I am not a physicist, I am looking for a non-technical explanation. Articles such as this one seem to hint at the fact that "macro reality" regulated by classical mechanics is somehow a pattern emerging out of quantum-level chaos. Is that correct? Can anything at a quantum level be defined as a complex system? I apologize if there are inaccuracies in the way the question is formulated, feel free to tear it apart and answer as it fits best.
There is no mention of chaos in the link you give. It is your description of what you read in the article using the everyday meaning of chaos.
Chaos theory, copying from wikipedia is defined mathematically and belongs to the framework of classical physics.
Small differences in initial conditions (such as those due to rounding errors in numerical computation) yield widely diverging outcomes for such dynamical systems, rendering long-term prediction impossible in general.This happens even though these systems are deterministic, meaning that their future behavior is fully determined by their initial conditions, with no random elements involved. In other words, the deterministic nature of these systems does not make them predictable.This behavior is known as deterministic chaos, or simply chaos.
Quantum mechanics is an underlying framework from which the classical frameworks emerge smoothly, i.e. it can be proven mathematically.
Chaos theory is a bit like thermodynamics:
The theory of thermodynamics has beautiful equations and solutions for them, describing and predicting the behavior of bulk matter which is described with quantities like temperature and pressure etc. A lot of our technological civilization is dependent on these solutions. Nevertheless, when it was found that bulk matter was composed of atoms, statistical mechanics was developed that showed that the thermodynamic quantities are averages and integrals over the behavior of many particle systems.
In a similar way, the classical fields emerge from the quantum mechanical solutions that describe the behavior of atoms and light at the microworld level.
Quantum mechanics is not chaotic, but probabilistic. It has strict solutions of the equations determining the behavior of particles and fields, but these solution describe and predict probabilities of observation. The wave nature of the underlying reality is a probability wave , i.e. the probability of finding a particle at (x,y,z) at time t fluctuates according to the solution of the equations. It is not a matter wave in the sense of a wave on water, where one can measure the energy carried at different (x,y,z)s . Whenever an elementary partice is measured it appears at a strict (x,y,z) as a point with all its mass. Atoms are composites of elementary particles and follow the same equations and behavior.
Thus the relationship of quantum mechanics with deterministic chaos ( which is the chaos used in physics) is similar to the relationship a Picasso picture has to the atoms and molecules composing it. Deterministic chaos paints a mathematically different view of many particle systems to the one of thermodynamics, and allows to explain and predict macroscopic behaviors of classical systems.
That's an interesting question, and just off the top of my head you might want to look at "Fractals, Chaos, and Power Laws" which discusses chaos in a wide variety of contexts, including quantum mechanics. I agree with the above response in that the difficulty of uniting chaos with quantum mechanics is that the former is deterministic (somewhat surprisingly) while the latter is probabilistic. One idea that seems interesting to me, although I'm not sure if there is any way to prove it, is that quantum mechanics may be deterministic in higher dimensions. This relates to Kaluza-Klein theory. David Bohm had his own interpretation of quantum theory, although he remains a black sheep among QM theorists.
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).
It has Fermi statistics emerging from some of the variables and also has a beautiful thermodynamic Hamiltonian ($\theta$). It has the maths and the Octave code. This is one dimensional; when done in three dimensions some amazing properties emerge. I guess this answers the question about SHM and chaos.