Does chaos theory occur in quantum mechanics? Or in any non-newtonian physics? Apart from perhaps thermodynamics?
In this PhD thesis (unfortunately in German :-/, a follow up paper can be found here), it is shown how for an action of a field theory containing even and Grassman fields, the renormalization group equation can be solved numerically (after expanding the action in derivatives and the fields). To investigate the corresponding renormalization group flow in the respective coupling space, analogous methods known to investigate trajectories in the phace space of a nonlinear dynamic system can be applied in this case too. A trajectory in this coupling space starting with certain inital values of the coupling constants at the high energy cutoff evolving in the course of renormalization time (which is related to the lenght or energy scale considered) can be considered as analogue to a trajectory in phase space, starting from an initial condition and evolving in the course of time.
As described in this paper, the coupling constants in a renormalizable field theory can not only flow to, bypass, spiral in/out or circle around isolated fixed points, but a chaotic behaviour of the renormalization group flow could occur too. This could have potential applications in different fields such as spin glasses, neural networks, or even string theory.
Since time evolution is linear for a quantum system, this rules out classical chaos in terms of hypersensivity to initial conditions. The question of how quantum theory could explain classical chaos is adressed from the perspective of decoherence. This interesting document may be of some help: http://www.iqc.ca/publications/tutorials/chaos.pdf
There is a SHM wave equation based on Schroedinger's equation and Maxwell's equation for an elliptical wave function which produces chaos (Langtons ant). I guess this answers the question about SHM and chaos.