods, or concisely ‘period 1 + 2 implies chaos’. More precisely we show that for a one-dimensional map defined on an interval, if there exist hidden orbits of periods p and q, then there exist hidden orbits of all periods ap + bq, for any a,b∈N. To better understand this result we show that, by smoothing or ‘regularising’ across the discontinuity, these hidden orbits become regular unstable orbits, and the familiar results for continuous maps (like ‘period 3 implies chaos’) are restored. We also show examples of hidden orbits in some applications of oscillators, and consider how these results can be used to approximate the behaviour of continuous maps with steep segments.