Stability of a three-dimensional system

Question

when I consider the three-dimensional ODE system such as $$ \dot{x} = \frac{25}{(1+y^2)(1+z^2)} - x\\ \dot{y} = \frac{25}{(1+x^2)(1+z^2)} - y\\ \dot{z} = \frac{5}{1+(x+y)^2} - z $$ There is an equilibrium state $$(2.78581, 2.78581, 0.15604).$$ When I substitute this equilibrium point into the corresponding Jacobian matrix. The eigenvalues at this equilibrium point are $$ [-2.82226, -0.949452, 0.771709] $$ Since there is a positive eigenvalue, this equilibrium should be defined as an unstable state. However, when I simulated this system in MATLAB, the simulation results show that this equilibrium state seems like a stable state. I was wondering how this thing happens? Is there anything I misunderstood?

Answer 1

This is a wrap up of the comments under the OP:

Your understanding of the eigenvalues is correct. Since we know that your numerical scheme indeed shows the correct behaviour for non-attractive eigenvector the numerics reflect the expected behaviour.

Now if you are on the equilibirium state and (numerical) pertubations occur only in the direction of the two stable eigenvectors, you stay at the equilibrium. If you get a pertubation in direction of the unstable eigenvector you should eventually(!) move away from it. Depending on the numerical scheme there might be the possibility of some artificial damping.