I have been trying to solve the following non-linear equation taking help from the book regarding perturbative technique by H. Nayfeh (chapter -4)
$$\ddot{x}+\frac{x}{(1-x^2)^2}=0$$ where $x<1$ with the initial condition assciated with it such as :
$x(0)=0$ and $\dot{x}(0)=0$.
$x$ and $t$ are two dimensionless parameters such that $x=\frac{y}{D}$ and $t=\omega_0 t$, where $D$ is the charateristic length and $\omega_0$ is the natural frequency of the system.
Following, chapter-4 as mentioned above, $f(x)=\frac{x}{(1-x^2)^2}$ is analytic at the neighbour of the point $x(0)=x_0=0$ and thus, can be expressed as $$f(x)=k_1(x-x_0)+k_2(x-x_0)^2+k_3(x-x_0)^3$$
where, $k_1=1$, $k_2=0$ and $k_3=\epsilon=2$. Plugging this in the original equation, I get $$\frac{d^2u}{dt^2}+u+\epsilon u^3=0$$ where, $u= x-x_0$ with the initial condition $u(0)=0$ and $\dot{u}(0)=0$ The above equation is a Duffing type equation with non-linearity factor greater than unity but in all the cases, which have been discussed in chapter-4, the non-linearity factor is very small. I can not understand how to solve this type of equation for both the fast and slow time scales.
Would you kindly suggest me any book or any other source from where I can get some help.