Exponential dichotomy
# 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

    ${\dot {\mathbf {x} }}=A(t)\mathbf {x}$

is a linear non-autonomous dynamical system in $R^n$ with fundamental solution matrix Φ(t), Φ(0) = I, then the equilibrium point 0 is said to have an exponential dichotomy if there exists a (constant) matrix P such that $P^2$ = P and positive constants K, L, α, and β such that

    $||\Phi (t)P\Phi ^{-1}(s)||\leq Ke^{-\alpha (t-s)}{\mbox{ for }}s\leq t<\infty$

and

    $||\Phi (t)(I-P)\Phi ^{-1}(s)||\leq Le^{-\beta (s-t)}{\mbox{ for }}s\geq t>-\infty .$

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, (−α, β).

## 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 $\scriptstyle P\oplus (I-P)=\mathbb {R} ^{n}$).

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.

