Local false nearest neighbors and dynamical dimensions from observed chaotic data

Methods for determining the integer-valued dimension of the state space of a system from observed scalar data using the idea of local false nearest neighbors are discussed.

Abstract

The time delay reconstruction of the state space of a system from observed scalar data requires a time lag and an integer embedding dimension. The minimum necessary global embedding dimension ${\mathit{d}}_{\mathit{E}}$ may still be larger than the actual dimension of the underlying dynamics ${\mathit{d}}_{\mathit{L}}$. The embedding theorem only guarantees that the attractor of the system is fully unfolded using ${\mathit{d}}_{\mathit{E}}$ greater than 2${\mathit{d}}_{\mathit{A}}$, with ${\mathit{d}}_{\mathit{A}}$ the fractal attractor dimension. Using the idea of local false nearest neighbors, we discuss methods for determining the integer-valued ${\mathit{d}}_{\mathit{L}}$.