Autonomous system - asymptotic behavior

  • Thread starter ramparts
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ramparts
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I have an autonomous system of two ODEs, i.e.,

dx/dt = f(x,y)
dy/dt = g(x,y)

I plotted the phase portrait in Mathematica and found that for y>0, all the solutions seemed to flow towards a constant value of y. The problem is I'm rusty on my ODEs and am not sure how to calculate that value (or even show that this behavior occurs) analytically.

I think that if f and g were linear, so I could write (x',y') as a matrix multiplied by (x,y), the eigenvector of that matrix would at least point in that direction (i.e., the constant y direction) but in my case, f and g are pretty non-linear. There are no (non-trivial) fixed points to work with; even if y asympotes to a constant, x should always be changing.

This is probably a basic ODE question so thanks for bearing with me!
 

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