Take for example the equation

$\dot x = \lambda x$

$\dot y = y^2$

For $\lambda < 0$. The origin is an unstable equilibrium point, however, its stable manifold is the whole lower semiplane (including the $y=0$ axis).

If there is one eigenvalue of real part bigger than $0$ for the derivative in the equilibrium point, the result is true. See for example here (proposition 4.1).

The proof is simple and based in the existence of center stable manifolds, this implies toghether with the eigenvalue with positive real part that the local stable set of the equilibrium has zero measure, after that, it is done since the stable set can be written as a countable union of sets diffeomorphic to this one).

Take for example the equation

$\dot x = \lambda x$

$\dot y = y^2$

For $\lambda < 0$. The origin is an unstable equilibrium point, however, its stable manifold is the whole lower semiplane (including the $y=0$ axis).

If there is one eigenvalue of real part bigger than $0$ for the derivative in the equilibrium point, it is then true that the set of points converging to $x_e$ has zero measure. See for example here (proposition 4.1).

The proof is simple and based in the existence of center stable manifolds, this implies (toghether with the eigenvalue with positive real part) that the local stable set of the equilibrium has zero measure, after that, it is done since the stable set can be written as a countable union of sets diffeomorphic to this one.

Take for example the equation

$\dot x = \lambda x$

$\dot y = y^2$

For $\lambda < 0$. The origin is an unstable equilibrium point, however, its stable manifold is the whole lower semiplane (including the $y=0$ axis).

If there is one eigenvalue of modulus bigger than $0$ for the derivative in the equilibrium point, the result is true. See for example here.

Take for example the equation

$\dot x = \lambda x$

$\dot y = y^2$

For $\lambda < 0$. The origin is an unstable equilibrium point, however, its stable manifold is the whole lower semiplane (including the $y=0$ axis).

If there is one eigenvalue of real part bigger than $0$ for the derivative in the equilibrium point, the result is true. See for example here (proposition 4.1).

The proof is simple and based in the existence of center stable manifolds, this implies toghether with the eigenvalue with positive real part that the local stable set of the equilibrium has zero measure, after that, it is done since the stable set can be written as a countable union of sets diffeomorphic to this one).