I suspect there may be periodic solutions.  For $A = 1$, numerically plotting the solution with initial conditions $$x(0)=0, \dot{x}(0) = 0.442091320614410, \ddot{x}(0) = 0.774949154843236$$ I get this:

[![enter image description here][1]][1]


This looks to me like an approximation of an unstable periodic orbit.


  [1]: https://i.sstatic.net/ocNn4.png