Hidden dynamics for piecewise smooth maps

Abstract

We develop a hidden dynamics formulation of regularisation for piecewise smooth maps. This involves blowing up the discontinuity into an interval, but in contrast to piecewise smooth flows every preimage of the discontinuity needs to be blown up as well. This results in a construction similar to classic approaches to the Denjoy counterexample.

Periods 1 + 2 imply chaos in steep or nonsmooth maps

“…This map is shown in figure 12, and was obtained by simulation, taking an initial point (u 1 , u 2 , u 3 ) = (0.1, 0.2635, 0.3371) near the attractor, and then varying initial points m n , to calculate the return point m n+1 . We note that this is only an approximation of the higher dimensional dynamics, but it was shown in [16] to be sufficient to understand the qualitative periodic behaviour of the four-dimensional model (17). The discontinuity in this map is due to the two qualitatively different return paths observable in figure 11: the left branch of the map describes orbits that make a loop above u 1 = 1 10 and thereby allow longer for m to grow between divisions, the right branch describes orbits with no loop, and the discontinuity occurs where an orbit forms a loop that grazes the threshold u 1 = 1 10 .…”

“…Consider the connected map x n+1 = f (x n ), from (2), to be the limit of a continuous map x n+1 = g(x n , ω) as some ω tends to infinity. A general method to define such 'regularisations' of connected maps was given in [17]. Essentially it replaces a vertical jump in (2) by a steep segment with approximate slope ±ω, and we will define such a function below.…”

“…Other works have attempted to extend Sharkovkii's theorem to maps with discontinuities in other ways. Recently in [17] it was shown that a discontinuous map can be blown-up into a continuous one, in a manner that preserves kneading invariants and hence periodicities, implying that the discontinuous map must also contain the cycles guaranteed by the Sharkovskii ordering.…”

“…It was shown in [15], for example, that by introducing hidden orbits, the bifurcations of discontinuous maps become qualitatively similar to those of continuous maps. In [17] it was shown in general how one can regularise a discontinuous map, by first connecting it with a vertical branch, and then blowing-up the pre-images of the discontinuity in a manner that preserves kneading invariants and hence periodicities. This implies that the discontinuous map must also contain the cycles guaranteed by the Sharkovskii ordering, and that 'period three implies chaos' should apply to discontinuous maps.…”

“…In section 5 we saw moreover an example of how, using the results of sections 2 and 3, hidden orbits can be used to infer the existence of regular orbits, and even to infer the existence of orbits of continuous maps with steep segments. Therefore, though hidden orbits are inherently unstable, and might not be readily observable, as with other unstable orbits this does not preclude their usefulness, whether seeking to understand the periodicities of a map as in this paper, to understand the unstable objects propping up their otherwise 'gappy' bifurcation diagrams as in [15,18], or to understand their relation to continuous maps as in [17]. It is not so much the hidden cycles themselves that have significance as physical trajectories, but rather the points outside the discontinuity that they connect into a hidden periodic structure, strung out through pre-images of the point of discontinuity, that plays a crucial role in organising the map's dynamics, both hidden and regular.…”