Exponential dichotomy

Exponential dichotomy is a concept from the theory of dynamical systems and differential equations, particularly in the study of linear systems. It describes the behavior of solutions to a linear differential equation in terms of their growth or decay rates over time. ### Definition An exponential dichotomy occurs for a linear system of the form: \[ \frac{dx}{dt} = Ax(t) \] where \( A \) is a linear operator (often represented by a matrix in finite dimensions).