Explanation
The matrix P is a projection onto the stable subspace and I − P is a projection onto the unstable subspace. What the exponential dichotomy says is that the norm of the projection onto the stable subspace of any orbit in the system decays exponentially as t → ∞ and the norm of the projection onto the unstable subspace of any orbit decays exponentially as t → −∞, and furthermore that the stable and unstable subspaces are conjugate (because ).
An equilibrium point with an exponential dichotomy has many of the properties of a hyperbolic equilibrium point in autonomous systems. In fact, it can be shown that a hyperbolic point has an exponential dichotomy.
Read more about this topic: Exponential Dichotomy
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