Abstract: Probably the most widely accepted definition of chaos is the one by Devaney, which we will call Devaney-chaos. It consists of three conditions, (1) the sensitive dependence upon the initial condition, (2) the topological transitivity, and (3) the dense distribution of the periodic orbits. The third condition is often omitted for being too stringent. The purpose of our research is to investigate how the first two characteristic properties of Devaney-chaos are affected by the presence of the discontinuity.
Devaney-chaos is more inclusive than most of the competing notions of chaos, especially when the dynamics includes singularity. Recent discoveries by Goetz and Buzzi on the discontinuous dynamical systems include that the piecewise isometric dynamical systems, which are partly inspired by the digital signal processing and Hamiltonian dynamics, can generate complicated orbit structure, even though their Lyapunov-exponents and topological entropies are 0. Consequently, neither Lyapunov-chaos nor topological chaos can be applied to explain the complex behavior of the piecewise isometric dynamics. Goetz also proved that Smale-chaos fails to apply as well. Devaney-chaos, on the other hand, proved to be a useful tool, at least for some special cases, as exemplified by some of the author’s earlier contributions.
We prove that Devaney-chaos can be used to successfully characterize the chaos that are generated by the discontinuity in general, if appropriate adjustments are made. Also, we show that the straightforward application of Devaney’s conditions is too inclusive, when the system is discontinuous, consequently necessitating the afore-mentioned adjustments.