Solving Nonlinear System using Matlab

[xsw001]

Homework Statement

Find the stable/unstable manifold for the nonlinear system dx/dt=y^2-(x+1)^2; dy/dt=-x

Homework Equations

The Attempt at a Solution

I'm trying to solve the below nonlinear system using Matlab, but got the following warning message. Any idea?

[x,y]=dsolve('Dx=y^2-(x+1)^2','Dy=-x')

Warning: Explicit solution could not be found.

I'm trying to find the stable/unstable manifolds. There are two critical pts for this nonlinear system (0,1) & (0,-1).

C.P.(0,1) is hyperbolic stable focus with complex eigenvalues => the eigenvectors are complex => there are No real manifolds.

C.P.(0,-1) is hyperbolic saddle point that has one positive eigenvalue, its eigenvector gives the direction of unstable manifold, there is another negative eigenvalue, its eigenvector gives the direction of stable manifold.

I'm trying to find the explicit solution for the nonlinear system so that I could find the stable/unstable manifolds.

Any suggestions would be appreciated!

 

[mighty2000]

Thank you for your post. I understand that you are trying to find the stable and unstable manifolds for the given nonlinear system. It seems like you have already identified the critical points and their corresponding eigenvalues and eigenvectors. However, you are facing difficulty in finding the explicit solution for the nonlinear system using Matlab, and hence are unable to determine the stable and unstable manifolds.

In this case, I would suggest trying to find the explicit solution using a different method or software. You can also try to linearize the system around the critical points and find the explicit solution for the linearized system. This can give you an approximation of the stable and unstable manifolds for the nonlinear system. Additionally, you can also try to plot the phase portrait of the system to visually identify the stable and unstable manifolds.

I hope this helps. Good luck with your research!