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

:$\dot = A(t)\mathbf$

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

:$, , \Phi(t) P \Phi^(s) , , \le Ke^\mboxs \le t < \infty$

and

:$, , \Phi(t) (I - P) \Phi^(s) , , \le Le^\mboxs \ge t > -\infty.$

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 $\scriptstyle P \oplus (I - P) = \mathbb^n$).

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),

Dynamical systems
Dichotomies

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