In modeling biologic rhythmic actions as limit-cycles, it has been reported that a negative Duffing term (i.e. -x^3) represents decreasing of variability near reversal points (or softening spring). The question is what type of "variability" we mean here? Spatial? Temporal? Any other type?
How could it be proved or visualized based on Duffing equations or any other method that variability decreases near reversal points?
Space-time, actually. You could, for example, see the influence of a x^3 term by creating a Hooke plot. That plot is position~aceleration. A completely straight lined plot means the oscillator (limit-cycle) reduces to a harmonic oscillator; one only with a linear stiffness term. On the other hand, a x^3 term means the angular frequency , or velocity if that's easier to imagine, in the cycle is not constant.
Now, the problem faced by your question is it's slightly under defined. Are you basing your question on an analytic model such as x"+b*x'+c*x+d*x^3=0 ? (Note this has no nonlinear damping. So it won't give you a limit cycle, per se, but a damped out cycle.). If you are, the variability means nothing because it's a noise free system. However, try simulating this an adding a Gaussian noise term ( I think), and then making the hooke plot. If there is less variability around the endpoint, this is I believe , as you might guess, due to a softening spring. This implies a slowing down as an endpoint is reached. This also predicts an increased deceleration in a Hooke plgot, which is where the 3rd order term comes from. See the Hooke plot I attached (it's my data): It is several cycles of a handheld pendulum being swung, with the average superimposed. Arrows are added to where the variability would be expected to start decreasing.
If you're familiar with Fitts' law, look up Mottet and Bootsma (1999) "Dynamics of goal-directed rhythmical aiming", I believe. They fit this type of model to movement data and give a good description.
P.s., I wrote this quickly, but if you have more questions (model fitting, etc...) let me know.
Good Luck:
Justin F.
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