Recently, I was exposed to SDEs, though I still did not fully understand the concepts. I want to solve a nonlinear undamped oscillator
$$\frac{d^2x}{dt^2}=-\alpha^2 g(x) + f(t)$$
where $g(x)= \sin(x), \cos(x), \exp(ix), \dots$ and $f(t)$ is the force due to diffusion, where $\langle f(t) \rangle = 0$ and $\langle f(t) f(\tau) \rangle = 2D \delta(t-\tau)$.
I have the following questions
- What methods are used to solve these type of SDE to get $x(t)$ (numerically or analytically)? I know the linear response theorem can be used, but I don't know how to obtain the response function.
- How does the random force affect the frequency of the oscillation, and how to obtain it? for example, in case of no random force, frequency can be obtained by multiplying the equation of motion by $x'$ and perform integration to obtain $x'=h(x)$.