Exponential dichotomy
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In the mathematical theory of dynamical systems, an exponential dichotomy is a property of an equilibrium point that extends the idea of hyperbolicity to non-autonomous systems.
[edit] Definition
If
is a linear non-autonomous dynamical system in 
with fundamental solution matrix Φ(t), Φ(0) = I, the equilibrium point 
is said to have an exponential dichotomy if there exists a (constant) matrix P such that P2 = P and positive constants K, L, α, and β such that
for 
and
for 
If furthermore, 
and β = α, is said to have a uniform exponential dichotomy.
The constants α and β allow us to define the spectral window of the equilibrium point, ( − α,β).
[edit] 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 
and the norm of the projection onto the unstable subspace of any orbit decays exponentially as 
, 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.
