How is a quartic oscillator solved in classical mechanics? Quantum mechanically, a quartic anharmonic oscillator with potential $$V(x)=\frac{1}{2}m\omega^2x^2+\lambda x^4$$ is dealt with perturbation theory- the approximate energies $E_n$ and energy eigenstates $|\phi_n\rangle$ are obtained using the  time-independent perturbation theory. Classically, the problem amounts to solving the trajectory $x(t)$. At this point, one is stuck with a nonlinear differential equation which cannot be solved in closed analytical form. How do we go about solving such problems, classically? Do we use similar perturbation technique to obtain corrections to the trajectory order by order? Any suggestions? In short. I am curious about the classical behaviour of this system.
 A: This problem is discussed as an example of secular perturbation theory in 

José, J. and Saletan, E., 2000. Classical dynamics: a contemporary approach.

For first order in $\epsilon$, the (rationalized) equation of motion
$$
\ddot{x}+\omega_0^2 x+\epsilon x^3=0
$$
has the approximate solution
$$
x(t)=a \cos(\omega_0 t)-\epsilon \frac{a^3}{8\omega_0^2}
\left(3\omega_0t \sin\omega_0 t +
\frac{1}{4}(\cos\omega_0 t- -\cos(3\omega_0t))\right)
$$
with the secular term (linear in $t$) appearing explicitly.
One can get rid of the secular term using Poincaré-Lindstedt theory, i.e. by introducing corrections to the unperturbed frequency $\omega_0$.  
The same problem is used in José and Saletan as an example of canonical perturbation theory (or Hamilton-Jacobi perturbation theory), where the solution is to find successive canonical transformations. 
using the unperturbed canonical variables
$$
\phi_0=\arctan(m\omega_0 q/p)\, ,\qquad J_0=\frac{1}{2}\left(\frac{p^2}{2\omega_0}+m\omega_0 q^2\right)\, .
$$
The first order correction to the frequency is $\omega\approx\omega_0+\epsilon
\frac{3a^2}{8\omega_0}$ when the initial conditions are $p(0)=0, q(0)=a$.
This problem, also as an example of the canonical perturbation approach, is similarly discussed in Example 8.3 of 

Percival, I.C. and Richards, D., 1982. Introduction to dynamics. Cambridge University Press.

