Autonomous system - asymptotic behavior

[ramparts]

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!

 

[jvicens]

Thank you for sharing your findings and questions about your autonomous system of two ODEs. As a scientist with expertise in ODEs, I would like to provide some insights and suggestions for further analysis.

Firstly, it is important to note that the phase portrait in Mathematica is a visual representation of the behavior of solutions to the given system of ODEs. It can give us a good understanding of the overall behavior, but it may not provide precise numerical values for certain parameters. Therefore, it is always recommended to perform further analytical calculations to confirm and understand the behavior observed in the phase portrait.

In this case, it seems that for y>0, all the solutions tend to approach a constant value of y. This behavior can be better understood by analyzing the system of ODEs. One approach is to consider the stability of the system. If the system is stable, it means that small perturbations in the initial conditions will not significantly affect the behavior of solutions. In other words, the solutions will converge to a particular value regardless of the initial conditions. On the other hand, if the system is unstable, small perturbations in the initial conditions can lead to significantly different behaviors of solutions.

To analyze the stability of the system, we can consider the eigenvalues of the Jacobian matrix associated with the system of ODEs. If all the eigenvalues have negative real parts, the system is stable, and if any of the eigenvalues have positive real parts, the system is unstable. In your case, since f and g are nonlinear, it may not be possible to find an analytical solution for the eigenvalues. However, you can use numerical methods to approximate the eigenvalues and determine the stability of the system.

Another approach to understanding the behavior of the system is to consider the equilibrium points. These are points where both dx/dt and dy/dt are equal to zero. In your case, it seems that there are no non-trivial equilibrium points, but there may be a trivial equilibrium point at (x,y)=(0,0). Analyzing the behavior of the system near this point can provide insights into the overall behavior of the system.

In summary, to better understand and confirm the behavior observed in the phase portrait, I suggest analyzing the stability of the system and considering the equilibrium points. This may require further numerical or analytical calculations, but it will provide a more in-depth understanding of the system. I hope this helps and please do