Consider the 3D Dynamical system $\dot{X} = F_{\mu}(X)$, Say that the stable and unstable manifolds of the fixed point at $\mu = 0$ intersect each other(Homoclinic tangency to a periodic orbit $C$ of period $\tau_{c}$).
I am referring this article - Homoclinic Orbits and Mixed-Mode Oscillations in Far-from-Equilibrium Systems by Gaspard and Wang.
Let $\lambda_{s},\lambda_{u}$ denote the eigenvalues of the stable and unstable manifolds, where we have the condition that $|\lambda_{s}|<1<|\lambda_{u}|<\frac{1}{|\lambda_{s}|}$.
Then I was trying to understand how to prove the following statement below -
"Since the flow preserves orientation in the phase space $\Bbb{R}^3$, either $\lambda_{s},\lambda_{u} <0$ OR $\lambda_{s},\lambda_{u}>0$" ?
Any reference or idea of the proof?