The hidden unstable orbits of maps with gaps

Piecewise-continuous maps consist of smooth branches separated by jumps, i.e. isolated discontinuities. They appear not to be constrained by the same rules that come with being continuous or differentiable, able to exhibit period incrementing and period adding bifurcations in which branches of attractors seem to appear ‘out of nowhere’, and able to break the rule that ‘period three implies chaos’. We will show here that piecewise maps are not actually so free of the rules governing their continuous cousins, once they are recognised as containing numerous unstable orbits that can only be found by explicitly including the ‘gap’ in the map’s definition. The addition of these ‘hidden’ orbits — which possess an iterate that lies on the discontinuity — bring the theory of piecewise-continuous maps closer to continuous maps. They restore the connections between branches of stable periodic orbits that are missing if the gap is not fully accounted for, showing that stability changes must occur in discontinuous maps via stability changes not so different to smooth maps, and bringing piecewise maps back under the powerful umbrella of Sharkovskii’s theorem. Hidden orbits are also
vital for understanding what happens if the discontinuity is smoothed out to render the map continuous and/or differentiable.