Abstract
We show how to formally identify chaotic attractors in continuous, piecewise-linear maps on . For such a map f, this is achieved by constructing three objects. First,
is trapping region for f. Second,
is a finite set of words that encodes the forward orbits of all points in
. Finally,
is an invariant expanding cone for derivatives of compositions of f formed by the words in
. The existence of
,
, and C implies f has a topological attractor with a positive Lyapunov exponent. We develop an algorithm that identifies these objects for two-dimensional homeomorphisms comprised of two affine pieces. The main effort is in the explicit construction of
and C. Their existence is equated to a set of computable conditions in a general way. This results in a computer-assisted proof of chaos throughout a relatively large region of parameter space. We also observe how the failure of C to be expanding can coincide with a bifurcation of f. Lyapunov exponents are evaluated using one-sided directional derivatives so that forward orbits that intersect a switching manifold (where f is not differentiable) can be included in the analysis.
Acknowledgements
The author thanks Paul Glendinning whose May 2017 visit to Massey University stimulated many of the ideas presented here.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Notes
1 To the left of , Figure contains 173779 red pixels and 21462 white pixels, giving
.
2 The Hausdorff distance between sets and
is defined as
3 Equation (Equation64(64)
(64) ) is a version of the well-known formula
for the sine of the angle between
.