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

{\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

||\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.

References

Coppel, W. A. Dichotomies in stability theory, Springer-Verlag (1978), [ISBN 978-3-540-08536-2] doi:10.1007/BFb0067780

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