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.
If
is a linear non-autonomous dynamical system in R<sup>n</sup> with fundamental solution matrix æ(t), æ(0) = I, then the equilibrium point 0 is said to have an exponential dichotomy if there exists a (constant) matrix P such that P<sup>2</sup> = P and positive constants K, L, ñ, and ò such that
and
If furthermore, L = 1/K and ò = ñ, then 0 is said to have a uniform exponential dichotomy.
The constants ñ and ò allow us to define the spectral window of the equilibrium point, (−ñ, ò).
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 â â and the norm of the projection onto the unstable subspace of any orbit decays exponentially as t â −âÂÂ, 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.