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 nonautonomous systems.

Definition

If

<math>\dot\mathbf{x} = A(t)\mathbf{x}<math>

is a linear nonautonomous dynamical system in <math>\mathbb{R}^n<math> with fundamental solution matrix <math>\Phi(t)<math>, <math>\Phi(0) = I<math>, the equilibrium point <math>\mathbf{0}<math> is said to have an exponential dichotomy if there exists a (constant) matrix <math>P<math>, <math>P^2 = P<math>, and positive constants <math>K<math>, <math>L<math>, <math>\alpha<math>, and <math>\beta<math> such that

<math>|| \Phi(t) P \Phi^{-1}(s) || \le Ke^{-\alpha(t – s)}<math> for <math>s \le t < \infty<math>

and

<math>|| \Phi(t) (I – P) \Phi^{-1}(s) || \le Le^{-\beta(s – t)}<math> for <math>s \ge t > -\infty<math>

If furthermore, <math>L = \frac{1}{K}<math> and <math>\beta = \alpha<math>, <math>\mathbf{0}<math> is said to have a uniform exponential dichotomy.

The constants <math>\alpha<math> and <math>\beta<math> allow us to define the spectral window of the equilibrium point, <math>(-\alpha,\beta)<math>.

Explanation

The matrix <math>P<math> is a projection onto the stable subspace and <math>I – P<math> 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 <math>t \to \infty<math> and the norm of the projection onto the unstable subspace of any orbit decays exponentially as <math>t \to -\infty<math>, and furthermore that the stable and unstable subspaces are conjugate (because <math>P \oplus (I – P) = \mathbb{R}^n<math>).

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.