Estimate the time spent along fluid flow I have a 2d flow field with a singular point
$$\dot x=y-px^2\\\dot y=-x-qy^3$$
where $p,q$ are small parameters.
How do I compute the cost time of a particle from $(x_0,y_0)$ to ambient of the singularity?
How does the time change when I slightly change parameters? 

Edit: I ask this question because I want to see how nonlinear effects cost time. For a linear system, I can estimate according to the eigenvalues of matrix. So I add a nonlinear perturbation. But it seems suddenly grow too hard for me.
 A: I don't know the relative strength of $p$ and $q$. I will assume they have the same order and both are small. 
Like the standard perturbation method in QM, suppose the solution could be arranged order by order,
\begin{equation}
x = x_0 + x_1 + \cdots \qquad y = y_0 + y_1 + \cdots
\end{equation}
and apply a series of initial conditions,
\begin{equation}
x_0(0) = x_0\quad x_1(0) = 0 \quad \cdots \qquad y_0(0) = y_0 \quad y_1(0) = 0 \cdots 
\end{equation}
At 0th order, 
\begin{equation}
\dot{x_0} =  y_0 \quad \dot{y_0} = -x_0
\end{equation}
The general solution is 
\begin{equation}
x_0(t) = R\cos(t+\phi_0) \quad y_0(t) = -R\sin(t+\phi_0) 
\end{equation}
where $R=\sqrt{x_0^2+y_0^2}$
At 1st order, 
\begin{equation}
\dot{x_0} + \dot{x_1} = y_0 + y_1 - px_0^2 \qquad \dot{y_0} + \dot{y_1} = -x_0 -x_1 -q y_0^3
\end{equation}
which could be reduced to 
\begin{equation}
  \ddot{x_1} + x_1 = -2px_0\dot{x_0} -qy_0^3 = pR\sin(2t+2\phi_0) + qR^3 \sin^3(t+\phi_0)
\end{equation}
The solution satisfying boundary conditions is
\begin{equation}
x_1 = -\frac{pR}{3}\sin(2t+2\phi_0) + qR^3[ \frac{1}{32}\sin(3t+ 3\phi_0) -\frac{3t}{8}\cos(t+\phi_0) ]
\end{equation}
$y_1$ can be computed similarly. 
You can also go ahead to calculate higher order terms. One caveat is that $x_1$ contains a term that is proportional to $t$. Once $qt \sim \mathcal{O}(1)$, this first order "perturbation" isn't small as assumed. You should include higher order terms, but it can't be guaranteed that series will converge. 
Anyway within the range when $qt$ is small, 1st order result enable you to compute the (1st order) correction of the period and the change of the trajectory.
