How the frequency of oscillation is determined in a multi-resonance oscillator circuit?
I agree with @TimWescott answer, however he missed one aspect of it which I expanded upon in this answer to a related question.

All linear oscillators are special cases of dynamical systems in which some constraints (namely Barkhausen Criteria) have been put in place to simplify the differential equations, analysis, and design. One assumption that is sometimes taught alongside those criteria, is that for every other possible oscillation frequency the loop gain must be strictly less than one. But in general these are just a set of behaviors that lie in a much wider parameter space in which chaos reigns.

Any dynamical system that is put together as an oscillator and can have more than one oscillation frequency (thus third-order or more) would have a parameter space that would follow a well-known path to chaos which includes period-doubling and complex oscillation patterns (e.g., squeging). As one of the defining characteristics of chaos is the exponential sensitivity to initial conditions and parameters, any minor parameter variation (e.g., temperature) or noise will alter the behavior making it completely unpredictable.