Are there any reasons why a system would exhibit chaotic behavior? Or is this something only found through numerical modelling or experimental testing?
For example, the simple forced, damped pendulum or the duffing oscillator. Were these experimented on and it was found that they were sensitive to initial conditions, and then examined further to prove the 3 chaotic properties and finally deemed to be chaotic? Or is there something physical about them that gives away a possibility to chaos?
If it is the former, how would we determine chaotic systems? Just trial and error until all 3 properties are proven?
Best Answer
There is a bit to disentangle here, so this may seem like a somewhat roundabout answer, but we will get to your questions eventually:
First of all, chaos is a property of deterministic and bounded dynamical systems, which in turn are mathematical constructs. The general mechanisms of chaos in such systems are generally well understood, but are nothing that I can explain within the scope of this answer. Moreover, for such systems, chaos is the norm, more specifically the non-chaotic regions of parameter space become increasingly small with increasing complexity of the system. So, in some sense the interesting question is rather why some (mathematical model) systems are not chaotic and why we can derive valuable insights from those. There are some necessary ingredients required for chaos¹, but these can be found in every real system.
All definitions of chaos employ criteria that are inherently mathematical, in particular employing infinitesimal or infinite quantities. As such, they cannot be applied to natural systems. The one aspect of chaos that most definitions agree upon is the sensitivity to initial conditions, which can be determined for natural (and simulated) systems in good approximation (see this answer of mine for some details). Other properties like topological transitivity and dense periodical orbits are not reasonably accessible in nature at all. Rigorously proving chaos in nature is as impossible as proving whether the ratio of the proton and electron mass is a rational number.
With that as a background, let’s turn to your specific questions:
As elaborated above, chaos is more or less the default. Rather, there are reasons why chaos may be absent in a model.
For some mathematical models, chaoticity according to some definition was rigorously proven. For most model systems, it is shown numerically (and not rigorously). For natural systems, you can only show the sensitivity to initial conditions experimentally, either by reproducing initial conditions and observe their divergence (example by me) or analysing data from the system. Alternatively, it may satisfy you to create a model for the system that explains the relevant observed behaviour and show that this model is chaotic. After all, showing chaos for its own sake is pretty pointless in nature, as you will always find always find chaos if you look closely enough, and the question is rather how to best describe the relevant properties of the system.
I am not exactly sure what three properties you are referring to, but as elaborated above only sensitivity to initial conditions is usually experimentally accessible.
¹ three dynamical variables (in continuous-time systems, which real physical systems are), non-linearities