Clock Systems in Stochastic System Theory

Closed, deterministic clock system

Definition.A closed, discrete, deterministic dynamical system is a set

X

along with an update function

φ : X \to X

For this slide, system = closed discrete deterministic dynamical system

Definition.A behavior of a system

(X, φ)

is a sequence

x_{1}, x_{2}, \dots : X

such that

φ (x_{i}) = x_{i + 1}

Definition.A system morphism from

(X, φ_{X})

(Y, φ_{Y})

is a function

f : X \to Y

such that

φ_{Y} (f (x)) = f (φ_{X} (x))

for all

x : X

Definition.Let

(N, s)

be the system with

s (n) = n + 1

Proposition. []. For any system

(X, φ)

, there is a bijection between behaviors of

(X, φ)

and morphisms

(N, s) \to (X, φ)