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
x ˙ = A ( t ) x {\displaystyle {\dot {\mathbf {x} }}=A(t)\mathbf {x} }
is a linear non-autonomous dynamical system in Rn 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 P2 = P and positive constants K, L, α, and β such that
| | Φ ( t ) P Φ − 1 ( s ) | | ≤ K e − α ( t − s ) for s ≤ t < ∞ {\displaystyle ||\Phi (t)P\Phi ^{-1}(s)||\leq Ke^{-\alpha (t-s)}{\mbox{ for }}s\leq t<\infty }
and
| | Φ ( t ) ( I − P ) Φ − 1 ( s ) | | ≤ L e − β ( s − t ) for s ≥ t > − ∞ . {\displaystyle ||\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 P ⊕ ( I − P ) = R n {\displaystyle \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.
References Coppel, W. A. Dichotomies in stability theory, Springer-Verlag (1978), ISBN 978-3-540-08536-2 doi:10.1007/BFb0067780