Consider the following simplest example:
$ $ \dot{x} = x(x-1)(x+1)$ $ $ [-1,1]$ is the ROA.
Now consider the two dimensional case:
\begin{equation} \begin{aligned} &\dot{x} = x(x-1)(x+1)\ &\dot{y} = y(y-1)(y+1) \end{aligned} \end{equation}Obviously, ROA is a square. However, if I consider the following coupled ODE:
\begin{equation} \begin{aligned} &\dot{x} = x(x-1)(x+1) + \epsilon (y-x)\ &\dot{y} = y(y-1)(y+1) + \epsilon (x-y) \end{aligned} \end{equation} where $ \epsilon$ is a very small number. Or
\begin{equation} \begin{aligned} &\dot{x} = x(x-1)(x+1) + \epsilon (-y+x)\ &\dot{y} = y(y-1)(y+1) + \epsilon (-x+y) \end{aligned} \end{equation} Then I have the following ROAs: (blue line-case three, black line-case two, red line-case one)
My questions are:
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There are two different tilt directions for case two and three. I know this is because of the slope of the coupling term (for case two, the slope of $ x$ and $ y$ in the coupling terms are $ -1$ ). But how could I analyze this formally?
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Is it a good way to analyze 1. by perturbation method? (observe the sign of the leading order term of the solution obtained from perturbation method?) and how could I proceed it for the coupling term?
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Are there any reference about my questions?
