Dear all members,
Anyone knows this non-linear differential equation related to what mechanic system.

x ̈+x ̇^3+x=0
(or dx/dt=z, dz/dt=-x-x^3)
Deeply appreciate to any help.

I assume the middle term is x-dot (it's not very clear, and it would mean you left the "dot" off the x^3 term in the second line). Please confirm that this is what you meant:

\frac {d^2x}{dt^2} + ( \frac {dx}{dt} )^3 + x = 0

That velocity cubed term is a puzzler. If that term was velocity squared I would say the system involves resistance from air or fluid flow, which is often modeled as being proportional to the square of velocity.