What is the importance of the Duffing Differential Equation?

[nonequilibrium]

Hello,

Is there any significance to it past its modelling of a damped and driven oscillator with a non-linear (anti-)restoring force? For example, apparently there are cases where the k in $$mx''+\delta x' + kx + \beta x^3 = F cos(\omega t)$$ is negative. What kind of physical situation is associated with this?

 

[Filip Larsen]

You have written the equation for a linear spring system, that is, written it like a "degenerated" version Duffings equations [1] where $\beta$ has been set to zero so that it no longer is non-linear and, thus, no longer Duffings equation.

If this is really the equation you mean to ask about, then I'd say that a negative k physically corresponds to the displacement from an unstable equilibrium of some kind. You can for instance imaging a heavy rod balancing vertically, frictionlessly hinged at the bottom, with x signifying the displacement of its top away from the unstable equilibrium at x = 0. This gives a positive feed-back on x (negative k) which is also fairly linear for small displacements relative to the rods length. Note however that this is not an ideal "negative spring" as the rod has mass that needs to be included in the dynamics. It may be that it is possible to make a "negative spring" within a small displacement range more purely in some other way, like using a buckling column or at least as some electronically controlled feed-back mechanism.

I might also add, as you probably already know or suspect, that a negative k in the equation you gave will give a solution that will blow up (go to infinity) unless $\left|F/k\right|$ is sufficiently large, so it the equation as it stands (without limits to x) does not correspond to a physical system when k is negative.


[1] http://mathworld.wolfram.com/DuffingDifferentialEquation.html

 

[nonequilibrium]

My apologies! I had simply forgotten the beta: I had intended to write it down, but apparently didn't.

 

[Andy Resnick]

We used Duffing's equation to model a liquid bridge- it behaves as a soft spring:

http://webserver.dmt.upm.es/~isidoro/lc1/EiF04/Martinez,%20Perales%20and%20Meseguer.htm

http://www.google.com/url?sa=t&sour...sg=AFQjCNEXQRn_4A9Bize5y9J-BhhKGibRAg&cad=rja

 

[Filip Larsen]

So, now the the beta-term back in the differential equation (and with beta != 0), you ask if it makes sense to have k < 0? If so, I'd more or less repeat my reply, suggesting that it could model (the central domain of) a mass in some sort of an unstable equilibrium, now only with the non-linearity back in.