nonlinear time series analysis (Kantz, Schreiber)

Comments on "Nonlinear Time Series Analysis," Holger Kantz and Thomas Schreiber, 2nd edn, Cambridge

Chapter 3 Phase Space Methods
Consider a dynamical system whose trajectory can be studied in finite dimensional vector space R^m, thus the state is specified by some vector x belonging to R^m. The said dynamical system can be described as xn+1 = F(xn), for discrete variable, and d/dt x(t) = f(x(t)) for a continuous case. Based on the initial condition for x0, the sequence of points for xn will portray a trajectory of the dynamica lsystem, which either will run to infinite or stay within some bounds.

the set of initial conditions leading to the same asymptotic behaviour of the trajectory is called the basin of attraction for this particular motion.
Since the dynamical equations (or the equations of motion) are defined in phase space, it is also most natural to yuse a phase space description for approximations.

3.2 delay reconstruction pg 35
sn = s(x(n5t)) - sequence of scale measurements of some scalar quantity taken at mutiples of fixed sampling time
a delay reconstruction in m dimensions is then formed by the vectors sn,
sn = (sn-(m-1)tao, sn-(m-2)tao, ...sn)
The attractor formed by above reconstruction is equivalent to the attractor in the unknown space in which the original system is living if the dimension m if the delay coordinate space is sufficiently large. precisely, this is guaranteed if m is larger than twice the box counting dimension Df of the attractor.

pg 36
for many practical purposes, the most important embedding parameter is the product mtao of the delay time, and the embedding dimension, rather than the embedding dimension m or the delay time tao alone.