# Phase space and quantization

Symbolic Dynamics&ECG papers explain how to generate a symbolic representation of a nonlinear chaotic system. Symbolic dynamics is a tool to explain the evolving behaviour and dynamics of a nonlinear system. Many other papers especially those related to applications of symbolic dynamics use this to represent a time series symbolically. Based on these examples I have the following fundamental queries

1. The theory of symbolic dynamics states to use "a phase space" for partitioning. A phase space means taking two variables into account, say $x_1$ and $x_2$ for a 2D nonlinear system or $x_1(t)$ vs $x_1(t+1)$. But the other papers which use time series i.e say $x_1$ vs t for symbolic dynamics do not use phase space reconstruction. So, does this mean that symbolic dynamics can be applied to a time series as well as to a phase space?Is this correct and what is the justification?
2. Is there a general rule for partitioning?
3. How is symbolic dynamics of time series different from quantization of signals and piece wise component analysis in signal processing ?

Will be grateful for any other pointers. Thank you

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