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 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

$||\mathrm{\Phi }(t)P{\mathrm{\Phi }}^{-1}(s)||\le K{e}^{-\alpha (t-s)}\text{ for }s\le t<\mathrm{\infty }$

and

$||\mathrm{\Phi }(t)(I-P){\mathrm{\Phi }}^{-1}(s)||\le L{e}^{-\beta (s-t)}\text{ for }s\ge t>-\mathrm{\infty }.$

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 $P\oplus (I-P)={\mathbb{ℝ}}^n$).

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