I'm having a hard time to find a scenario in physics (or in any science) where I can meet the duffing equation with all coefficients not zero (the most general case)
The equation is $x''+\delta x'+\alpha x+\beta x^3=\gamma \cos(\omega t)$
for example I read on wikipedia that I can modelize a "pendulum with a spring that doesn't exactly obey the Hooke's law", but what would the comeback force be then?
Of course, any help/hint would be really appreciate.
Thanks, have a good day.
As a student of physics, the real question should not be, "Why is $\beta \ne 0$?" but rather "Why should we take $\beta = 0$?". Most "springs" don't really obey Hooke's law; elastic deformation isn't really linear. The second order term vanishes near equilibrium because it is a local minimum of the potential energy, so after the linear term, the cubic term in the potential force is the most significant in general.
Physicists like the approximation $\beta = 0$ because the resulting linear equation is explicitly solvable, and because it is reasonably accurate in many cases. However, if we measure the forces more accurately, we often see the $\beta \ne 0$ term.
Mathematicians, however, like to take $\beta \ne 0$, because otherwise the linear system is too easy to solve. Also, the phenonomena associated with this nonlinearity include fun stuff like chaos.
Note that the pendulum equation without linearization is
$\ddot \theta + \mu \dot \theta + \lambda \sin \theta = 0; \tag 1$
also,
$\sin \theta = \theta - \dfrac{\theta^3}{6} + \ldots, \tag 2$
so to third order in $\theta$ the force "looks like" that of a spring with a cubic correction term, a spring which "doesn't exactly obey Hooke's law"; in the case of (2), the "comeback force" is apparently $-\lambda \sin \theta$; to third order in $\theta$ this is $-\lambda \theta - \lambda \theta^3 / 6$
But hell, even some physicists like chaos. Reminds me of my desk . . .