Does anyone knows how to average function in three dimensions

Question

$$ \begin{split} \frac{dr}{dt} &= -\epsilon \sin^2(\theta)z \\ \frac{d \theta}{dt} &= -1-\epsilon \cos\theta \sin\theta z\\ \frac{dz}{dt} &= \epsilon(r^2-T) \end{split} $$

Does anyone know how to do function averaging by using Fourier analysis in cylindrical coordinates or have some helpful reference?

Answer 1

I would refer to F. Verhulst, Nonlinear Differential Equations and Dynamical Systems (Springer, 1996), chapter 11, section 11.6. Quoting equation (11.37) therein, you have \begin{align} \dot{x} &= \epsilon\, X(\phi,x) + \mathcal{O}(\epsilon^2),\quad x\in D\subset \mathbb{R}^n, \tag{1a}\\ \dot{\phi} &= \Omega(x) + \mathcal{O}(\epsilon),\quad \phi \in S^1,\tag{1b} \end{align} with $X(\phi,x)$ periodic in $\phi$; in your case, $\phi = \theta$, $\Omega(x) = -1$, $x = (r,z)$ and $X(\phi,x) = (-\sin(\theta)^2 z, r^2 - T)$. Then, Theorem 11.4 (Verhulst, p.151) says that

Consider system $(1)$ with initial values $x(0) = x_0$, $\phi(0) = \phi_0$ and suppose that

a. the righthand sides are $C^1$ in $D \times S^1$;

b.) the solution of $$ \dot{y} = \epsilon\, X^0(y),\quad y(0) = x_0 $$ with $$ X^0(y) = \int_{S^1} X(\phi,x)\,\text{d}\phi $$ is contained in an interior subset of $D$ in which $\Omega(x)$ is bounded away from zero by a constant independent of $\epsilon$;

then $x(t) - y(t) = \mathcal{O}(\epsilon)$ on the time-scale $1/\epsilon$.

In your case, this yields the system

\begin{align} \dot{r}_\text{av} &= -\frac{1}{2} z_\text{av},\\ \dot{z}_\text{av} &= r_\text{av}^2 - T, \end{align} for the averaged values of $r$ and $z$, which is Hamiltonian and hence integrable.