Exponential dichotomy


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.

Definition

If
is a linear non-autonomous dynamical system in Rn with fundamental solution matrix Φ, Φ = I, then the equilibrium point 0 is said to have an exponential dichotomy if there exists a matrix P such that P2 = P and positive constants K, L, α, and β such that
and
If furthermore, L = 1/K and β = α, then 0 is said to have a uniform exponential dichotomy.
The constants α and β allow us to define the spectral window of the equilibrium point,.

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.
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.