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.

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    $\begingroup$ Can you elaborate more on where did this problem rise from and what you have tried so far? $\endgroup$ – Ali Dec 17 '13 at 13:19

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.


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