We usually only consider the first order term for an oscillation, are there any papers on extending that model and including third and fifth order terms (since only odd power terms would cause a periodic motion)? The ODE would look like x''=-αx-βx^3+O(x^5)
here is a one,but you may not like it http://link.springer.com/article/10.1007%2FBF01962591?LI=true#page-1
Look up the Duffing equation and its solution. Your equation would be a special case of that if only the terms up to O(x^{3}) are considered.
Added even terms would let oscillate as well. They are negligible if your system has an odd response, that is, its transfer function is odd, and then the even components of the series expansion are zero. Many materials behave symmetrically hence build an odd transfer function, but this approximation would be grossly false in an electronic oscillator for instance.